Add inner-face down-apex colour-sequence experiment (n=9 sweep)
Mirror of the up-tooth experiment with the distinguished valid face moved from the outer face to an inner non-tooth face (root or bite inner-gap). For each (M(T), inner face) config holding m singleton down-tooth apexes, record the apex colour sequence (cyclic order, mod colour permutation) over Kempe-balanced colourings and group configs by their sequence-set. Runs for m=3,4,5,6 with per-sequence notes, figures, and a config atlas. Finding: inner faces realise the same parity-admissible sequence vocabulary and the same distinct-sequence counts (1/4/10/28) as the outer face, i.e. the Kempe-parity law acts uniformly on every valid face. At m=6 the configs are the U<->D embedding mirror of the up-m=6 graphs (matching 7 configs, 28 sequences, 127 colourings). Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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"""Singleton-down-apex colour sequences of Kempe-balanced 3-colourings.
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Inner-face counterpart of ``kempe_up_tooth_sequences.py``. There the role of
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the distinguished valid face was the unique *outer* face, which carries every
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up-tooth apex; here we play the same game on an *inner* non-tooth face of B(T)
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(the root face, or a bite's inner-gap face), which carries the singleton
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down-tooth apexes assigned to it.
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For a fixed annular size ``n`` and a fixed count ``m`` we:
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1. take every full medial tire graph M(T) with |A(T)| = n (one representative
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per dihedral symmetry class) that has an inner non-tooth face F holding
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exactly ``m`` singleton down-tooth apexes -- by Remark 3.8 every such face
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holds 0 or >= 3, so m >= 3. (At n = 9 each M(T) has at most one inner face
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bearing singletons, so a graph and its face coincide one-to-one.)
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2. enumerate the Kempe-balanced (``valid``) proper 3-colourings of M(T) and
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read off the colour sequence of F's singleton down-tooth apexes d_i in
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increasing annular-edge order (their cyclic order along F's arc);
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3. reduce each sequence modulo the six colour permutations -- NOT modulo the
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dihedral symmetry of the cycle -- to a canonical sequence;
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4. group the configurations by the *set* of canonical down-apex sequences
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they realise, and report how many share each set.
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The Kempe-balanced rule is global, so the same valid colourings are used as in
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the up-tooth experiment; only the apex set we read changes. Note that the rule
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forces, on every valid face, each colour pair to meet the counted apexes an even
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number of times -- so the down-apex sequences obey the same equal-parity law as
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the up-tooth sequences.
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Run: python3 kempe_down_face_sequences.py --n 9 --m 3
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"""
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from __future__ import annotations
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import argparse
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import math
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import os
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from collections import defaultdict
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from full_medial_tire_generator import (
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FullMedialTireGraph,
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generate,
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innermost_bite,
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)
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from kempe_valid_colorings import classify_colorings
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from kempe_up_tooth_sequences import (
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PALETTE,
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PALETTE_NAME,
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_parity_partitions,
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_positions,
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canonical_sequence,
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compact_coloring,
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seq_str,
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)
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HERE = os.path.dirname(os.path.abspath(__file__))
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Coloring = dict[str, int]
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Bite = tuple[int, int]
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FaceKey = Bite | None # None = root face
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def face_name(face: FaceKey) -> str:
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return "root" if face is None else f"bite({face[0]},{face[1]})"
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def inner_face_singletons(graph: FullMedialTireGraph) -> dict[FaceKey, list[int]]:
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"""Map each inner non-tooth face to its singleton down-tooth edges (sorted)."""
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perface: dict[FaceKey, list[int]] = defaultdict(list)
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for e in graph.singleton_down_edges:
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perface[innermost_bite(e, graph.bites)].append(e)
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return {face: sorted(edges) for face, edges in perface.items()}
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def config_id(idx: int) -> str:
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return f"C{idx:02d}"
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def describe_config(graph: FullMedialTireGraph, face: FaceKey, edges) -> str:
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bites = ",".join(f"({i},{j})" for i, j in sorted(graph.bites)) or "-"
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apexes = ",".join(f"d{e}" for e in edges)
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return (f"word={graph.tooth_word} bites={bites} "
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f"face={face_name(face)} apexes=[{apexes}]")
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# ---------------------------------------------------------------------------
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# Data collection.
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# ---------------------------------------------------------------------------
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class Config:
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"""A single (M(T), inner face) specimen with m singleton down apexes."""
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def __init__(self, graph: FullMedialTireGraph, face: FaceKey, edges: list[int]):
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self.graph = graph
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self.face = face
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self.edges = edges # singleton down edges on the face
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self.apexes = [f"d{e}" for e in edges] # their apex vertex names
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def sequence(self, coloring: Coloring) -> tuple[int, ...]:
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return tuple(coloring[v] for v in self.apexes)
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class Experiment:
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def __init__(self, n: int, m: int):
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self.n = n
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self.m = m
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self.configs: list[Config] = []
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for g in generate(n, min_up_teeth=3, dedup=True):
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for face, edges in inner_face_singletons(g).items():
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if len(edges) == m:
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self.configs.append(Config(g, face, edges))
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# per config: list of (coloring, canonical down-apex sequence)
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self.colorings: list[list[tuple[Coloring, tuple[int, ...]]]] = []
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# per config: set of canonical sequences it realises
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self.config_seq_sets: list[frozenset[tuple[int, ...]]] = []
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# canonical sequence -> list of (config_idx, coloring)
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self.by_sequence: dict[tuple[int, ...], list[tuple[int, Coloring]]] = defaultdict(list)
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for cidx, cfg in enumerate(self.configs):
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entries: list[tuple[Coloring, tuple[int, ...]]] = []
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seqs: set[tuple[int, ...]] = set()
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for coloring, verdict in classify_colorings(cfg.graph, dedup_colors=True):
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if not verdict.valid:
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continue
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cseq = canonical_sequence(cfg.sequence(coloring))
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entries.append((coloring, cseq))
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seqs.add(cseq)
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self.by_sequence[cseq].append((cidx, coloring))
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self.colorings.append(entries)
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self.config_seq_sets.append(frozenset(seqs))
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def groups(self):
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groups: dict[frozenset[tuple[int, ...]], list[int]] = defaultdict(list)
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for cidx, sset in enumerate(self.config_seq_sets):
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groups[sset].append(cidx)
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return sorted(groups.items(), key=lambda kv: (-len(kv[1]), len(kv[0])))
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def sequences(self) -> list[tuple[int, ...]]:
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return sorted(self.by_sequence)
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# ---------------------------------------------------------------------------
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# Drawing (rings the singleton down apexes of the chosen inner face).
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# ---------------------------------------------------------------------------
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def _draw(ax, cfg: Config, coloring, title):
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graph = cfg.graph
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pos = _positions(graph)
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for u, v in graph.edges():
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ax.plot([pos[u][0], pos[v][0]], [pos[u][1], pos[v][1]],
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color="#bbbbbb", lw=0.5, zorder=1)
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for k in range(graph.n):
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a, b = f"a{k}", f"a{(k + 1) % graph.n}"
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ax.plot([pos[a][0], pos[b][0]], [pos[a][1], pos[b][1]],
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color="#666666", lw=1.0, zorder=2)
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for v, (x, y) in pos.items():
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is_bite = v.startswith("p")
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ax.scatter([x], [y], s=34 if is_bite else 24, color=PALETTE[coloring[v]],
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edgecolors="black", linewidths=0.5 if is_bite else 0.3, zorder=3)
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# ring the singleton down apexes of this inner face
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dx = [pos[v][0] for v in cfg.apexes]
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dy = [pos[v][1] for v in cfg.apexes]
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ax.scatter(dx, dy, s=120, facecolors="none", edgecolors="#222222",
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linewidths=1.4, zorder=4)
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ax.set_xlim(-1.65, 1.65)
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ax.set_ylim(-1.85, 1.65)
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ax.set_aspect("equal")
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ax.axis("off")
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ax.set_title(title, fontsize=6, pad=1.5)
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def draw_sequence(exp: Experiment, seq, out_png, out_pdf):
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import matplotlib
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matplotlib.use("Agg")
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import matplotlib.pyplot as plt
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entries = exp.by_sequence[seq]
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cols = 10
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rows = math.ceil(len(entries) / cols)
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fig, axes = plt.subplots(rows, cols, figsize=(cols * 1.5, rows * 1.7), squeeze=False)
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for idx in range(rows * cols):
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ax = axes[idx // cols][idx % cols]
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if idx < len(entries):
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cidx, coloring = entries[idx]
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cfg = exp.configs[cidx]
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dseq = seq_str(cfg.sequence(coloring))
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_draw(ax, cfg, coloring, f"{config_id(cidx)} d={dseq}")
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else:
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ax.axis("off")
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fig.suptitle(
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f"Kempe-balanced colourings with inner-face singleton-down-apex "
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f"sequence {seq_str(seq)} (mod colour permutation)\n"
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f"n={exp.n}, m={exp.m} down apexes on the face — {len(entries)} "
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f"colourings on {len({c for c, _ in entries})} configs; "
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f"black rings mark the face's down apexes",
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fontsize=11, y=0.998,
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)
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fig.tight_layout(rect=(0, 0, 1, 0.96))
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fig.savefig(out_png, dpi=170)
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fig.savefig(out_pdf)
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plt.close(fig)
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print(f"wrote {out_png}")
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# ---------------------------------------------------------------------------
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# Markdown notes.
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# ---------------------------------------------------------------------------
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def write_sequence_note(exp: Experiment, seq, path, fig_name):
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s = seq_str(seq)
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by_config: dict[int, list[Coloring]] = defaultdict(list)
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for cidx, coloring in exp.by_sequence[seq]:
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by_config[cidx].append(coloring)
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cm: dict[int, int] = {}
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for c in seq:
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cm[c] = cm.get(c, 0) + 1
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counts = ", ".join(f"{v}×colour{k}" for k, v in sorted(cm.items()))
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lines = []
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lines.append(f"# Inner-face down-apex sequence `{s}`")
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lines.append("")
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lines.append(
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f"Canonical colour sequence of the singleton down-tooth apexes on a "
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f"single inner non-tooth face (read in increasing annular-edge order, "
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f"reduced modulo the six colour permutations) for Kempe-balanced "
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f"3-colourings of M(T) with **n = {exp.n}**, **m = {exp.m} singleton "
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f"down apexes on the face**."
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)
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lines.append("")
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lines.append(f"- Colour multiset: {counts}.")
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lines.append(f"- Realised by **{len(by_config)}** of {len(exp.configs)} "
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f"configs (M(T), inner face).")
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lines.append(f"- **{len(exp.by_sequence[seq])}** Kempe-balanced colourings "
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f"(mod colour permutation) produce it.")
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lines.append(f"- Figure: `{fig_name}` (black rings mark the face's down apexes).")
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lines.append("")
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lines.append("Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` "
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"up-tooth apexes; `D[...]` singleton down apexes `d` and bite "
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"apexes `p`. Colours 0/1/2 = "
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+ ", ".join(f"{c}:{PALETTE_NAME[c]}" for c in (0, 1, 2)) + ".")
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lines.append("")
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for cidx in sorted(by_config):
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cfg = exp.configs[cidx]
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cols = by_config[cidx]
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lines.append(f"## {config_id(cidx)} — {describe_config(cfg.graph, cfg.face, cfg.edges)}")
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lines.append("")
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lines.append(f"{len(cols)} colouring(s) with down-apex sequence `{s}`:")
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lines.append("")
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for coloring in cols:
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raw = seq_str(cfg.sequence(coloring))
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lines.append(f"- face apexes (raw labels) `{raw}` → canonical `{s}` · "
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f"`{compact_coloring(cfg.graph, coloring)}`")
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lines.append("")
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with open(path, "w") as fh:
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fh.write("\n".join(lines) + "\n")
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print(f"wrote {path}")
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def write_summary(exp: Experiment, path):
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lines = []
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lines.append(f"# Inner-face singleton-down-apex sequences of Kempe-balanced "
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f"colourings (n={exp.n}, m={exp.m})")
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lines.append("")
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lines.append(
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f"Every full medial tire graph M(T) with |A(T)| = {exp.n} (one "
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f"representative per dihedral class) that has an inner non-tooth face "
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f"holding exactly {exp.m} singleton down-tooth apexes: "
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f"**{len(exp.configs)} configs (M(T), inner face)**. For each we "
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f"enumerate the Kempe-balanced (valid) proper 3-colourings (modulo "
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f"colour permutation), read the down-apex colour sequence in increasing "
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f"annular-edge order, and reduce it modulo colour permutation (NOT "
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f"dihedral symmetry)."
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)
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lines.append("")
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total = sum(len(c) for c in exp.colorings)
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lines.append(f"- Total Kempe-balanced colourings (mod colour permutation): "
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f"**{total}**.")
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lines.append(f"- Distinct canonical down-apex sequences overall: "
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f"**{len(exp.by_sequence)}**.")
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lines.append("")
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lines.append("## Distinct canonical down-apex sequences")
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lines.append("")
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lines.append("| sequence | colour multiset | #configs realising | #colourings |")
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lines.append("|---|---|---|---|")
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for seq in exp.sequences():
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cm: dict[int, int] = {}
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for c in seq:
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cm[c] = cm.get(c, 0) + 1
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cms = "+".join(str(v) for v in sorted(cm.values(), reverse=True))
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ncfg = len({c for c, _ in exp.by_sequence[seq]})
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lines.append(f"| `{seq_str(seq)}` | {cms} | {ncfg} | "
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f"{len(exp.by_sequence[seq])} |")
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lines.append("")
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parity = "even" if exp.m % 2 == 0 else "odd"
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allowed = sorted(
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{"+".join(str(v) for v in sorted(p, reverse=True))
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for p in _parity_partitions(exp.m)}
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)
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lines.append("Note: every realised sequence has its three colour-counts of "
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"**equal parity** — exactly the Kempe-parity constraint on the "
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"inner face (each colour pair meets its singleton down apexes an "
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f"even number of times). With m = {exp.m} apexes (m is "
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f"{'even' if exp.m % 2 == 0 else 'odd'}) every count must be "
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f"**{parity}**, so the only admissible colour multisets are "
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+ ", ".join(allowed) + ".")
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lines.append("")
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lines.append("## Step 4 — grouping configs by their set of unique down-apex "
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"sequences")
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lines.append("")
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groups = exp.groups()
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lines.append(f"The {len(exp.configs)} configs fall into **{len(groups)}** "
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f"groups by the set of canonical down-apex sequences they "
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f"realise:")
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lines.append("")
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lines.append("| #configs | set of down-apex sequences | config ids |")
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lines.append("|---|---|---|")
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for sset, cidxs in groups:
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seqs = ", ".join(f"`{seq_str(s)}`" for s in sorted(sset))
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ids = ", ".join(config_id(i) for i in cidxs)
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lines.append(f"| {len(cidxs)} | {{ {seqs} }} | {ids} |")
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lines.append("")
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lines.append("## Config atlas (ids)")
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lines.append("")
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lines.append("| id | word / bites / face / apexes | #Kempe-balanced | "
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"down-apex sequence set |")
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lines.append("|---|---|---|---|")
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for cidx, cfg in enumerate(exp.configs):
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sset = exp.config_seq_sets[cidx]
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seqs = ", ".join(f"`{seq_str(s)}`" for s in sorted(sset))
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lines.append(f"| {config_id(cidx)} | "
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f"{describe_config(cfg.graph, cfg.face, cfg.edges)} | "
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f"{len(exp.colorings[cidx])} | {{ {seqs} }} |")
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lines.append("")
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lines.append("## Per-sequence notes")
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lines.append("")
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for seq in exp.sequences():
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lines.append(f"- [`{seq_str(seq)}`](seq_{seq_str(seq)}.md) — "
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f"figure `seq_{seq_str(seq)}.png`")
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lines.append("")
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with open(path, "w") as fh:
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fh.write("\n".join(lines) + "\n")
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print(f"wrote {path}")
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# ---------------------------------------------------------------------------
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# Driver.
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# ---------------------------------------------------------------------------
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def run(args):
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exp = Experiment(args.n, args.m)
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print(f"n={args.n}, m={args.m}: {len(exp.configs)} configs (M(T), inner face)")
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print(f"distinct canonical down-apex sequences: {len(exp.by_sequence)}")
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for seq in exp.sequences():
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nc = len({c for c, _ in exp.by_sequence[seq]})
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print(f" {seq_str(seq)}: {nc} configs, {len(exp.by_sequence[seq])} colourings")
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print(f"groups by sequence-set: {len(exp.groups())}")
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notes_dir = os.path.join(HERE, f"kempe_down_face_sequences_n{args.n}_m{args.m}")
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os.makedirs(notes_dir, exist_ok=True)
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write_summary(exp, os.path.join(notes_dir, "summary.md"))
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for seq in exp.sequences():
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s = seq_str(seq)
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fig_name = f"seq_{s}.png"
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write_sequence_note(exp, seq, os.path.join(notes_dir, f"seq_{s}.md"), fig_name)
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if not args.no_figures:
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draw_sequence(exp, seq,
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os.path.join(notes_dir, fig_name),
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os.path.join(notes_dir, f"seq_{s}.pdf"))
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def main():
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parser = argparse.ArgumentParser(description=__doc__)
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parser.add_argument("--n", type=int, default=9)
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parser.add_argument("--m", type=int, default=3,
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help="singleton down apexes on the inner face (>=3)")
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parser.add_argument("--no-figures", action="store_true")
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run(parser.parse_args())
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||||
|
||||
if __name__ == "__main__":
|
||||
main()
|
||||
+382
@@ -0,0 +1,382 @@
|
||||
# Inner-face down-apex sequence `012`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 3 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 1×colour0, 1×colour1, 1×colour2.
|
||||
- Realised by **26** of 26 configs (M(T), inner face).
|
||||
- **241** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_012.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C00 — word=UUUUUUDDD bites=- face=root apexes=[d6,d7,d8]
|
||||
|
||||
22 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u2:2 u3:2 u4:2 u5:2] D[d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010101021 U[u0:2 u1:2 u2:2 u3:2 u4:2 u5:2] D[d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010102012 U[u0:2 u1:2 u2:2 u3:2 u4:1 u5:1] D[d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u4:1 u5:1] D[d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u4:0 u5:2] D[d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010121021 U[u0:2 u1:2 u2:2 u3:0 u4:0 u5:2] D[d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u4:2 u5:2] D[d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010201021 U[u0:2 u1:2 u2:1 u3:1 u4:2 u5:2] D[d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010202012 U[u0:2 u1:2 u2:1 u3:1 u4:1 u5:1] D[d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u4:1 u5:1] D[d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010212012 U[u0:2 u1:2 u2:1 u3:0 u4:0 u5:1] D[d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010212021 U[u0:2 u1:2 u2:1 u3:0 u4:0 u5:1] D[d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012012012 U[u0:2 u1:0 u2:1 u3:2 u4:0 u5:1] D[d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012012021 U[u0:2 u1:0 u2:1 u3:2 u4:0 u5:1] D[d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012021012 U[u0:2 u1:0 u2:1 u3:1 u4:0 u5:2] D[d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012021021 U[u0:2 u1:0 u2:1 u3:1 u4:0 u5:2] D[d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012101012 U[u0:2 u1:0 u2:0 u3:2 u4:2 u5:2] D[d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012101021 U[u0:2 u1:0 u2:0 u3:2 u4:2 u5:2] D[d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012102012 U[u0:2 u1:0 u2:0 u3:2 u4:1 u5:1] D[d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012102021 U[u0:2 u1:0 u2:0 u3:2 u4:1 u5:1] D[d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012121012 U[u0:2 u1:0 u2:0 u3:0 u4:0 u5:2] D[d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012121021 U[u0:2 u1:0 u2:0 u3:0 u4:0 u5:2] D[d6:1 d7:0 d8:2]`
|
||||
|
||||
## C01 — word=UUUUUDUDD bites=- face=root apexes=[d5,d7,d8]
|
||||
|
||||
11 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u2:2 u3:2 u4:2 u6:2] D[d5:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u4:1 u6:1] D[d5:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u4:0 u6:2] D[d5:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u4:2 u6:2] D[d5:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u4:1 u6:1] D[d5:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010212021 U[u0:2 u1:2 u2:1 u3:0 u4:0 u6:1] D[d5:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012012021 U[u0:2 u1:0 u2:1 u3:2 u4:0 u6:1] D[d5:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012021012 U[u0:2 u1:0 u2:1 u3:1 u4:0 u6:2] D[d5:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012101012 U[u0:2 u1:0 u2:0 u3:2 u4:2 u6:2] D[d5:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012102021 U[u0:2 u1:0 u2:0 u3:2 u4:1 u6:1] D[d5:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012121012 U[u0:2 u1:0 u2:0 u3:0 u4:0 u6:2] D[d5:2 d7:0 d8:1]`
|
||||
|
||||
## C02 — word=UUUUDUUDD bites=- face=root apexes=[d4,d7,d8]
|
||||
|
||||
17 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u2:2 u3:2 u5:2 u6:2] D[d4:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101212 U[u0:2 u1:2 u2:2 u3:2 u5:0 u6:0] D[d4:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u5:1 u6:1] D[d4:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010102121 U[u0:2 u1:2 u2:2 u3:2 u5:0 u6:0] D[d4:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u5:2 u6:0] D[d4:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u5:2 u6:2] D[d4:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010201212 U[u0:2 u1:2 u2:1 u3:1 u5:0 u6:0] D[d4:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u5:1 u6:1] D[d4:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202121 U[u0:2 u1:2 u2:1 u3:1 u5:0 u6:0] D[d4:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010210212 U[u0:2 u1:2 u2:1 u3:0 u5:1 u6:0] D[d4:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012010212 U[u0:2 u1:0 u2:1 u3:2 u5:1 u6:0] D[d4:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012020121 U[u0:2 u1:0 u2:1 u3:1 u5:2 u6:0] D[d4:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012101012 U[u0:2 u1:0 u2:0 u3:2 u5:2 u6:2] D[d4:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012101212 U[u0:2 u1:0 u2:0 u3:2 u5:0 u6:0] D[d4:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012102021 U[u0:2 u1:0 u2:0 u3:2 u5:1 u6:1] D[d4:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012102121 U[u0:2 u1:0 u2:0 u3:2 u5:0 u6:0] D[d4:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012120121 U[u0:2 u1:0 u2:0 u3:0 u5:2 u6:0] D[d4:1 d7:0 d8:2]`
|
||||
|
||||
## C03 — word=UUUUDUDUD bites=- face=root apexes=[d4,d6,d8]
|
||||
|
||||
26 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101212 U[u0:2 u1:2 u2:2 u3:2 u5:0 u7:0] D[d4:2 d6:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010102121 U[u0:2 u1:2 u2:2 u3:2 u5:0 u7:0] D[d4:1 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:1 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:0 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=010121021 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:0 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=010121201 U[u0:2 u1:2 u2:2 u3:0 u5:0 u7:2] D[d4:0 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010201212 U[u0:2 u1:2 u2:1 u3:1 u5:0 u7:0] D[d4:2 d6:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202121 U[u0:2 u1:2 u2:1 u3:1 u5:0 u7:0] D[d4:1 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010210212 U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0] D[d4:2 d6:0 d8:1]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=010212012 U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0] D[d4:0 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=010212021 U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0] D[d4:0 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=010212102 U[u0:2 u1:2 u2:1 u3:0 u5:0 u7:1] D[d4:0 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012010212 U[u0:2 u1:0 u2:1 u3:2 u5:1 u7:0] D[d4:2 d6:0 d8:1]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=012012012 U[u0:2 u1:0 u2:1 u3:2 u5:1 u7:0] D[d4:0 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=012012021 U[u0:2 u1:0 u2:1 u3:2 u5:1 u7:0] D[d4:0 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=012012102 U[u0:2 u1:0 u2:1 u3:2 u5:0 u7:1] D[d4:0 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012020121 U[u0:2 u1:0 u2:1 u3:1 u5:2 u7:0] D[d4:1 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=012021012 U[u0:2 u1:0 u2:1 u3:1 u5:2 u7:0] D[d4:0 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=012021021 U[u0:2 u1:0 u2:1 u3:1 u5:2 u7:0] D[d4:0 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=012021201 U[u0:2 u1:0 u2:1 u3:1 u5:0 u7:2] D[d4:0 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012101212 U[u0:2 u1:0 u2:0 u3:2 u5:0 u7:0] D[d4:2 d6:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012102121 U[u0:2 u1:0 u2:0 u3:2 u5:0 u7:0] D[d4:1 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012120121 U[u0:2 u1:0 u2:0 u3:0 u5:2 u7:0] D[d4:1 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=012121012 U[u0:2 u1:0 u2:0 u3:0 u5:2 u7:0] D[d4:0 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=012121021 U[u0:2 u1:0 u2:0 u3:0 u5:2 u7:0] D[d4:0 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=012121201 U[u0:2 u1:0 u2:0 u3:0 u5:0 u7:2] D[d4:0 d6:1 d8:2]`
|
||||
|
||||
## C04 — word=UUUUDDDDD bites=(4,8) face=bite(4,8) apexes=[d5,d6,d7]
|
||||
|
||||
12 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `210` → canonical `012` · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d5:2 d6:1 d7:0 p4_8:2]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d5:0 d6:1 d7:2 p4_8:2]`
|
||||
- face apexes (raw labels) `120` → canonical `012` · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d5:1 d6:2 d7:0 p4_8:1]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d5:0 d6:2 d7:1 p4_8:1]`
|
||||
- face apexes (raw labels) `210` → canonical `012` · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d5:2 d6:1 d7:0 p4_8:2]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d5:0 d6:1 d7:2 p4_8:2]`
|
||||
- face apexes (raw labels) `120` → canonical `012` · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d5:1 d6:2 d7:0 p4_8:1]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d5:0 d6:2 d7:1 p4_8:1]`
|
||||
- face apexes (raw labels) `210` → canonical `012` · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d5:2 d6:1 d7:0 p4_8:2]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d5:0 d6:1 d7:2 p4_8:2]`
|
||||
- face apexes (raw labels) `120` → canonical `012` · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d5:1 d6:2 d7:0 p4_8:1]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d5:0 d6:2 d7:1 p4_8:1]`
|
||||
|
||||
## C05 — word=UUUDUUUDD bites=- face=root apexes=[d3,d7,d8]
|
||||
|
||||
13 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u2:2 u4:2 u5:2 u6:2] D[d3:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101212 U[u0:2 u1:2 u2:2 u4:2 u5:0 u6:0] D[d3:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010102012 U[u0:2 u1:2 u2:2 u4:1 u5:1 u6:2] D[d3:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010201021 U[u0:2 u1:2 u2:1 u4:2 u5:2 u6:1] D[d3:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u2:1 u4:1 u5:1 u6:1] D[d3:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202121 U[u0:2 u1:2 u2:1 u4:1 u5:0 u6:0] D[d3:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012010212 U[u0:2 u1:0 u2:1 u4:2 u5:1 u6:0] D[d3:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012012012 U[u0:2 u1:0 u2:1 u4:0 u5:1 u6:2] D[d3:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012020121 U[u0:2 u1:0 u2:1 u4:1 u5:2 u6:0] D[d3:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012021021 U[u0:2 u1:0 u2:1 u4:0 u5:2 u6:1] D[d3:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012101012 U[u0:2 u1:0 u2:0 u4:2 u5:2 u6:2] D[d3:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012101212 U[u0:2 u1:0 u2:0 u4:2 u5:0 u6:0] D[d3:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012102012 U[u0:2 u1:0 u2:0 u4:1 u5:1 u6:2] D[d3:2 d7:0 d8:1]`
|
||||
|
||||
## C06 — word=UUUDUUDUD bites=- face=root apexes=[d3,d6,d8]
|
||||
|
||||
20 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101212 U[u0:2 u1:2 u2:2 u4:2 u5:0 u7:0] D[d3:2 d6:0 d8:1]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=010120102 U[u0:2 u1:2 u2:2 u4:1 u5:2 u7:1] D[d3:0 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=010120201 U[u0:2 u1:2 u2:2 u4:1 u5:1 u7:2] D[d3:0 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=010121012 U[u0:2 u1:2 u2:2 u4:0 u5:2 u7:0] D[d3:0 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=010121021 U[u0:2 u1:2 u2:2 u4:0 u5:2 u7:0] D[d3:0 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=010121201 U[u0:2 u1:2 u2:2 u4:0 u5:0 u7:2] D[d3:0 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202121 U[u0:2 u1:2 u2:1 u4:1 u5:0 u7:0] D[d3:1 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=010210102 U[u0:2 u1:2 u2:1 u4:2 u5:2 u7:1] D[d3:0 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=010210201 U[u0:2 u1:2 u2:1 u4:2 u5:1 u7:2] D[d3:0 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=010212012 U[u0:2 u1:2 u2:1 u4:0 u5:1 u7:0] D[d3:0 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=010212021 U[u0:2 u1:2 u2:1 u4:0 u5:1 u7:0] D[d3:0 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=010212102 U[u0:2 u1:2 u2:1 u4:0 u5:0 u7:1] D[d3:0 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012010212 U[u0:2 u1:0 u2:1 u4:2 u5:1 u7:0] D[d3:2 d6:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012020121 U[u0:2 u1:0 u2:1 u4:1 u5:2 u7:0] D[d3:1 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012101212 U[u0:2 u1:0 u2:0 u4:2 u5:0 u7:0] D[d3:2 d6:0 d8:1]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=012120102 U[u0:2 u1:0 u2:0 u4:1 u5:2 u7:1] D[d3:0 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=012120201 U[u0:2 u1:0 u2:0 u4:1 u5:1 u7:2] D[d3:0 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=012121012 U[u0:2 u1:0 u2:0 u4:0 u5:2 u7:0] D[d3:0 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=012121021 U[u0:2 u1:0 u2:0 u4:0 u5:2 u7:0] D[d3:0 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=012121201 U[u0:2 u1:0 u2:0 u4:0 u5:0 u7:2] D[d3:0 d6:1 d8:2]`
|
||||
|
||||
## C07 — word=UUUDUDDDD bites=(3,5) face=root apexes=[d6,d7,d8]
|
||||
|
||||
6 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d6:2 d7:0 d8:1 p3_5:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d6:1 d7:0 d8:2 p3_5:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d6:2 d7:0 d8:1 p3_5:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d6:1 d7:0 d8:2 p3_5:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d6:2 d7:0 d8:1 p3_5:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d6:1 d7:0 d8:2 p3_5:2]`
|
||||
|
||||
## C08 — word=UUUDUDDDD bites=(3,8) face=bite(3,8) apexes=[d5,d6,d7]
|
||||
|
||||
6 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `210` → canonical `012` · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d5:2 d6:1 d7:0 p3_8:2]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d5:0 d6:1 d7:2 p3_8:2]`
|
||||
- face apexes (raw labels) `120` → canonical `012` · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d5:1 d6:2 d7:0 p3_8:1]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d5:0 d6:2 d7:1 p3_8:1]`
|
||||
- face apexes (raw labels) `210` → canonical `012` · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d5:2 d6:1 d7:0 p3_8:2]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d5:0 d6:1 d7:2 p3_8:2]`
|
||||
|
||||
## C09 — word=UUUDDUDDD bites=(4,6) face=root apexes=[d3,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d7:0 d8:1 p4_6:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d7:0 d8:2 p4_6:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d7:0 d8:1 p4_6:2]`
|
||||
|
||||
## C10 — word=UUUDDUDDD bites=(3,8) face=bite(3,8) apexes=[d4,d6,d7]
|
||||
|
||||
3 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `210` → canonical `012` · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d4:2 d6:1 d7:0 p3_8:2]`
|
||||
- face apexes (raw labels) `120` → canonical `012` · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d4:1 d6:2 d7:0 p3_8:1]`
|
||||
- face apexes (raw labels) `210` → canonical `012` · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d4:2 d6:1 d7:0 p3_8:2]`
|
||||
|
||||
## C11 — word=UUDUUDUUD bites=- face=root apexes=[d2,d5,d8]
|
||||
|
||||
18 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101202 U[u0:2 u1:2 u3:2 u4:2 u6:1 u7:1] D[d2:2 d5:0 d8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101212 U[u0:2 u1:2 u3:2 u4:2 u6:0 u7:0] D[d2:2 d5:0 d8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010102102 U[u0:2 u1:2 u3:2 u4:1 u6:2 u7:1] D[d2:2 d5:0 d8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010121202 U[u0:2 u1:2 u3:0 u4:0 u6:1 u7:1] D[d2:2 d5:0 d8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010121212 U[u0:2 u1:2 u3:0 u4:0 u6:0 u7:0] D[d2:2 d5:0 d8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010201201 U[u0:2 u1:2 u3:1 u4:2 u6:1 u7:2] D[d2:1 d5:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202101 U[u0:2 u1:2 u3:1 u4:1 u6:2 u7:2] D[d2:1 d5:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202121 U[u0:2 u1:2 u3:1 u4:1 u6:0 u7:0] D[d2:1 d5:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010212101 U[u0:2 u1:2 u3:0 u4:0 u6:2 u7:2] D[d2:1 d5:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010212121 U[u0:2 u1:2 u3:0 u4:0 u6:0 u7:0] D[d2:1 d5:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012012101 U[u0:2 u1:0 u3:2 u4:0 u6:2 u7:2] D[d2:1 d5:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012012121 U[u0:2 u1:0 u3:2 u4:0 u6:0 u7:0] D[d2:1 d5:0 d8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012021201 U[u0:2 u1:0 u3:1 u4:0 u6:1 u7:2] D[d2:1 d5:0 d8:2]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=012101012 U[u0:2 u1:0 u3:2 u4:2 u6:2 u7:0] D[d2:0 d5:2 d8:1]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=012102021 U[u0:2 u1:0 u3:2 u4:1 u6:1 u7:0] D[d2:0 d5:1 d8:2]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=012120102 U[u0:2 u1:0 u3:0 u4:1 u6:2 u7:1] D[d2:0 d5:2 d8:1]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=012120201 U[u0:2 u1:0 u3:0 u4:1 u6:1 u7:2] D[d2:0 d5:1 d8:2]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=012121012 U[u0:2 u1:0 u3:0 u4:0 u6:2 u7:0] D[d2:0 d5:2 d8:1]`
|
||||
|
||||
## C12 — word=UUDUUDDDD bites=(2,5) face=root apexes=[d6,d7,d8]
|
||||
|
||||
10 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d6:2 d7:0 d8:1 p2_5:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d6:1 d7:0 d8:2 p2_5:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d6:2 d7:0 d8:1 p2_5:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d6:1 d7:0 d8:2 p2_5:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d6:2 d7:0 d8:1 p2_5:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d6:1 d7:0 d8:2 p2_5:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d6:2 d7:0 d8:1 p2_5:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d6:1 d7:0 d8:2 p2_5:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d6:2 d7:0 d8:1 p2_5:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d6:1 d7:0 d8:2 p2_5:1]`
|
||||
|
||||
## C13 — word=UUDUDUDDD bites=(4,6) face=root apexes=[d2,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d7:0 d8:1 p4_6:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d7:0 d8:1 p4_6:0]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d7:0 d8:2 p4_6:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d7:0 d8:2 p4_6:0]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d7:0 d8:2 p4_6:0]`
|
||||
|
||||
## C14 — word=UUDUDUDDD bites=(2,4) face=root apexes=[d6,d7,d8]
|
||||
|
||||
6 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d6:2 d7:0 d8:1 p2_4:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d6:1 d7:0 d8:2 p2_4:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d6:2 d7:0 d8:1 p2_4:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d6:1 d7:0 d8:2 p2_4:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d6:2 d7:0 d8:1 p2_4:0]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d6:1 d7:0 d8:2 p2_4:0]`
|
||||
|
||||
## C15 — word=UUDUDUDDD bites=(2,8) face=bite(2,8) apexes=[d4,d6,d7]
|
||||
|
||||
5 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `210` → canonical `012` · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d4:2 d6:1 d7:0 p2_8:2]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d4:0 d6:1 d7:2 p2_8:2]`
|
||||
- face apexes (raw labels) `120` → canonical `012` · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d4:1 d6:2 d7:0 p2_8:1]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d4:0 d6:2 d7:1 p2_8:1]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d4:0 d6:2 d7:1 p2_8:1]`
|
||||
|
||||
## C16 — word=UUDUDDUDD bites=(5,7) face=root apexes=[d2,d4,d8]
|
||||
|
||||
6 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d8:1 p5_7:0]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d8:2 p5_7:0]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d8:2 p5_7:0]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d8:1 p5_7:0]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d8:2 p5_7:0]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d8:2 p5_7:2]`
|
||||
|
||||
## C17 — word=UUDUDDUDD bites=(2,4) face=root apexes=[d5,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d5:2 d7:0 d8:1 p2_4:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d5:1 d7:0 d8:2 p2_4:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d5:2 d7:0 d8:1 p2_4:0]`
|
||||
|
||||
## C18 — word=UUDUDDUDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d7]
|
||||
|
||||
5 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d4:1 d5:0 d7:2 p2_8:2]`
|
||||
- face apexes (raw labels) `120` → canonical `012` · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d4:1 d5:2 d7:0 p2_8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d4:2 d5:0 d7:1 p2_8:1]`
|
||||
- face apexes (raw labels) `210` → canonical `012` · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d4:2 d5:1 d7:0 p2_8:1]`
|
||||
- face apexes (raw labels) `210` → canonical `012` · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d4:2 d5:1 d7:0 p2_8:1]`
|
||||
|
||||
## C19 — word=UUDUDDDUD bites=(6,8) face=root apexes=[d2,d4,d5]
|
||||
|
||||
3 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `210` → canonical `012` · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 p6_8:2]`
|
||||
- face apexes (raw labels) `120` → canonical `012` · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 p6_8:1]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 p6_8:2]`
|
||||
|
||||
## C20 — word=UUDUDDDUD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6]
|
||||
|
||||
10 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d4:2 d5:0 d6:1 p2_8:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d4:1 d5:0 d6:2 p2_8:2]`
|
||||
- face apexes (raw labels) `120` → canonical `012` · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d4:1 d5:2 d6:0 p2_8:2]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d4:0 d5:2 d6:1 p2_8:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d4:2 d5:0 d6:1 p2_8:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d4:1 d5:0 d6:2 p2_8:1]`
|
||||
- face apexes (raw labels) `210` → canonical `012` · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d4:2 d5:1 d6:0 p2_8:1]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d4:0 d5:1 d6:2 p2_8:1]`
|
||||
- face apexes (raw labels) `210` → canonical `012` · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d4:2 d5:1 d6:0 p2_8:1]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d4:0 d5:1 d6:2 p2_8:1]`
|
||||
|
||||
## C21 — word=UUDDUUDDD bites=(3,6) face=root apexes=[d2,d7,d8]
|
||||
|
||||
9 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d7:0 d8:1 p3_6:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d7:0 d8:1 p3_6:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d7:0 d8:1 p3_6:0]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d7:0 d8:1 p3_6:0]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d7:0 d8:2 p3_6:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d7:0 d8:2 p3_6:1]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d7:0 d8:2 p3_6:0]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d7:0 d8:2 p3_6:0]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d7:0 d8:2 p3_6:1]`
|
||||
|
||||
## C22 — word=UUDDUDUDD bites=(5,7) face=root apexes=[d2,d3,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d8:1 p5_7:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d8:1 p5_7:0]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d8:2 p5_7:2]`
|
||||
- face apexes (raw labels) `102` → canonical `012` · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d8:2 p5_7:0]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d8:1 p5_7:0]`
|
||||
|
||||
## C23 — word=UUDDUDUDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d7]
|
||||
|
||||
8 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `210` → canonical `012` · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d3:2 d5:1 d7:0 p2_8:2]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d3:0 d5:1 d7:2 p2_8:2]`
|
||||
- face apexes (raw labels) `120` → canonical `012` · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d3:1 d5:2 d7:0 p2_8:1]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d3:0 d5:2 d7:1 p2_8:1]`
|
||||
- face apexes (raw labels) `210` → canonical `012` · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d3:2 d5:1 d7:0 p2_8:1]`
|
||||
- face apexes (raw labels) `210` → canonical `012` · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d3:2 d5:1 d7:0 p2_8:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d3:2 d5:0 d7:1 p2_8:1]`
|
||||
- face apexes (raw labels) `120` → canonical `012` · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d3:1 d5:2 d7:0 p2_8:1]`
|
||||
|
||||
## C24 — word=UDUDUDUDD bites=(5,7) face=root apexes=[d1,d3,d8]
|
||||
|
||||
6 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d8:1 p5_7:1]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d8:1 p5_7:0]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d8:1 p5_7:1]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d8:1 p5_7:1]`
|
||||
- face apexes (raw labels) `012` → canonical `012` · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d8:2 p5_7:2]`
|
||||
- face apexes (raw labels) `021` → canonical `012` · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d8:1 p5_7:0]`
|
||||
|
||||
## C25 — word=UDUDUDUDD bites=(3,5) face=root apexes=[d1,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `012`:
|
||||
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d7:0 d8:1 p3_5:2]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d7:0 d8:1 p3_5:0]`
|
||||
- face apexes (raw labels) `201` → canonical `012` · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d7:0 d8:1 p3_5:1]`
|
||||
|
||||
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|
||||
# Inner-face singleton-down-apex sequences of Kempe-balanced colourings (n=9, m=3)
|
||||
|
||||
Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 3 singleton down-tooth apexes: **26 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in increasing annular-edge order, and reduce it modulo colour permutation (NOT dihedral symmetry).
|
||||
|
||||
- Total Kempe-balanced colourings (mod colour permutation): **241**.
|
||||
- Distinct canonical down-apex sequences overall: **1**.
|
||||
|
||||
## Distinct canonical down-apex sequences
|
||||
|
||||
| sequence | colour multiset | #configs realising | #colourings |
|
||||
|---|---|---|---|
|
||||
| `012` | 1+1+1 | 26 | 241 |
|
||||
|
||||
Note: every realised sequence has its three colour-counts of **equal parity** — exactly the Kempe-parity constraint on the inner face (each colour pair meets its singleton down apexes an even number of times). With m = 3 apexes (m is odd) every count must be **odd**, so the only admissible colour multisets are 1+1+1.
|
||||
|
||||
## Step 4 — grouping configs by their set of unique down-apex sequences
|
||||
|
||||
The 26 configs fall into **1** groups by the set of canonical down-apex sequences they realise:
|
||||
|
||||
| #configs | set of down-apex sequences | config ids |
|
||||
|---|---|---|
|
||||
| 26 | { `012` } | C00, C01, C02, C03, C04, C05, C06, C07, C08, C09, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19, C20, C21, C22, C23, C24, C25 |
|
||||
|
||||
## Config atlas (ids)
|
||||
|
||||
| id | word / bites / face / apexes | #Kempe-balanced | down-apex sequence set |
|
||||
|---|---|---|---|
|
||||
| C00 | word=UUUUUUDDD bites=- face=root apexes=[d6,d7,d8] | 22 | { `012` } |
|
||||
| C01 | word=UUUUUDUDD bites=- face=root apexes=[d5,d7,d8] | 11 | { `012` } |
|
||||
| C02 | word=UUUUDUUDD bites=- face=root apexes=[d4,d7,d8] | 17 | { `012` } |
|
||||
| C03 | word=UUUUDUDUD bites=- face=root apexes=[d4,d6,d8] | 26 | { `012` } |
|
||||
| C04 | word=UUUUDDDDD bites=(4,8) face=bite(4,8) apexes=[d5,d6,d7] | 12 | { `012` } |
|
||||
| C05 | word=UUUDUUUDD bites=- face=root apexes=[d3,d7,d8] | 13 | { `012` } |
|
||||
| C06 | word=UUUDUUDUD bites=- face=root apexes=[d3,d6,d8] | 20 | { `012` } |
|
||||
| C07 | word=UUUDUDDDD bites=(3,5) face=root apexes=[d6,d7,d8] | 6 | { `012` } |
|
||||
| C08 | word=UUUDUDDDD bites=(3,8) face=bite(3,8) apexes=[d5,d6,d7] | 6 | { `012` } |
|
||||
| C09 | word=UUUDDUDDD bites=(4,6) face=root apexes=[d3,d7,d8] | 3 | { `012` } |
|
||||
| C10 | word=UUUDDUDDD bites=(3,8) face=bite(3,8) apexes=[d4,d6,d7] | 3 | { `012` } |
|
||||
| C11 | word=UUDUUDUUD bites=- face=root apexes=[d2,d5,d8] | 18 | { `012` } |
|
||||
| C12 | word=UUDUUDDDD bites=(2,5) face=root apexes=[d6,d7,d8] | 10 | { `012` } |
|
||||
| C13 | word=UUDUDUDDD bites=(4,6) face=root apexes=[d2,d7,d8] | 5 | { `012` } |
|
||||
| C14 | word=UUDUDUDDD bites=(2,4) face=root apexes=[d6,d7,d8] | 6 | { `012` } |
|
||||
| C15 | word=UUDUDUDDD bites=(2,8) face=bite(2,8) apexes=[d4,d6,d7] | 5 | { `012` } |
|
||||
| C16 | word=UUDUDDUDD bites=(5,7) face=root apexes=[d2,d4,d8] | 6 | { `012` } |
|
||||
| C17 | word=UUDUDDUDD bites=(2,4) face=root apexes=[d5,d7,d8] | 3 | { `012` } |
|
||||
| C18 | word=UUDUDDUDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d7] | 5 | { `012` } |
|
||||
| C19 | word=UUDUDDDUD bites=(6,8) face=root apexes=[d2,d4,d5] | 3 | { `012` } |
|
||||
| C20 | word=UUDUDDDUD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6] | 10 | { `012` } |
|
||||
| C21 | word=UUDDUUDDD bites=(3,6) face=root apexes=[d2,d7,d8] | 9 | { `012` } |
|
||||
| C22 | word=UUDDUDUDD bites=(5,7) face=root apexes=[d2,d3,d8] | 5 | { `012` } |
|
||||
| C23 | word=UUDDUDUDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d7] | 8 | { `012` } |
|
||||
| C24 | word=UDUDUDUDD bites=(5,7) face=root apexes=[d1,d3,d8] | 6 | { `012` } |
|
||||
| C25 | word=UDUDUDUDD bites=(3,5) face=root apexes=[d1,d7,d8] | 3 | { `012` } |
|
||||
|
||||
## Per-sequence notes
|
||||
|
||||
- [`012`](seq_012.md) — figure `seq_012.png`
|
||||
|
||||
+180
@@ -0,0 +1,180 @@
|
||||
# Inner-face down-apex sequence `0000`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 4×colour0.
|
||||
- Realised by **21** of 23 configs (M(T), inner face).
|
||||
- **64** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_0000.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C00 — word=UUUUUDDDD bites=- face=root apexes=[d5,d6,d7,d8]
|
||||
|
||||
10 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=010120101 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=010120202 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=010210101 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=010210202 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012010202 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=012020101 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=012120101 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012120202 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:1 d7:1 d8:1]`
|
||||
|
||||
## C01 — word=UUUUDUDDD bites=- face=root apexes=[d4,d6,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=010120202 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=010210101 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012120202 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:1 d7:1 d8:1]`
|
||||
|
||||
## C02 — word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=010120202 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:1 d5:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=010210101 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:2 d5:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:2 d5:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:1 d5:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012120202 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:1 d5:1 d7:1 d8:1]`
|
||||
|
||||
## C03 — word=UUUDUUDDD bites=- face=root apexes=[d3,d6,d7,d8]
|
||||
|
||||
7 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=010102101 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=010201202 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=012012101 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012021202 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=012102101 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:2 d7:2 d8:2]`
|
||||
|
||||
## C04 — word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u6:2] D[d3:2 d5:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u2:1 u4:1 u6:1] D[d3:1 d5:1 d7:1 d8:1]`
|
||||
|
||||
## C05 — word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u7:2] D[d3:2 d5:2 d6:2 d8:2]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u2:1 u4:1 u7:1] D[d3:1 d5:1 d6:1 d8:1]`
|
||||
|
||||
## C06 — word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8]
|
||||
|
||||
7 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=010101201 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:2 d4:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=010202102 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:1 d4:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=012010101 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:2 d4:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=012010201 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020102 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:1 d4:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=012101201 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:2 d4:2 d7:2 d8:2]`
|
||||
|
||||
## C07 — word=UUUDDDDDD bites=(3,8) face=bite(3,8) apexes=[d4,d5,d6,d7]
|
||||
|
||||
4 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=012010101 U[u0:2 u1:0 u2:1] D[d4:2 d5:2 d6:2 d7:2 p3_8:2]`
|
||||
- face apexes (raw labels) `0000` → canonical `0000` · `A=012012121 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:0 d7:0 p3_8:2]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u2:1] D[d4:1 d5:1 d6:1 d7:1 p3_8:1]`
|
||||
- face apexes (raw labels) `0000` → canonical `0000` · `A=012021212 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:0 d7:0 p3_8:1]`
|
||||
|
||||
## C08 — word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8]
|
||||
|
||||
4 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=010120101 U[u0:2 u1:2 u3:0 u4:1 u6:2] D[d2:2 d5:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=010210202 U[u0:2 u1:2 u3:0 u4:2 u6:1] D[d2:1 d5:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012010202 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:1 d5:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u3:1 u4:1 u6:1] D[d2:1 d5:1 d7:1 d8:1]`
|
||||
|
||||
## C09 — word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8]
|
||||
|
||||
4 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=010101201 U[u0:2 u1:2 u3:2 u5:0 u6:1] D[d2:2 d4:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=010202102 U[u0:2 u1:2 u3:1 u5:0 u6:2] D[d2:1 d4:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020102 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u3:1 u5:1 u6:1] D[d2:1 d4:1 d7:1 d8:1]`
|
||||
|
||||
## C10 — word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u3:1 u5:1 u7:1] D[d2:1 d4:1 d6:1 d8:1]`
|
||||
|
||||
## C11 — word=UUDUDDDDD bites=(2,4) face=root apexes=[d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=012020101 U[u0:2 u1:0 u3:1] D[d5:2 d6:2 d7:2 d8:2 p2_4:1]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u3:1] D[d5:1 d6:1 d7:1 d8:1 p2_4:1]`
|
||||
|
||||
## C12 — word=UUDUDDDDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6,d7]
|
||||
|
||||
2 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u3:1] D[d4:1 d5:1 d6:1 d7:1 p2_8:1]`
|
||||
- face apexes (raw labels) `0000` → canonical `0000` · `A=012021212 U[u0:2 u1:0 u3:1] D[d4:0 d5:0 d6:0 d7:0 p2_8:1]`
|
||||
|
||||
## C13 — word=UUDDUDDDD bites=(3,5) face=root apexes=[d2,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u4:1] D[d2:1 d6:1 d7:1 d8:1 p3_5:1]`
|
||||
|
||||
## C14 — word=UUDDUDDDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d6,d7]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u4:1] D[d3:1 d5:1 d6:1 d7:1 p2_8:1]`
|
||||
|
||||
## C15 — word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d7:1 d8:1 p4_6:1]`
|
||||
|
||||
## C16 — word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u5:1] D[d3:1 d4:1 d6:1 d7:1 p2_8:1]`
|
||||
|
||||
## C17 — word=UDUDUDDDD bites=(3,5) face=root apexes=[d1,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=010212101 U[u0:2 u2:1 u4:0] D[d1:2 d6:2 d7:2 d8:2 p3_5:0]`
|
||||
|
||||
## C18 — word=UDUDUDDDD bites=(1,3) face=root apexes=[d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `2222` → canonical `0000` · `A=012120101 U[u0:2 u2:0 u4:1] D[d5:2 d6:2 d7:2 d8:2 p1_3:0]`
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012120202 U[u0:2 u2:0 u4:1] D[d5:1 d6:1 d7:1 d8:1 p1_3:0]`
|
||||
|
||||
## C20 — word=UDUDDUDDD bites=(1,3) face=root apexes=[d4,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `1111` → canonical `0000` · `A=012120202 U[u0:2 u2:0 u5:1] D[d4:1 d6:1 d7:1 d8:1 p1_3:0]`
|
||||
|
||||
## C21 — word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0000`:
|
||||
|
||||
- face apexes (raw labels) `0000` → canonical `0000` · `A=010212121 U[u0:2 u2:1 u5:0] D[d3:0 d4:0 d6:0 d7:0 p1_8:2]`
|
||||
|
||||
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|
||||
# Inner-face down-apex sequence `0011`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 2×colour0, 2×colour1.
|
||||
- Realised by **23** of 23 configs (M(T), inner face).
|
||||
- **108** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_0011.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C00 — word=UUUUUDDDD bites=- face=root apexes=[d5,d6,d7,d8]
|
||||
|
||||
10 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=010120102 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=010210201 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012010201 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=012020102 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=012120102 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:1 d7:2 d8:2]`
|
||||
|
||||
## C01 — word=UUUUDUDDD bites=- face=root apexes=[d4,d6,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:1 d7:2 d8:2]`
|
||||
|
||||
## C02 — word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8]
|
||||
|
||||
15 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:1 d5:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=010121201 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=010121202 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:0 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:2 d5:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=010212101 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=010212102 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:0 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:2 d5:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=012012101 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=012012102 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:0 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:1 d5:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=012021201 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=012021202 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:0 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:1 d5:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=012121201 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=012121202 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:0 d7:1 d8:1]`
|
||||
|
||||
## C03 — word=UUUDUUDDD bites=- face=root apexes=[d3,d6,d7,d8]
|
||||
|
||||
7 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=010102102 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=010201201 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=012010102 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=012012102 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012021201 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=012102102 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:2 d7:1 d8:1]`
|
||||
|
||||
## C04 — word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8]
|
||||
|
||||
8 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=010121201 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=010121202 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:0 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=010212101 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=010212102 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:0 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=012010102 U[u0:2 u1:0 u2:1 u4:2 u6:2] D[d3:2 d5:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u2:1 u4:1 u6:1] D[d3:1 d5:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=012121201 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=012121202 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:0 d7:1 d8:1]`
|
||||
|
||||
## C05 — word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=010121202 U[u0:2 u1:2 u2:2 u4:0 u7:1] D[d3:0 d5:0 d6:1 d8:1]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=010212101 U[u0:2 u1:2 u2:1 u4:0 u7:2] D[d3:0 d5:0 d6:2 d8:2]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=012121202 U[u0:2 u1:0 u2:0 u4:0 u7:1] D[d3:0 d5:0 d6:1 d8:1]`
|
||||
|
||||
## C06 — word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8]
|
||||
|
||||
13 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=010101202 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:2 d4:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=010121201 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:0 d4:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=010121202 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:0 d4:0 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=010202101 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:1 d4:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=010212101 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:0 d4:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=010212102 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:0 d4:0 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=012010102 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:2 d4:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=012010202 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012020101 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:1 d4:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=012101202 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:2 d4:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=012121201 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:0 d4:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=012121202 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:0 d4:0 d7:1 d8:1]`
|
||||
|
||||
## C07 — word=UUUDDDDDD bites=(3,8) face=bite(3,8) apexes=[d4,d5,d6,d7]
|
||||
|
||||
4 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `2200` → canonical `0011` · `A=012010121 U[u0:2 u1:0 u2:1] D[d4:2 d5:2 d6:0 d7:0 p3_8:2]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=012012101 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:2 d7:2 p3_8:2]`
|
||||
- face apexes (raw labels) `1100` → canonical `0011` · `A=012020212 U[u0:2 u1:0 u2:1] D[d4:1 d5:1 d6:0 d7:0 p3_8:1]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=012021202 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:1 d7:1 p3_8:1]`
|
||||
|
||||
## C08 — word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8]
|
||||
|
||||
10 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=010120102 U[u0:2 u1:2 u3:0 u4:1 u6:2] D[d2:2 d5:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=010210201 U[u0:2 u1:2 u3:0 u4:2 u6:1] D[d2:1 d5:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012010201 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:1 d5:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u3:1 u4:1 u6:1] D[d2:1 d5:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=012101201 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=012101202 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:0 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=012102101 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=012102102 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:0 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=012121201 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=012121202 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:0 d7:1 d8:1]`
|
||||
|
||||
## C09 — word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8]
|
||||
|
||||
6 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=010101202 U[u0:2 u1:2 u3:2 u5:0 u6:1] D[d2:2 d4:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=010202101 U[u0:2 u1:2 u3:1 u5:0 u6:2] D[d2:1 d4:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012020101 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u3:1 u5:1 u6:1] D[d2:1 d4:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=012121201 U[u0:2 u1:0 u3:0 u5:0 u6:1] D[d2:0 d4:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=012121202 U[u0:2 u1:0 u3:0 u5:0 u6:1] D[d2:0 d4:0 d7:1 d8:1]`
|
||||
|
||||
## C10 — word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8]
|
||||
|
||||
4 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=010101202 U[u0:2 u1:2 u3:2 u5:0 u7:1] D[d2:2 d4:2 d6:1 d8:1]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=010202101 U[u0:2 u1:2 u3:1 u5:0 u7:2] D[d2:1 d4:1 d6:2 d8:2]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012020101 U[u0:2 u1:0 u3:1 u5:2 u7:2] D[d2:1 d4:1 d6:2 d8:2]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=012121202 U[u0:2 u1:0 u3:0 u5:0 u7:1] D[d2:0 d4:0 d6:1 d8:1]`
|
||||
|
||||
## C11 — word=UUDUDDDDD bites=(2,4) face=root apexes=[d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=012020102 U[u0:2 u1:0 u3:1] D[d5:2 d6:2 d7:1 d8:1 p2_4:1]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u3:1] D[d5:1 d6:1 d7:2 d8:2 p2_4:1]`
|
||||
|
||||
## C12 — word=UUDUDDDDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6,d7]
|
||||
|
||||
2 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `1100` → canonical `0011` · `A=012020212 U[u0:2 u1:0 u3:1] D[d4:1 d5:1 d6:0 d7:0 p2_8:1]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=012021202 U[u0:2 u1:0 u3:1] D[d4:0 d5:0 d6:1 d7:1 p2_8:1]`
|
||||
|
||||
## C13 — word=UUDDUDDDD bites=(3,5) face=root apexes=[d2,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d6:1 d7:2 d8:2 p3_5:1]`
|
||||
|
||||
## C14 — word=UUDDUDDDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d6,d7]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `1100` → canonical `0011` · `A=012020212 U[u0:2 u1:0 u4:1] D[d3:1 d5:1 d6:0 d7:0 p2_8:1]`
|
||||
|
||||
## C15 — word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d7:2 d8:2 p4_6:1]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=012120201 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d7:2 d8:2 p4_6:1]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d7:1 d8:1 p4_6:1]`
|
||||
|
||||
## C16 — word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7]
|
||||
|
||||
3 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=012010202 U[u0:2 u1:0 u5:1] D[d3:2 d4:2 d6:1 d7:1 p2_8:1]`
|
||||
- face apexes (raw labels) `2200` → canonical `0011` · `A=012010212 U[u0:2 u1:0 u5:1] D[d3:2 d4:2 d6:0 d7:0 p2_8:1]`
|
||||
- face apexes (raw labels) `1100` → canonical `0011` · `A=012020212 U[u0:2 u1:0 u5:1] D[d3:1 d4:1 d6:0 d7:0 p2_8:1]`
|
||||
|
||||
## C17 — word=UDUDUDDDD bites=(3,5) face=root apexes=[d1,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=010212102 U[u0:2 u2:1 u4:0] D[d1:2 d6:2 d7:1 d8:1 p3_5:0]`
|
||||
|
||||
## C18 — word=UDUDUDDDD bites=(1,3) face=root apexes=[d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=012120102 U[u0:2 u2:0 u4:1] D[d5:2 d6:2 d7:1 d8:1 p1_3:0]`
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012120201 U[u0:2 u2:0 u4:1] D[d5:1 d6:1 d7:2 d8:2 p1_3:0]`
|
||||
|
||||
## C19 — word=UDUDDUDDD bites=(4,6) face=root apexes=[d1,d3,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=012120201 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d7:2 d8:2 p4_6:1]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d7:1 d8:1 p4_6:1]`
|
||||
|
||||
## C20 — word=UDUDDUDDD bites=(1,3) face=root apexes=[d4,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=012120201 U[u0:2 u2:0 u5:1] D[d4:1 d6:1 d7:2 d8:2 p1_3:0]`
|
||||
|
||||
## C21 — word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7]
|
||||
|
||||
3 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `1122` → canonical `0011` · `A=010202101 U[u0:2 u2:1 u5:0] D[d3:1 d4:1 d6:2 d7:2 p1_8:2]`
|
||||
- face apexes (raw labels) `1100` → canonical `0011` · `A=010202121 U[u0:2 u2:1 u5:0] D[d3:1 d4:1 d6:0 d7:0 p1_8:2]`
|
||||
- face apexes (raw labels) `0022` → canonical `0011` · `A=010212101 U[u0:2 u2:1 u5:0] D[d3:0 d4:0 d6:2 d7:2 p1_8:2]`
|
||||
|
||||
## C22 — word=UDDUDDUDD bites=(5,7) face=root apexes=[d1,d2,d4,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `0011`:
|
||||
|
||||
- face apexes (raw labels) `2211` → canonical `0011` · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d8:1 p5_7:1]`
|
||||
- face apexes (raw labels) `0011` → canonical `0011` · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d8:1 p5_7:1]`
|
||||
|
||||
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|
||||
# Inner-face down-apex sequence `0101`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 2×colour0, 2×colour1.
|
||||
- Realised by **12** of 23 configs (M(T), inner face).
|
||||
- **53** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_0101.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C02 — word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `0101`:
|
||||
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=010121021 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:2 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=010212012 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:1 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=012012012 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:1 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=012021021 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:2 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=012121021 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:2 d7:0 d8:2]`
|
||||
|
||||
## C04 — word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8]
|
||||
|
||||
9 colouring(s) with down-apex sequence `0101`:
|
||||
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=010120121 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[d3:0 d5:2 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=010120212 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[d3:0 d5:1 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=010121021 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:2 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=010210121 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[d3:0 d5:2 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=010210212 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[d3:0 d5:1 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=010212012 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:1 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=012120121 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:0 d5:2 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=012120212 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:0 d5:1 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=012121021 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:2 d7:0 d8:2]`
|
||||
|
||||
## C05 — word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8]
|
||||
|
||||
11 colouring(s) with down-apex sequence `0101`:
|
||||
|
||||
- face apexes (raw labels) `2121` → canonical `0101` · `A=010102012 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:1 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=010120121 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:0 d5:2 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=010120212 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:0 d5:1 d6:0 d8:1]`
|
||||
- face apexes (raw labels) `1212` → canonical `0101` · `A=010201021 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:2 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=010210121 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:0 d5:2 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=010210212 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:0 d5:1 d6:0 d8:1]`
|
||||
- face apexes (raw labels) `2121` → canonical `0101` · `A=012012012 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:1 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `1212` → canonical `0101` · `A=012021021 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:2 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `2121` → canonical `0101` · `A=012102012 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:1 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=012120121 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:0 d5:2 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=012120212 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:0 d5:1 d6:0 d8:1]`
|
||||
|
||||
## C06 — word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `0101`:
|
||||
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=010120212 U[u0:2 u1:2 u2:2 u5:1 u6:0] D[d3:0 d4:1 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=010210121 U[u0:2 u1:2 u2:1 u5:2 u6:0] D[d3:0 d4:2 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=012120212 U[u0:2 u1:0 u2:0 u5:1 u6:0] D[d3:0 d4:1 d7:0 d8:1]`
|
||||
|
||||
## C08 — word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `0101`:
|
||||
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=012101021 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:2 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=012102012 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:1 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=012120121 U[u0:2 u1:0 u3:0 u4:1 u6:0] D[d2:0 d5:2 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=012120212 U[u0:2 u1:0 u3:0 u4:1 u6:0] D[d2:0 d5:1 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=012121021 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:2 d7:0 d8:2]`
|
||||
|
||||
## C09 — word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `0101`:
|
||||
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=012101021 U[u0:2 u1:0 u3:2 u5:2 u6:1] D[d2:0 d4:2 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=012102012 U[u0:2 u1:0 u3:2 u5:1 u6:2] D[d2:0 d4:1 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=012120212 U[u0:2 u1:0 u3:0 u5:1 u6:0] D[d2:0 d4:1 d7:0 d8:1]`
|
||||
|
||||
## C10 — word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8]
|
||||
|
||||
8 colouring(s) with down-apex sequence `0101`:
|
||||
|
||||
- face apexes (raw labels) `2121` → canonical `0101` · `A=010102012 U[u0:2 u1:2 u3:2 u5:1 u7:0] D[d2:2 d4:1 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `2121` → canonical `0101` · `A=010102102 U[u0:2 u1:2 u3:2 u5:0 u7:1] D[d2:2 d4:1 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `2121` → canonical `0101` · `A=010120102 U[u0:2 u1:2 u3:0 u5:2 u7:1] D[d2:2 d4:1 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `1212` → canonical `0101` · `A=010201021 U[u0:2 u1:2 u3:1 u5:2 u7:0] D[d2:1 d4:2 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `1212` → canonical `0101` · `A=010201201 U[u0:2 u1:2 u3:1 u5:0 u7:2] D[d2:1 d4:2 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `1212` → canonical `0101` · `A=010210201 U[u0:2 u1:2 u3:0 u5:1 u7:2] D[d2:1 d4:2 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `1212` → canonical `0101` · `A=012010201 U[u0:2 u1:0 u3:2 u5:1 u7:2] D[d2:1 d4:2 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=012120212 U[u0:2 u1:0 u3:0 u5:1 u7:0] D[d2:0 d4:1 d6:0 d8:1]`
|
||||
|
||||
## C15 — word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0101`:
|
||||
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=012102021 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d7:0 d8:2 p4_6:1]`
|
||||
|
||||
## C16 — word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0101`:
|
||||
|
||||
- face apexes (raw labels) `2020` → canonical `0101` · `A=012012012 U[u0:2 u1:0 u5:1] D[d3:2 d4:0 d6:2 d7:0 p2_8:1]`
|
||||
|
||||
## C19 — word=UDUDDUDDD bites=(4,6) face=root apexes=[d1,d3,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `0101`:
|
||||
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=012012121 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d7:0 d8:2 p4_6:0]`
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=012021212 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d7:0 d8:1 p4_6:0]`
|
||||
- face apexes (raw labels) `0202` → canonical `0101` · `A=012102021 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d7:0 d8:2 p4_6:1]`
|
||||
|
||||
## C21 — word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0101`:
|
||||
|
||||
- face apexes (raw labels) `1212` → canonical `0101` · `A=010201201 U[u0:2 u2:1 u5:0] D[d3:1 d4:2 d6:1 d7:2 p1_8:2]`
|
||||
|
||||
## C22 — word=UDDUDDUDD bites=(5,7) face=root apexes=[d1,d2,d4,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `0101`:
|
||||
|
||||
- face apexes (raw labels) `2121` → canonical `0101` · `A=010201212 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:2 d8:1 p5_7:0]`
|
||||
- face apexes (raw labels) `2121` → canonical `0101` · `A=010210202 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d8:1 p5_7:1]`
|
||||
- face apexes (raw labels) `0101` → canonical `0101` · `A=012021212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:0 d8:1 p5_7:0]`
|
||||
|
||||
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|
||||
# Inner-face down-apex sequence `0110`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 2×colour0, 2×colour1.
|
||||
- Realised by **19** of 23 configs (M(T), inner face).
|
||||
- **73** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_0110.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C00 — word=UUUUUDDDD bites=- face=root apexes=[d5,d6,d7,d8]
|
||||
|
||||
10 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=010120212 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=010210121 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=010210212 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=012010121 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012010212 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=012020121 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012020212 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=012120121 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012120212 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## C01 — word=UUUUDUDDD bites=- face=root apexes=[d4,d6,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=010120212 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=010210121 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=012010121 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012020212 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012120212 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## C03 — word=UUUDUUDDD bites=- face=root apexes=[d3,d6,d7,d8]
|
||||
|
||||
7 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=012010121 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012020212 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:0 d7:0 d8:2]`
|
||||
|
||||
## C04 — word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[d3:2 d5:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[d3:1 d5:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[d3:2 d5:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[d3:1 d5:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:2 d5:0 d7:0 d8:2]`
|
||||
|
||||
## C05 — word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8]
|
||||
|
||||
12 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `2112` → canonical `0110` · `A=010102021 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:1 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:0 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `1221` → canonical `0110` · `A=010201012 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:2 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:0 d6:0 d8:1]`
|
||||
- face apexes (raw labels) `2112` → canonical `0110` · `A=012010201 U[u0:2 u1:0 u2:1 u4:2 u7:2] D[d3:2 d5:1 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `2112` → canonical `0110` · `A=012012021 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:1 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:0 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `1221` → canonical `0110` · `A=012020102 U[u0:2 u1:0 u2:1 u4:1 u7:1] D[d3:1 d5:2 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `1221` → canonical `0110` · `A=012021012 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:2 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:0 d6:0 d8:1]`
|
||||
- face apexes (raw labels) `2112` → canonical `0110` · `A=012102021 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:1 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:0 d6:0 d8:2]`
|
||||
|
||||
## C06 — word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8]
|
||||
|
||||
4 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=012012021 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=012012121 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[d3:2 d4:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012021012 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012021212 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[d3:1 d4:0 d7:0 d8:1]`
|
||||
|
||||
## C07 — word=UUUDDDDDD bites=(3,8) face=bite(3,8) apexes=[d4,d5,d6,d7]
|
||||
|
||||
4 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `2112` → canonical `0110` · `A=012010201 U[u0:2 u1:0 u2:1] D[d4:2 d5:1 d6:1 d7:2 p3_8:2]`
|
||||
- face apexes (raw labels) `0110` → canonical `0110` · `A=012012021 U[u0:2 u1:0 u2:1] D[d4:0 d5:1 d6:1 d7:0 p3_8:2]`
|
||||
- face apexes (raw labels) `1221` → canonical `0110` · `A=012020102 U[u0:2 u1:0 u2:1] D[d4:1 d5:2 d6:2 d7:1 p3_8:1]`
|
||||
- face apexes (raw labels) `0220` → canonical `0110` · `A=012021012 U[u0:2 u1:0 u2:1] D[d4:0 d5:2 d6:2 d7:0 p3_8:1]`
|
||||
|
||||
## C08 — word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=010102121 U[u0:2 u1:2 u3:2 u4:1 u6:0] D[d2:2 d5:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=010201212 U[u0:2 u1:2 u3:1 u4:2 u6:0] D[d2:1 d5:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012021212 U[u0:2 u1:0 u3:1 u4:0 u6:0] D[d2:1 d5:0 d7:0 d8:1]`
|
||||
|
||||
## C09 — word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=010121021 U[u0:2 u1:2 u3:0 u5:2 u6:1] D[d2:2 d4:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=010212012 U[u0:2 u1:2 u3:0 u5:1 u6:2] D[d2:1 d4:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012012012 U[u0:2 u1:0 u3:2 u5:1 u6:2] D[d2:1 d4:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012021012 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u6:0] D[d2:1 d4:0 d7:0 d8:1]`
|
||||
|
||||
## C10 — word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8]
|
||||
|
||||
6 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `2112` → canonical `0110` · `A=010102021 U[u0:2 u1:2 u3:2 u5:1 u7:0] D[d2:2 d4:1 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `2112` → canonical `0110` · `A=010120201 U[u0:2 u1:2 u3:0 u5:1 u7:2] D[d2:2 d4:1 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `1221` → canonical `0110` · `A=010201012 U[u0:2 u1:2 u3:1 u5:2 u7:0] D[d2:1 d4:2 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `1221` → canonical `0110` · `A=010210102 U[u0:2 u1:2 u3:0 u5:2 u7:1] D[d2:1 d4:2 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `1221` → canonical `0110` · `A=012010102 U[u0:2 u1:0 u3:2 u5:2 u7:1] D[d2:1 d4:2 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u7:0] D[d2:1 d4:0 d6:0 d8:1]`
|
||||
|
||||
## C11 — word=UUDUDDDDD bites=(2,4) face=root apexes=[d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=012020121 U[u0:2 u1:0 u3:1] D[d5:2 d6:0 d7:0 d8:2 p2_4:1]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012020212 U[u0:2 u1:0 u3:1] D[d5:1 d6:0 d7:0 d8:1 p2_4:1]`
|
||||
|
||||
## C12 — word=UUDUDDDDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6,d7]
|
||||
|
||||
2 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `1221` → canonical `0110` · `A=012020102 U[u0:2 u1:0 u3:1] D[d4:1 d5:2 d6:2 d7:1 p2_8:1]`
|
||||
- face apexes (raw labels) `0220` → canonical `0110` · `A=012021012 U[u0:2 u1:0 u3:1] D[d4:0 d5:2 d6:2 d7:0 p2_8:1]`
|
||||
|
||||
## C13 — word=UUDDUDDDD bites=(3,5) face=root apexes=[d2,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d6:0 d7:0 d8:1 p3_5:1]`
|
||||
|
||||
## C14 — word=UUDDUDDDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d6,d7]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `1221` → canonical `0110` · `A=012020102 U[u0:2 u1:0 u4:1] D[d3:1 d5:2 d6:2 d7:1 p2_8:1]`
|
||||
|
||||
## C17 — word=UDUDUDDDD bites=(3,5) face=root apexes=[d1,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=010212121 U[u0:2 u2:1 u4:0] D[d1:2 d6:0 d7:0 d8:2 p3_5:0]`
|
||||
|
||||
## C18 — word=UDUDUDDDD bites=(1,3) face=root apexes=[d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=012120121 U[u0:2 u2:0 u4:1] D[d5:2 d6:0 d7:0 d8:2 p1_3:0]`
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012120212 U[u0:2 u2:0 u4:1] D[d5:1 d6:0 d7:0 d8:1 p1_3:0]`
|
||||
|
||||
## C19 — word=UDUDDUDDD bites=(4,6) face=root apexes=[d1,d3,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `2002` → canonical `0110` · `A=010212121 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d7:0 d8:2 p4_6:0]`
|
||||
|
||||
## C20 — word=UDUDDUDDD bites=(1,3) face=root apexes=[d4,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `1001` → canonical `0110` · `A=012120212 U[u0:2 u2:0 u5:1] D[d4:1 d6:0 d7:0 d8:1 p1_3:0]`
|
||||
|
||||
## C22 — word=UDDUDDUDD bites=(5,7) face=root apexes=[d1,d2,d4,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `0110`:
|
||||
|
||||
- face apexes (raw labels) `2112` → canonical `0110` · `A=010202121 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:1 d8:2 p5_7:0]`
|
||||
|
||||
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|
||||
# Inner-face singleton-down-apex sequences of Kempe-balanced colourings (n=9, m=4)
|
||||
|
||||
Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 4 singleton down-tooth apexes: **23 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in increasing annular-edge order, and reduce it modulo colour permutation (NOT dihedral symmetry).
|
||||
|
||||
- Total Kempe-balanced colourings (mod colour permutation): **298**.
|
||||
- Distinct canonical down-apex sequences overall: **4**.
|
||||
|
||||
## Distinct canonical down-apex sequences
|
||||
|
||||
| sequence | colour multiset | #configs realising | #colourings |
|
||||
|---|---|---|---|
|
||||
| `0000` | 4 | 21 | 64 |
|
||||
| `0011` | 2+2 | 23 | 108 |
|
||||
| `0101` | 2+2 | 12 | 53 |
|
||||
| `0110` | 2+2 | 19 | 73 |
|
||||
|
||||
Note: every realised sequence has its three colour-counts of **equal parity** — exactly the Kempe-parity constraint on the inner face (each colour pair meets its singleton down apexes an even number of times). With m = 4 apexes (m is even) every count must be **even**, so the only admissible colour multisets are 2+2, 4.
|
||||
|
||||
## Step 4 — grouping configs by their set of unique down-apex sequences
|
||||
|
||||
The 23 configs fall into **4** groups by the set of canonical down-apex sequences they realise:
|
||||
|
||||
| #configs | set of down-apex sequences | config ids |
|
||||
|---|---|---|
|
||||
| 11 | { `0000`, `0011`, `0110` } | C00, C01, C03, C07, C11, C12, C13, C14, C17, C18, C20 |
|
||||
| 6 | { `0000`, `0011`, `0101`, `0110` } | C04, C05, C06, C08, C09, C10 |
|
||||
| 4 | { `0000`, `0011`, `0101` } | C02, C15, C16, C21 |
|
||||
| 2 | { `0011`, `0101`, `0110` } | C19, C22 |
|
||||
|
||||
## Config atlas (ids)
|
||||
|
||||
| id | word / bites / face / apexes | #Kempe-balanced | down-apex sequence set |
|
||||
|---|---|---|---|
|
||||
| C00 | word=UUUUUDDDD bites=- face=root apexes=[d5,d6,d7,d8] | 30 | { `0000`, `0011`, `0110` } |
|
||||
| C01 | word=UUUUDUDDD bites=- face=root apexes=[d4,d6,d7,d8] | 15 | { `0000`, `0011`, `0110` } |
|
||||
| C02 | word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8] | 25 | { `0000`, `0011`, `0101` } |
|
||||
| C03 | word=UUUDUUDDD bites=- face=root apexes=[d3,d6,d7,d8] | 21 | { `0000`, `0011`, `0110` } |
|
||||
| C04 | word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8] | 24 | { `0000`, `0011`, `0101`, `0110` } |
|
||||
| C05 | word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8] | 28 | { `0000`, `0011`, `0101`, `0110` } |
|
||||
| C06 | word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8] | 27 | { `0000`, `0011`, `0101`, `0110` } |
|
||||
| C07 | word=UUUDDDDDD bites=(3,8) face=bite(3,8) apexes=[d4,d5,d6,d7] | 12 | { `0000`, `0011`, `0110` } |
|
||||
| C08 | word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8] | 22 | { `0000`, `0011`, `0101`, `0110` } |
|
||||
| C09 | word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8] | 18 | { `0000`, `0011`, `0101`, `0110` } |
|
||||
| C10 | word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8] | 19 | { `0000`, `0011`, `0101`, `0110` } |
|
||||
| C11 | word=UUDUDDDDD bites=(2,4) face=root apexes=[d5,d6,d7,d8] | 6 | { `0000`, `0011`, `0110` } |
|
||||
| C12 | word=UUDUDDDDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6,d7] | 6 | { `0000`, `0011`, `0110` } |
|
||||
| C13 | word=UUDDUDDDD bites=(3,5) face=root apexes=[d2,d6,d7,d8] | 3 | { `0000`, `0011`, `0110` } |
|
||||
| C14 | word=UUDDUDDDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d6,d7] | 3 | { `0000`, `0011`, `0110` } |
|
||||
| C15 | word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8] | 5 | { `0000`, `0011`, `0101` } |
|
||||
| C16 | word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7] | 5 | { `0000`, `0011`, `0101` } |
|
||||
| C17 | word=UDUDUDDDD bites=(3,5) face=root apexes=[d1,d6,d7,d8] | 3 | { `0000`, `0011`, `0110` } |
|
||||
| C18 | word=UDUDUDDDD bites=(1,3) face=root apexes=[d5,d6,d7,d8] | 6 | { `0000`, `0011`, `0110` } |
|
||||
| C19 | word=UDUDDUDDD bites=(4,6) face=root apexes=[d1,d3,d7,d8] | 6 | { `0011`, `0101`, `0110` } |
|
||||
| C20 | word=UDUDDUDDD bites=(1,3) face=root apexes=[d4,d6,d7,d8] | 3 | { `0000`, `0011`, `0110` } |
|
||||
| C21 | word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7] | 5 | { `0000`, `0011`, `0101` } |
|
||||
| C22 | word=UDDUDDUDD bites=(5,7) face=root apexes=[d1,d2,d4,d8] | 6 | { `0011`, `0101`, `0110` } |
|
||||
|
||||
## Per-sequence notes
|
||||
|
||||
- [`0000`](seq_0000.md) — figure `seq_0000.png`
|
||||
- [`0011`](seq_0011.md) — figure `seq_0011.png`
|
||||
- [`0101`](seq_0101.md) — figure `seq_0101.png`
|
||||
- [`0110`](seq_0110.md) — figure `seq_0110.png`
|
||||
|
||||
+91
@@ -0,0 +1,91 @@
|
||||
# Inner-face down-apex sequence `00012`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
|
||||
- Realised by **10** of 10 configs (M(T), inner face).
|
||||
- **30** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_00012.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8]
|
||||
|
||||
6 colouring(s) with down-apex sequence `00012`:
|
||||
|
||||
- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `11102` → canonical `00012` · `A=010102021 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `22201` → canonical `00012` · `A=010201012 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `11102` → canonical `00012` · `A=010202021 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `22201` → canonical `00012` · `A=012101012 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `11102` → canonical `00012` · `A=012102021 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
|
||||
|
||||
## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `00012`:
|
||||
|
||||
- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `11102` → canonical `00012` · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `22201` → canonical `00012` · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]`
|
||||
|
||||
## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `00012`:
|
||||
|
||||
- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `11102` → canonical `00012` · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `22201` → canonical `00012` · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]`
|
||||
|
||||
## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `00012`:
|
||||
|
||||
- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `22201` → canonical `00012` · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `11102` → canonical `00012` · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `11102` → canonical `00012` · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `11102` → canonical `00012` · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
|
||||
|
||||
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `00012`:
|
||||
|
||||
- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `11102` → canonical `00012` · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:1 d7:0 d8:2]`
|
||||
|
||||
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `00012`:
|
||||
|
||||
- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:2 d5:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `11102` → canonical `00012` · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:1 d5:1 d7:0 d8:2]`
|
||||
|
||||
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `00012`:
|
||||
|
||||
- face apexes (raw labels) `00012` → canonical `00012` · `A=012121201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:0 d5:0 d6:1 d8:2]`
|
||||
|
||||
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `00012`:
|
||||
|
||||
- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `22201` → canonical `00012` · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `11102` → canonical `00012` · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `11102` → canonical `00012` · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `11102` → canonical `00012` · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
|
||||
|
||||
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `00012`:
|
||||
|
||||
- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u1:2 u4:2 u6:2] D[d2:2 d3:2 d5:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `11102` → canonical `00012` · `A=010202021 U[u0:2 u1:2 u4:1 u6:1] D[d2:1 d3:1 d5:1 d7:0 d8:2]`
|
||||
|
||||
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `00012`:
|
||||
|
||||
- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d3:2 d5:2 d7:0 d8:1]`
|
||||
|
||||
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|
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+54
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|
||||
# Inner-face down-apex sequence `00102`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
|
||||
- Realised by **6** of 10 configs (M(T), inner face).
|
||||
- **13** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_00102.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
|
||||
|
||||
6 colouring(s) with down-apex sequence `00102`:
|
||||
|
||||
- face apexes (raw labels) `00201` → canonical `00102` · `A=010121012 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `00102` → canonical `00102` · `A=010121021 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `00201` → canonical `00102` · `A=010212012 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `00102` → canonical `00102` · `A=010212021 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `00201` → canonical `00102` · `A=012121012 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `00102` → canonical `00102` · `A=012121021 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
|
||||
|
||||
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `00102`:
|
||||
|
||||
- face apexes (raw labels) `00201` → canonical `00102` · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `00102` → canonical `00102` · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:1 d7:0 d8:2]`
|
||||
|
||||
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `00102`:
|
||||
|
||||
- face apexes (raw labels) `00201` → canonical `00102` · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:0 d5:2 d7:0 d8:1]`
|
||||
|
||||
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `00102`:
|
||||
|
||||
- face apexes (raw labels) `00201` → canonical `00102` · `A=012121012 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `00102` → canonical `00102` · `A=012121021 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:1 d7:0 d8:2]`
|
||||
|
||||
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `00102`:
|
||||
|
||||
- face apexes (raw labels) `00201` → canonical `00102` · `A=012121012 U[u0:2 u1:0 u4:0 u6:2] D[d2:0 d3:0 d5:2 d7:0 d8:1]`
|
||||
|
||||
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `00102`:
|
||||
|
||||
- face apexes (raw labels) `00201` → canonical `00102` · `A=012121012 U[u0:2 u2:0 u4:0 u6:2] D[d1:0 d3:0 d5:2 d7:0 d8:1]`
|
||||
|
||||
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|
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+86
@@ -0,0 +1,86 @@
|
||||
# Inner-face down-apex sequence `00120`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
|
||||
- Realised by **9** of 10 configs (M(T), inner face).
|
||||
- **30** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_00120.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8]
|
||||
|
||||
6 colouring(s) with down-apex sequence `00120`:
|
||||
|
||||
- face apexes (raw labels) `22102` → canonical `00120` · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `11201` → canonical `00120` · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `22102` → canonical `00120` · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `11201` → canonical `00120` · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `22102` → canonical `00120` · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `11201` → canonical `00120` · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
|
||||
|
||||
## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `00120`:
|
||||
|
||||
- face apexes (raw labels) `22102` → canonical `00120` · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `11201` → canonical `00120` · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `22102` → canonical `00120` · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]`
|
||||
|
||||
## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `00120`:
|
||||
|
||||
- face apexes (raw labels) `22102` → canonical `00120` · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `11201` → canonical `00120` · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `22102` → canonical `00120` · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]`
|
||||
|
||||
## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `00120`:
|
||||
|
||||
- face apexes (raw labels) `22102` → canonical `00120` · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `22102` → canonical `00120` · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `11201` → canonical `00120` · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `11201` → canonical `00120` · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `11201` → canonical `00120` · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
|
||||
|
||||
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `00120`:
|
||||
|
||||
- face apexes (raw labels) `22102` → canonical `00120` · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `11201` → canonical `00120` · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:2 d7:0 d8:1]`
|
||||
|
||||
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `00120`:
|
||||
|
||||
- face apexes (raw labels) `22012` → canonical `00120` · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:2 d5:0 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `11021` → canonical `00120` · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:1 d5:0 d6:2 d8:1]`
|
||||
|
||||
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `00120`:
|
||||
|
||||
- face apexes (raw labels) `22102` → canonical `00120` · `A=010101021 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `22102` → canonical `00120` · `A=010102021 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `11201` → canonical `00120` · `A=010201012 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `11201` → canonical `00120` · `A=010202012 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `11201` → canonical `00120` · `A=012021012 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
|
||||
|
||||
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `00120`:
|
||||
|
||||
- face apexes (raw labels) `22102` → canonical `00120` · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:2 d5:1 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `11201` → canonical `00120` · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `11201` → canonical `00120` · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]`
|
||||
|
||||
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `00120`:
|
||||
|
||||
- face apexes (raw labels) `22102` → canonical `00120` · `A=010102021 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:2 d5:1 d7:0 d8:2]`
|
||||
|
||||
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+62
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|
||||
# Inner-face down-apex sequence `01002`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
|
||||
- Realised by **7** of 10 configs (M(T), inner face).
|
||||
- **16** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_01002.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `01002`:
|
||||
|
||||
- face apexes (raw labels) `01002` → canonical `01002` · `A=010120121 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `02001` → canonical `01002` · `A=010210212 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `01002` → canonical `01002` · `A=012120121 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]`
|
||||
|
||||
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `01002`:
|
||||
|
||||
- face apexes (raw labels) `02001` → canonical `01002` · `A=012101212 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `01002` → canonical `01002` · `A=012102121 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `01002` → canonical `01002` · `A=012120121 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:0 d7:0 d8:2]`
|
||||
|
||||
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `01002`:
|
||||
|
||||
- face apexes (raw labels) `02001` → canonical `01002` · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d5:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `01002` → canonical `01002` · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d5:0 d7:0 d8:2]`
|
||||
|
||||
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `01002`:
|
||||
|
||||
- face apexes (raw labels) `20221` → canonical `01002` · `A=010121012 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `10112` → canonical `01002` · `A=010212021 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `10112` → canonical `01002` · `A=012012021 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `02001` → canonical `01002` · `A=012101212 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:0 d6:0 d8:1]`
|
||||
- face apexes (raw labels) `01002` → canonical `01002` · `A=012102121 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:0 d6:0 d8:2]`
|
||||
|
||||
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `01002`:
|
||||
|
||||
- face apexes (raw labels) `02001` → canonical `01002` · `A=012101212 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:0 d7:0 d8:1]`
|
||||
|
||||
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `01002`:
|
||||
|
||||
- face apexes (raw labels) `02001` → canonical `01002` · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d5:0 d7:0 d8:1]`
|
||||
|
||||
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `01002`:
|
||||
|
||||
- face apexes (raw labels) `02001` → canonical `01002` · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d5:0 d7:0 d8:1]`
|
||||
|
||||
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|
||||
# Inner-face down-apex sequence `01020`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
|
||||
- Realised by **4** of 10 configs (M(T), inner face).
|
||||
- **12** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_01020.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `01020`:
|
||||
|
||||
- face apexes (raw labels) `21202` → canonical `01020` · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:2 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `12101` → canonical `01020` · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `12101` → canonical `01020` · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]`
|
||||
|
||||
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
|
||||
|
||||
6 colouring(s) with down-apex sequence `01020`:
|
||||
|
||||
- face apexes (raw labels) `21202` → canonical `01020` · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:2 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `20212` → canonical `01020` · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `12101` → canonical `01020` · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]`
|
||||
- face apexes (raw labels) `10121` → canonical `01020` · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `12101` → canonical `01020` · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]`
|
||||
- face apexes (raw labels) `10121` → canonical `01020` · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]`
|
||||
|
||||
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `01020`:
|
||||
|
||||
- face apexes (raw labels) `12101` → canonical `01020` · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:1 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `12101` → canonical `01020` · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:1 d7:0 d8:1]`
|
||||
|
||||
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `01020`:
|
||||
|
||||
- face apexes (raw labels) `21202` → canonical `01020` · `A=010201021 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:2 d7:0 d8:2]`
|
||||
|
||||
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+89
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|
||||
# Inner-face down-apex sequence `01112`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 1×colour0, 3×colour1, 1×colour2.
|
||||
- Realised by **9** of 10 configs (M(T), inner face).
|
||||
- **33** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_01112.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8]
|
||||
|
||||
6 colouring(s) with down-apex sequence `01112`:
|
||||
|
||||
- face apexes (raw labels) `20001` → canonical `01112` · `A=010101212 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `10002` → canonical `01112` · `A=010102121 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `20001` → canonical `01112` · `A=010201212 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `10002` → canonical `01112` · `A=010202121 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `20001` → canonical `01112` · `A=012101212 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `10002` → canonical `01112` · `A=012102121 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
|
||||
|
||||
## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `01112`:
|
||||
|
||||
- face apexes (raw labels) `20001` → canonical `01112` · `A=010101212 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `10002` → canonical `01112` · `A=010202121 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `20001` → canonical `01112` · `A=012101212 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]`
|
||||
|
||||
## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `01112`:
|
||||
|
||||
- face apexes (raw labels) `20001` → canonical `01112` · `A=010101212 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `20001` → canonical `01112` · `A=010121212 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `10002` → canonical `01112` · `A=010202121 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `10002` → canonical `01112` · `A=010212121 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `10002` → canonical `01112` · `A=012012121 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
|
||||
|
||||
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `01112`:
|
||||
|
||||
- face apexes (raw labels) `20001` → canonical `01112` · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `10002` → canonical `01112` · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `10002` → canonical `01112` · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]`
|
||||
|
||||
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `01112`:
|
||||
|
||||
- face apexes (raw labels) `20001` → canonical `01112` · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d5:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `10002` → canonical `01112` · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `10002` → canonical `01112` · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]`
|
||||
|
||||
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
|
||||
|
||||
6 colouring(s) with down-apex sequence `01112`:
|
||||
|
||||
- face apexes (raw labels) `20001` → canonical `01112` · `A=010121212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:0 d6:0 d8:1]`
|
||||
- face apexes (raw labels) `10002` → canonical `01112` · `A=010212121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `10002` → canonical `01112` · `A=012012121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `02221` → canonical `01112` · `A=012101012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:2 d8:1]`
|
||||
- face apexes (raw labels) `01112` → canonical `01112` · `A=012102021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `01112` → canonical `01112` · `A=012120201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:1 d6:1 d8:2]`
|
||||
|
||||
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
|
||||
|
||||
4 colouring(s) with down-apex sequence `01112`:
|
||||
|
||||
- face apexes (raw labels) `20001` → canonical `01112` · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `20001` → canonical `01112` · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `10002` → canonical `01112` · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `10002` → canonical `01112` · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:0 d7:0 d8:2]`
|
||||
|
||||
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `01112`:
|
||||
|
||||
- face apexes (raw labels) `20001` → canonical `01112` · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d5:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `10002` → canonical `01112` · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d5:0 d7:0 d8:2]`
|
||||
|
||||
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `01112`:
|
||||
|
||||
- face apexes (raw labels) `20001` → canonical `01112` · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d5:0 d7:0 d8:1]`
|
||||
|
||||
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+43
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|
||||
# Inner-face down-apex sequence `01121`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 1×colour0, 3×colour1, 1×colour2.
|
||||
- Realised by **4** of 10 configs (M(T), inner face).
|
||||
- **12** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_01121.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `01121`:
|
||||
|
||||
- face apexes (raw labels) `21101` → canonical `01121` · `A=010102012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:1 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `21101` → canonical `01121` · `A=010120212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:1 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `12202` → canonical `01121` · `A=010201021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `12202` → canonical `01121` · `A=010210121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `12202` → canonical `01121` · `A=012010121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
|
||||
|
||||
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `01121`:
|
||||
|
||||
- face apexes (raw labels) `21101` → canonical `01121` · `A=010120212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:1 d6:0 d8:1]`
|
||||
- face apexes (raw labels) `12202` → canonical `01121` · `A=010210121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `12202` → canonical `01121` · `A=012010121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]`
|
||||
- face apexes (raw labels) `02212` → canonical `01121` · `A=012101021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:1 d8:2]`
|
||||
- face apexes (raw labels) `01121` → canonical `01121` · `A=012102012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:2 d8:1]`
|
||||
|
||||
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `01121`:
|
||||
|
||||
- face apexes (raw labels) `12202` → canonical `01121` · `A=012010121 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:2 d7:0 d8:2]`
|
||||
|
||||
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `01121`:
|
||||
|
||||
- face apexes (raw labels) `21101` → canonical `01121` · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:1 d7:0 d8:1]`
|
||||
|
||||
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|
||||
# Inner-face down-apex sequence `01200`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
|
||||
- Realised by **9** of 10 configs (M(T), inner face).
|
||||
- **32** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_01200.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8]
|
||||
|
||||
6 colouring(s) with down-apex sequence `01200`:
|
||||
|
||||
- face apexes (raw labels) `20122` → canonical `01200` · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `10211` → canonical `01200` · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `20122` → canonical `01200` · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `10211` → canonical `01200` · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `20122` → canonical `01200` · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `10211` → canonical `01200` · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
|
||||
|
||||
## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `01200`:
|
||||
|
||||
- face apexes (raw labels) `20122` → canonical `01200` · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `10211` → canonical `01200` · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `20122` → canonical `01200` · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]`
|
||||
|
||||
## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `01200`:
|
||||
|
||||
- face apexes (raw labels) `20122` → canonical `01200` · `A=010101201 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `20122` → canonical `01200` · `A=010121201 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `10211` → canonical `01200` · `A=010202102 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `10211` → canonical `01200` · `A=010212102 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `10211` → canonical `01200` · `A=012012102 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
|
||||
|
||||
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `01200`:
|
||||
|
||||
- face apexes (raw labels) `20122` → canonical `01200` · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `10211` → canonical `01200` · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `10211` → canonical `01200` · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]`
|
||||
|
||||
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `01200`:
|
||||
|
||||
- face apexes (raw labels) `21022` → canonical `01200` · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `12011` → canonical `01200` · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:1 d8:1]`
|
||||
|
||||
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `01200`:
|
||||
|
||||
- face apexes (raw labels) `21022` → canonical `01200` · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 d6:2 d8:2]`
|
||||
- face apexes (raw labels) `12011` → canonical `01200` · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 d6:1 d8:1]`
|
||||
|
||||
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
|
||||
|
||||
4 colouring(s) with down-apex sequence `01200`:
|
||||
|
||||
- face apexes (raw labels) `20122` → canonical `01200` · `A=010120201 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `20122` → canonical `01200` · `A=010121201 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `10211` → canonical `01200` · `A=010210102 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `10211` → canonical `01200` · `A=010212102 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:1 d8:1]`
|
||||
|
||||
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `01200`:
|
||||
|
||||
- face apexes (raw labels) `20122` → canonical `01200` · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `10211` → canonical `01200` · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `12011` → canonical `01200` · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:1 d8:1]`
|
||||
|
||||
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
|
||||
|
||||
4 colouring(s) with down-apex sequence `01200`:
|
||||
|
||||
- face apexes (raw labels) `20122` → canonical `01200` · `A=010120201 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `21022` → canonical `01200` · `A=010201201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `21022` → canonical `01200` · `A=010202101 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `20122` → canonical `01200` · `A=010210201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]`
|
||||
|
||||
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|
||||
# Inner-face down-apex sequence `01211`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 1×colour0, 3×colour1, 1×colour2.
|
||||
- Realised by **6** of 10 configs (M(T), inner face).
|
||||
- **15** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_01211.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `01211`:
|
||||
|
||||
- face apexes (raw labels) `01211` → canonical `01211` · `A=010120102 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `02122` → canonical `01211` · `A=010210201 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `01211` → canonical `01211` · `A=012120102 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]`
|
||||
|
||||
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `01211`:
|
||||
|
||||
- face apexes (raw labels) `02122` → canonical `01211` · `A=012101201 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `01211` → canonical `01211` · `A=012102102 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `01211` → canonical `01211` · `A=012120102 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:1 d8:1]`
|
||||
|
||||
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `01211`:
|
||||
|
||||
- face apexes (raw labels) `21011` → canonical `01211` · `A=010102102 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `12022` → canonical `01211` · `A=010201201 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `01211` → canonical `01211` · `A=012120102 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:1 d8:1]`
|
||||
|
||||
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `01211`:
|
||||
|
||||
- face apexes (raw labels) `02122` → canonical `01211` · `A=012101201 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:2 d8:2]`
|
||||
|
||||
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `01211`:
|
||||
|
||||
- face apexes (raw labels) `12022` → canonical `01211` · `A=012012101 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:2 d8:2]`
|
||||
|
||||
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
|
||||
|
||||
4 colouring(s) with down-apex sequence `01211`:
|
||||
|
||||
- face apexes (raw labels) `21011` → canonical `01211` · `A=010201202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `21011` → canonical `01211` · `A=010202102 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `02122` → canonical `01211` · `A=012010201 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `01211` → canonical `01211` · `A=012020102 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:1 d8:1]`
|
||||
|
||||
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|
||||
# Inner-face down-apex sequence `01222`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 1×colour0, 1×colour1, 3×colour2.
|
||||
- Realised by **10** of 10 configs (M(T), inner face).
|
||||
- **36** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_01222.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8]
|
||||
|
||||
6 colouring(s) with down-apex sequence `01222`:
|
||||
|
||||
- face apexes (raw labels) `20111` → canonical `01222` · `A=010101202 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `10222` → canonical `01222` · `A=010102101 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `20111` → canonical `01222` · `A=010201202 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `10222` → canonical `01222` · `A=010202101 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `20111` → canonical `01222` · `A=012101202 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `10222` → canonical `01222` · `A=012102101 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
|
||||
|
||||
## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `01222`:
|
||||
|
||||
- face apexes (raw labels) `20111` → canonical `01222` · `A=010101202 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `10222` → canonical `01222` · `A=010202101 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `20111` → canonical `01222` · `A=012101202 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]`
|
||||
|
||||
## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `01222`:
|
||||
|
||||
- face apexes (raw labels) `01222` → canonical `01222` · `A=010120101 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `02111` → canonical `01222` · `A=010210202 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `01222` → canonical `01222` · `A=012120101 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]`
|
||||
|
||||
## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `01222`:
|
||||
|
||||
- face apexes (raw labels) `20111` → canonical `01222` · `A=010101202 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `20111` → canonical `01222` · `A=010121202 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `10222` → canonical `01222` · `A=010202101 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `10222` → canonical `01222` · `A=010212101 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `10222` → canonical `01222` · `A=012012101 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
|
||||
|
||||
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
|
||||
|
||||
6 colouring(s) with down-apex sequence `01222`:
|
||||
|
||||
- face apexes (raw labels) `20111` → canonical `01222` · `A=010121202 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `10222` → canonical `01222` · `A=010212101 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `10222` → canonical `01222` · `A=012012101 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `02111` → canonical `01222` · `A=012101202 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `01222` → canonical `01222` · `A=012102101 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `01222` → canonical `01222` · `A=012120101 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:2 d8:2]`
|
||||
|
||||
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `01222`:
|
||||
|
||||
- face apexes (raw labels) `01222` → canonical `01222` · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:2 d8:2]`
|
||||
|
||||
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `01222`:
|
||||
|
||||
- face apexes (raw labels) `01222` → canonical `01222` · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 d6:2 d8:2]`
|
||||
|
||||
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
|
||||
|
||||
5 colouring(s) with down-apex sequence `01222`:
|
||||
|
||||
- face apexes (raw labels) `20111` → canonical `01222` · `A=010120202 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `20111` → canonical `01222` · `A=010121202 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `10222` → canonical `01222` · `A=010210101 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `10222` → canonical `01222` · `A=010212101 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `02111` → canonical `01222` · `A=012101202 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:1 d8:1]`
|
||||
|
||||
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `01222`:
|
||||
|
||||
- face apexes (raw labels) `20111` → canonical `01222` · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `10222` → canonical `01222` · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:2 d8:2]`
|
||||
|
||||
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
|
||||
|
||||
4 colouring(s) with down-apex sequence `01222`:
|
||||
|
||||
- face apexes (raw labels) `20111` → canonical `01222` · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `20111` → canonical `01222` · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `02111` → canonical `01222` · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `01222` → canonical `01222` · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:2 d8:2]`
|
||||
|
||||
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|
||||
# Inner-face singleton-down-apex sequences of Kempe-balanced colourings (n=9, m=5)
|
||||
|
||||
Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 5 singleton down-tooth apexes: **10 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in increasing annular-edge order, and reduce it modulo colour permutation (NOT dihedral symmetry).
|
||||
|
||||
- Total Kempe-balanced colourings (mod colour permutation): **229**.
|
||||
- Distinct canonical down-apex sequences overall: **10**.
|
||||
|
||||
## Distinct canonical down-apex sequences
|
||||
|
||||
| sequence | colour multiset | #configs realising | #colourings |
|
||||
|---|---|---|---|
|
||||
| `00012` | 3+1+1 | 10 | 30 |
|
||||
| `00102` | 3+1+1 | 6 | 13 |
|
||||
| `00120` | 3+1+1 | 9 | 30 |
|
||||
| `01002` | 3+1+1 | 7 | 16 |
|
||||
| `01020` | 3+1+1 | 4 | 12 |
|
||||
| `01112` | 3+1+1 | 9 | 33 |
|
||||
| `01121` | 3+1+1 | 4 | 12 |
|
||||
| `01200` | 3+1+1 | 9 | 32 |
|
||||
| `01211` | 3+1+1 | 6 | 15 |
|
||||
| `01222` | 3+1+1 | 10 | 36 |
|
||||
|
||||
Note: every realised sequence has its three colour-counts of **equal parity** — exactly the Kempe-parity constraint on the inner face (each colour pair meets its singleton down apexes an even number of times). With m = 5 apexes (m is odd) every count must be **odd**, so the only admissible colour multisets are 3+1+1.
|
||||
|
||||
## Step 4 — grouping configs by their set of unique down-apex sequences
|
||||
|
||||
The 10 configs fall into **6** groups by the set of canonical down-apex sequences they realise:
|
||||
|
||||
| #configs | set of down-apex sequences | config ids |
|
||||
|---|---|---|
|
||||
| 3 | { `00012`, `00120`, `01112`, `01200`, `01222` } | C00, C01, C03 |
|
||||
| 2 | { `00012`, `00102`, `00120`, `01002`, `01112`, `01200`, `01211`, `01222` } | C04, C07 |
|
||||
| 2 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } | C08, C09 |
|
||||
| 1 | { `00012`, `00102`, `00120`, `01002`, `01211`, `01222` } | C02 |
|
||||
| 1 | { `00012`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01222` } | C06 |
|
||||
| 1 | { `00012`, `00102`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } | C05 |
|
||||
|
||||
## Config atlas (ids)
|
||||
|
||||
| id | word / bites / face / apexes | #Kempe-balanced | down-apex sequence set |
|
||||
|---|---|---|---|
|
||||
| C00 | word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8] | 30 | { `00012`, `00120`, `01112`, `01200`, `01222` } |
|
||||
| C01 | word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8] | 15 | { `00012`, `00120`, `01112`, `01200`, `01222` } |
|
||||
| C02 | word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8] | 21 | { `00012`, `00102`, `00120`, `01002`, `01211`, `01222` } |
|
||||
| C03 | word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8] | 25 | { `00012`, `00120`, `01112`, `01200`, `01222` } |
|
||||
| C04 | word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8] | 24 | { `00012`, `00102`, `00120`, `01002`, `01112`, `01200`, `01211`, `01222` } |
|
||||
| C05 | word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8] | 22 | { `00012`, `00102`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } |
|
||||
| C06 | word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8] | 28 | { `00012`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01222` } |
|
||||
| C07 | word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8] | 27 | { `00012`, `00102`, `00120`, `01002`, `01112`, `01200`, `01211`, `01222` } |
|
||||
| C08 | word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8] | 18 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } |
|
||||
| C09 | word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8] | 19 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } |
|
||||
|
||||
## Per-sequence notes
|
||||
|
||||
- [`00012`](seq_00012.md) — figure `seq_00012.png`
|
||||
- [`00102`](seq_00102.md) — figure `seq_00102.png`
|
||||
- [`00120`](seq_00120.md) — figure `seq_00120.png`
|
||||
- [`01002`](seq_01002.md) — figure `seq_01002.png`
|
||||
- [`01020`](seq_01020.md) — figure `seq_01020.png`
|
||||
- [`01112`](seq_01112.md) — figure `seq_01112.png`
|
||||
- [`01121`](seq_01121.md) — figure `seq_01121.png`
|
||||
- [`01200`](seq_01200.md) — figure `seq_01200.png`
|
||||
- [`01211`](seq_01211.md) — figure `seq_01211.png`
|
||||
- [`01222`](seq_01222.md) — figure `seq_01222.png`
|
||||
|
||||
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|
||||
# Inner-face down-apex sequence `000000`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 6×colour0.
|
||||
- Realised by **4** of 7 configs (M(T), inner face).
|
||||
- **5** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_000000.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `000000`:
|
||||
|
||||
- face apexes (raw labels) `222222` → canonical `000000` · `A=012010101 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `111111` → canonical `000000` · `A=012020202 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:1 d7:1 d8:1]`
|
||||
|
||||
## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `000000`:
|
||||
|
||||
- face apexes (raw labels) `111111` → canonical `000000` · `A=012020202 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:1 d7:1 d8:1]`
|
||||
|
||||
## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `000000`:
|
||||
|
||||
- face apexes (raw labels) `111111` → canonical `000000` · `A=012020202 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:1 d7:1 d8:1]`
|
||||
|
||||
## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `000000`:
|
||||
|
||||
- face apexes (raw labels) `111111` → canonical `000000` · `A=012020202 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:1 d7:1 d8:1]`
|
||||
|
||||
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|
||||
# Inner-face down-apex sequence `000011`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **5** of 7 configs (M(T), inner face).
|
||||
- **7** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_000011.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `000011`:
|
||||
|
||||
- face apexes (raw labels) `222211` → canonical `000011` · `A=012010102 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `111122` → canonical `000011` · `A=012020201 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:1 d7:2 d8:2]`
|
||||
|
||||
## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `000011`:
|
||||
|
||||
- face apexes (raw labels) `111122` → canonical `000011` · `A=012020201 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:1 d7:2 d8:2]`
|
||||
|
||||
## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `000011`:
|
||||
|
||||
- face apexes (raw labels) `111122` → canonical `000011` · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:1 d7:2 d8:2]`
|
||||
|
||||
## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `000011`:
|
||||
|
||||
- face apexes (raw labels) `111122` → canonical `000011` · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:1 d7:2 d8:2]`
|
||||
|
||||
## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `000011`:
|
||||
|
||||
- face apexes (raw labels) `000022` → canonical `000011` · `A=012121201 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `000011` → canonical `000011` · `A=012121202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:1 d8:1]`
|
||||
|
||||
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|
||||
# Inner-face down-apex sequence `000101`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **1** of 7 configs (M(T), inner face).
|
||||
- **1** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_000101.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `000101`:
|
||||
|
||||
- face apexes (raw labels) `000202` → canonical `000101` · `A=012121021 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:2 d7:0 d8:2]`
|
||||
|
||||
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|
||||
# Inner-face down-apex sequence `000110`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **4** of 7 configs (M(T), inner face).
|
||||
- **5** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_000110.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `000110`:
|
||||
|
||||
- face apexes (raw labels) `222002` → canonical `000110` · `A=012010121 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `111001` → canonical `000110` · `A=012020212 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `000110`:
|
||||
|
||||
- face apexes (raw labels) `111001` → canonical `000110` · `A=012020212 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `000110`:
|
||||
|
||||
- face apexes (raw labels) `111001` → canonical `000110` · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `000110`:
|
||||
|
||||
- face apexes (raw labels) `111001` → canonical `000110` · `A=012020212 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
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|
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+37
@@ -0,0 +1,37 @@
|
||||
# Inner-face down-apex sequence `001001`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **4** of 7 configs (M(T), inner face).
|
||||
- **6** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_001001.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `001001`:
|
||||
|
||||
- face apexes (raw labels) `002002` → canonical `001001` · `A=012120121 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `001001` → canonical `001001` · `A=012120212 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `001001`:
|
||||
|
||||
- face apexes (raw labels) `001001` → canonical `001001` · `A=012120212 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `001001`:
|
||||
|
||||
- face apexes (raw labels) `002002` → canonical `001001` · `A=012120121 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `001001` → canonical `001001` · `A=012120212 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `001001`:
|
||||
|
||||
- face apexes (raw labels) `001001` → canonical `001001` · `A=012120212 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
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|
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+17
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|
||||
# Inner-face down-apex sequence `001010`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **1** of 7 configs (M(T), inner face).
|
||||
- **1** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_001010.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `001010`:
|
||||
|
||||
- face apexes (raw labels) `220202` → canonical `001010` · `A=010121021 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:2 d7:0 d8:2]`
|
||||
|
||||
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|
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+37
@@ -0,0 +1,37 @@
|
||||
# Inner-face down-apex sequence `001100`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **4** of 7 configs (M(T), inner face).
|
||||
- **6** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_001100.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `001100`:
|
||||
|
||||
- face apexes (raw labels) `221122` → canonical `001100` · `A=012010201 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `112211` → canonical `001100` · `A=012020102 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:1 d8:1]`
|
||||
|
||||
## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `001100`:
|
||||
|
||||
- face apexes (raw labels) `112211` → canonical `001100` · `A=012020102 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:1 d8:1]`
|
||||
|
||||
## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `001100`:
|
||||
|
||||
- face apexes (raw labels) `112211` → canonical `001100` · `A=012020102 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:1 d8:1]`
|
||||
|
||||
## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `001100`:
|
||||
|
||||
- face apexes (raw labels) `221122` → canonical `001100` · `A=010120201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `220022` → canonical `001100` · `A=010121201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:0 d7:2 d8:2]`
|
||||
|
||||
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|
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+58
@@ -0,0 +1,58 @@
|
||||
# Inner-face down-apex sequence `001111`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 2×colour0, 4×colour1.
|
||||
- Realised by **7** of 7 configs (M(T), inner face).
|
||||
- **12** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_001111.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `001111`:
|
||||
|
||||
- face apexes (raw labels) `221111` → canonical `001111` · `A=012010202 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `112222` → canonical `001111` · `A=012020101 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:2 d8:2]`
|
||||
|
||||
## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `001111`:
|
||||
|
||||
- face apexes (raw labels) `112222` → canonical `001111` · `A=012020101 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:2 d8:2]`
|
||||
|
||||
## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `001111`:
|
||||
|
||||
- face apexes (raw labels) `112222` → canonical `001111` · `A=012020101 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `002222` → canonical `001111` · `A=012120101 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `001111` → canonical `001111` · `A=012120202 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:1 d7:1 d8:1]`
|
||||
|
||||
## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `001111`:
|
||||
|
||||
- face apexes (raw labels) `001111` → canonical `001111` · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:1 d7:1 d8:1]`
|
||||
|
||||
## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `001111`:
|
||||
|
||||
- face apexes (raw labels) `002222` → canonical `001111` · `A=012120101 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `001111` → canonical `001111` · `A=012120202 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:1 d7:1 d8:1]`
|
||||
|
||||
## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `001111`:
|
||||
|
||||
- face apexes (raw labels) `001111` → canonical `001111` · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:1 d7:1 d8:1]`
|
||||
|
||||
## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `001111`:
|
||||
|
||||
- face apexes (raw labels) `221111` → canonical `001111` · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `001111` → canonical `001111` · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d5:1 d7:1 d8:1]`
|
||||
|
||||
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|
After Width: | Height: | Size: 161 KiB |
+44
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|
||||
# Inner-face down-apex sequence `001122`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
|
||||
- Realised by **5** of 7 configs (M(T), inner face).
|
||||
- **8** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_001122.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `001122`:
|
||||
|
||||
- face apexes (raw labels) `002211` → canonical `001122` · `A=012120102 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `001122` → canonical `001122` · `A=012120201 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:1 d7:2 d8:2]`
|
||||
|
||||
## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `001122`:
|
||||
|
||||
- face apexes (raw labels) `001122` → canonical `001122` · `A=012120201 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:1 d7:2 d8:2]`
|
||||
|
||||
## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `001122`:
|
||||
|
||||
- face apexes (raw labels) `002211` → canonical `001122` · `A=012120102 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `001122` → canonical `001122` · `A=012120201 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:1 d7:2 d8:2]`
|
||||
|
||||
## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `001122`:
|
||||
|
||||
- face apexes (raw labels) `001122` → canonical `001122` · `A=012120201 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:1 d7:2 d8:2]`
|
||||
|
||||
## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `001122`:
|
||||
|
||||
- face apexes (raw labels) `220011` → canonical `001122` · `A=010121202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:0 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `001122` → canonical `001122` · `A=012120201 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d5:1 d7:2 d8:2]`
|
||||
|
||||
BIN
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BIN
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|
After Width: | Height: | Size: 101 KiB |
+30
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|
||||
# Inner-face down-apex sequence `001221`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
|
||||
- Realised by **3** of 7 configs (M(T), inner face).
|
||||
- **4** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_001221.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `001221`:
|
||||
|
||||
- face apexes (raw labels) `221001` → canonical `001221` · `A=012010212 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `112002` → canonical `001221` · `A=012020121 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:0 d7:0 d8:2]`
|
||||
|
||||
## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `001221`:
|
||||
|
||||
- face apexes (raw labels) `112002` → canonical `001221` · `A=012020121 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:0 d7:0 d8:2]`
|
||||
|
||||
## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `001221`:
|
||||
|
||||
- face apexes (raw labels) `112002` → canonical `001221` · `A=012020121 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:0 d7:0 d8:2]`
|
||||
|
||||
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|
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+38
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|
||||
# Inner-face down-apex sequence `010001`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **4** of 7 configs (M(T), inner face).
|
||||
- **7** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_010001.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `010001`:
|
||||
|
||||
- face apexes (raw labels) `020002` → canonical `010001` · `A=012102121 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:0 d7:0 d8:2]`
|
||||
|
||||
## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `010001`:
|
||||
|
||||
- face apexes (raw labels) `020002` → canonical `010001` · `A=012012121 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `010001` → canonical `010001` · `A=012021212 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:0 d7:0 d8:1]`
|
||||
- face apexes (raw labels) `020002` → canonical `010001` · `A=012102121 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]`
|
||||
|
||||
## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `010001`:
|
||||
|
||||
- face apexes (raw labels) `020002` → canonical `010001` · `A=012012121 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:0 d7:0 d8:2]`
|
||||
- face apexes (raw labels) `010001` → canonical `010001` · `A=012021212 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:0 d7:0 d8:1]`
|
||||
|
||||
## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `010001`:
|
||||
|
||||
- face apexes (raw labels) `010001` → canonical `010001` · `A=012021212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:0 d5:0 d7:0 d8:1]`
|
||||
|
||||
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|
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+23
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|
||||
# Inner-face down-apex sequence `010100`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **2** of 7 configs (M(T), inner face).
|
||||
- **2** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_010100.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `010100`:
|
||||
|
||||
- face apexes (raw labels) `212122` → canonical `010100` · `A=010201201 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:2 d8:2]`
|
||||
|
||||
## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `010100`:
|
||||
|
||||
- face apexes (raw labels) `212122` → canonical `010100` · `A=010210201 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:2 d8:2]`
|
||||
|
||||
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|
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+39
@@ -0,0 +1,39 @@
|
||||
# Inner-face down-apex sequence `010111`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 2×colour0, 4×colour1.
|
||||
- Realised by **4** of 7 configs (M(T), inner face).
|
||||
- **8** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_010111.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `010111`:
|
||||
|
||||
- face apexes (raw labels) `020222` → canonical `010111` · `A=012102101 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:2 d8:2]`
|
||||
|
||||
## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `010111`:
|
||||
|
||||
- face apexes (raw labels) `020222` → canonical `010111` · `A=012012101 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `010111` → canonical `010111` · `A=012021202 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `020222` → canonical `010111` · `A=012102101 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]`
|
||||
|
||||
## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `010111`:
|
||||
|
||||
- face apexes (raw labels) `212111` → canonical `010111` · `A=010201202 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `020222` → canonical `010111` · `A=012012101 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `010111` → canonical `010111` · `A=012021202 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:1 d8:1]`
|
||||
|
||||
## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `010111`:
|
||||
|
||||
- face apexes (raw labels) `212111` → canonical `010111` · `A=010210202 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:1 d8:1]`
|
||||
|
||||
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|
After Width: | Height: | Size: 101 KiB |
+32
@@ -0,0 +1,32 @@
|
||||
# Inner-face down-apex sequence `010122`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
|
||||
- Realised by **3** of 7 configs (M(T), inner face).
|
||||
- **6** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_010122.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `010122`:
|
||||
|
||||
- face apexes (raw labels) `020211` → canonical `010122` · `A=012102102 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:1 d8:1]`
|
||||
|
||||
## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
|
||||
|
||||
3 colouring(s) with down-apex sequence `010122`:
|
||||
|
||||
- face apexes (raw labels) `020211` → canonical `010122` · `A=012012102 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `010122` → canonical `010122` · `A=012021201 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `020211` → canonical `010122` · `A=012102102 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]`
|
||||
|
||||
## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `010122`:
|
||||
|
||||
- face apexes (raw labels) `020211` → canonical `010122` · `A=012012102 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:1 d8:1]`
|
||||
- face apexes (raw labels) `010122` → canonical `010122` · `A=012021201 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:2 d8:2]`
|
||||
|
||||
BIN
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|
After Width: | Height: | Size: 86 KiB |
+23
@@ -0,0 +1,23 @@
|
||||
# Inner-face down-apex sequence `010221`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
|
||||
- Realised by **2** of 7 configs (M(T), inner face).
|
||||
- **2** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_010221.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `010221`:
|
||||
|
||||
- face apexes (raw labels) `212001` → canonical `010221` · `A=010201212 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:0 d7:0 d8:1]`
|
||||
|
||||
## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `010221`:
|
||||
|
||||
- face apexes (raw labels) `212001` → canonical `010221` · `A=010201212 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:2 d5:0 d7:0 d8:1]`
|
||||
|
||||
BIN
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BIN
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|
After Width: | Height: | Size: 56 KiB |
+43
@@ -0,0 +1,43 @@
|
||||
# Inner-face down-apex sequence `011000`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **5** of 7 configs (M(T), inner face).
|
||||
- **7** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_011000.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `011000`:
|
||||
|
||||
- face apexes (raw labels) `200222` → canonical `011000` · `A=012012101 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `100111` → canonical `011000` · `A=012021202 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:1 d7:1 d8:1]`
|
||||
|
||||
## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `011000`:
|
||||
|
||||
- face apexes (raw labels) `100111` → canonical `011000` · `A=012021202 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:1 d7:1 d8:1]`
|
||||
|
||||
## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `011000`:
|
||||
|
||||
- face apexes (raw labels) `122111` → canonical `011000` · `A=012010202 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:1 d8:1]`
|
||||
|
||||
## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `011000`:
|
||||
|
||||
- face apexes (raw labels) `200222` → canonical `011000` · `A=010212101 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:2 d7:2 d8:2]`
|
||||
|
||||
## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
|
||||
|
||||
2 colouring(s) with down-apex sequence `011000`:
|
||||
|
||||
- face apexes (raw labels) `211222` → canonical `011000` · `A=010202101 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:2 d8:2]`
|
||||
- face apexes (raw labels) `200222` → canonical `011000` · `A=010212101 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:2 d7:2 d8:2]`
|
||||
|
||||
BIN
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BIN
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|
After Width: | Height: | Size: 92 KiB |
+23
@@ -0,0 +1,23 @@
|
||||
# Inner-face down-apex sequence `011011`
|
||||
|
||||
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**.
|
||||
|
||||
- Colour multiset: 2×colour0, 4×colour1.
|
||||
- Realised by **2** of 7 configs (M(T), inner face).
|
||||
- **2** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_011011.png` (black rings mark the face's down apexes).
|
||||
|
||||
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `011011`:
|
||||
|
||||
- face apexes (raw labels) `122122` → canonical `011011` · `A=012010201 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:2 d8:2]`
|
||||
|
||||
## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
|
||||
|
||||
1 colouring(s) with down-apex sequence `011011`:
|
||||
|
||||
- face apexes (raw labels) `211211` → canonical `011011` · `A=010202102 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:1 d8:1]`
|
||||
|
||||
BIN
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BIN
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|
After Width: | Height: | Size: 57 KiB |
Some files were not shown because too many files have changed in this diff Show More
Reference in New Issue
Block a user