Add up-tooth apex colour-sequence experiment over n=9 up-teeth sweep
Enumerate Kempe-balanced 3-colourings of every M(T) with |A(T)|=9 and a fixed number m of up teeth, record the up-tooth apex colour sequence (cyclic order, mod colour permutation only), and group the M(T) by their set of unique sequences. Runs for m=3,4,5,6 with per-sequence notes and figures plus a summary atlas. Finding: realised sequences obey outer-face Kempe parity (all three colour-counts share m's parity). Distinct sequences grow 1/4/10/28 while M(T) count falls 23/29/18/7 across m=3..6. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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"""Up-tooth apex colour sequences of Kempe-balanced 3-colourings.
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Companion experiment to ``kempe_valid_colorings.py``. For a fixed annular
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size ``n`` and a fixed number ``m`` of up teeth we:
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1. generate every full medial tire graph M(T) with |A(T)| = n and exactly
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``m`` up teeth (one representative per dihedral symmetry class);
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2. enumerate the Kempe-balanced (``valid``) proper 3-colourings of each M(T)
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and read off the colour sequence of the up-tooth apex vertices u0 < u1 <
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... in increasing annular-edge order (i.e. cyclic order around A(T));
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3. reduce each up-tooth sequence modulo the six colour permutations -- but
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NOT modulo the dihedral symmetry of the cycle -- to a canonical sequence;
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4. group the M(T) by the *set* of canonical up-tooth sequences they realise,
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and report how many M(T) share each set.
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For every canonical up-tooth sequence we emit a markdown note listing every
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M(T) that realises it together with every Kempe-balanced colouring (modulo
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colour permutation) on that M(T) producing the sequence, and a figure drawing
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those colourings. A summary note records the step-4 grouping.
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Run: python3 kempe_up_tooth_sequences.py --n 9 --m 4
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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 FullMedialTireGraph, generate
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from kempe_valid_colorings import (
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adjacency,
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classify_colorings,
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kempe_components,
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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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PALETTE = {0: "#e6550d", 1: "#3182bd", 2: "#31a354"}
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PALETTE_NAME = {0: "orange", 1: "blue", 2: "green"}
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def canonical_sequence(seq: tuple[int, ...]) -> tuple[int, ...]:
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"""Relabel a colour sequence by first appearance (mod colour permutation)."""
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remap: dict[int, int] = {}
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out = []
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for c in seq:
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if c not in remap:
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remap[c] = len(remap)
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out.append(remap[c])
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return tuple(out)
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def seq_str(seq: tuple[int, ...]) -> str:
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return "".join(str(c) for c in seq)
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def _parity_partitions(m: int) -> set[tuple[int, ...]]:
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"""Colour multisets (n0,n1,n2) of m up-tooth apexes admissible under the
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outer-face Kempe-parity rule: all three counts share m's parity. Returned
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as tuples of the nonzero parts (descending), matching the table format."""
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out: set[tuple[int, ...]] = set()
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for a in range(m + 1):
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for b in range(a + 1):
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c = m - a - b
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if c < 0 or c > b:
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continue
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if a % 2 == b % 2 == c % 2 == m % 2:
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out.add(tuple(v for v in (a, b, c) if v > 0))
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return out
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def up_tooth_sequence(graph: FullMedialTireGraph, coloring: Coloring) -> tuple[int, ...]:
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"""Colours of the up-tooth apexes in increasing annular-edge order."""
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return tuple(coloring[f"u{i}"] for i in graph.up_edges)
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def graph_id(idx: int) -> str:
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return f"G{idx:02d}"
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def describe_graph(graph: FullMedialTireGraph) -> str:
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bites = ",".join(f"({i},{j})" for i, j in sorted(graph.bites)) or "-"
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return f"word={graph.tooth_word} bites={bites}"
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# ---------------------------------------------------------------------------
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# Data collection.
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# ---------------------------------------------------------------------------
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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.graphs: list[FullMedialTireGraph] = [
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g for g in generate(n, min_up_teeth=3, dedup=True) if len(g.up_edges) == m
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]
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# per graph: list of (coloring, canonical up-tooth sequence)
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self.colorings: list[list[tuple[Coloring, tuple[int, ...]]]] = []
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# per graph: set of canonical up-tooth sequences it realises
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self.graph_seq_sets: list[frozenset[tuple[int, ...]]] = []
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# canonical sequence -> list of (graph_idx, coloring)
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self.by_sequence: dict[tuple[int, ...], list[tuple[int, Coloring]]] = defaultdict(list)
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for gidx, g in enumerate(self.graphs):
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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(g, dedup_colors=True):
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if not verdict.valid:
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continue
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cseq = canonical_sequence(up_tooth_sequence(g, 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((gidx, coloring))
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self.colorings.append(entries)
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self.graph_seq_sets.append(frozenset(seqs))
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def groups(self):
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"""Map each set-of-sequences to the list of graph indices realising it."""
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groups: dict[frozenset[tuple[int, ...]], list[int]] = defaultdict(list)
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for gidx, sset in enumerate(self.graph_seq_sets):
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groups[sset].append(gidx)
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# sort by descending membership, then by set size
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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 (adapted from kempe_valid_colorings._draw_colored).
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# ---------------------------------------------------------------------------
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def _positions(graph: FullMedialTireGraph) -> dict[str, tuple[float, float]]:
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n = graph.n
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def ann_xy(k):
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ang = math.pi / 2 - 2 * math.pi * k / n
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return math.cos(ang), math.sin(ang)
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def mid_ang(i):
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return math.pi / 2 - 2 * math.pi * (i + 0.5) / n
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pos = {f"a{k}": ann_xy(k) for k in range(n)}
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matched = graph.bite_edges
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for i, tooth in enumerate(graph.tooth_word):
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if tooth == "U":
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pos[f"u{i}"] = (1.42 * math.cos(mid_ang(i)), 1.42 * math.sin(mid_ang(i)))
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elif i not in matched:
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pos[f"d{i}"] = (0.58 * math.cos(mid_ang(i)), 0.58 * math.sin(mid_ang(i)))
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for i, j in sorted(graph.bites):
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corners = [ann_xy(i), ann_xy((i + 1) % n), ann_xy(j), ann_xy((j + 1) % n)]
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cx = sum(p[0] for p in corners) / 4.0
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cy = sum(p[1] for p in corners) / 4.0
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pos[f"p{i}_{j}"] = (cx * 0.82, cy * 0.82)
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return pos
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def _draw(ax, graph, coloring, title):
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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 up-tooth apexes whose colours form the recorded sequence
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ux = [pos[f"u{i}"][0] for i in graph.up_edges]
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uy = [pos[f"u{i}"][1] for i in graph.up_edges]
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ax.scatter(ux, uy, 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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gidx, coloring = entries[idx]
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g = exp.graphs[gidx]
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useq = seq_str(up_tooth_sequence(g, coloring))
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_draw(ax, g, coloring, f"{graph_id(gidx)} u={useq}")
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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 up-tooth apex sequence "
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f"{seq_str(seq)} (mod colour permutation)\n"
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f"n={exp.n}, m={exp.m} up teeth — {len(entries)} colourings on "
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f"{len({g for g, _ in entries})} M(T); black rings mark up-tooth 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 compact_coloring(graph: FullMedialTireGraph, coloring: Coloring) -> str:
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"""A readable one-line dump of the full colouring, grouped by vertex kind."""
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parts = []
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parts.append("A=" + "".join(str(coloring[f"a{k}"]) for k in range(graph.n)))
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up = " ".join(f"u{i}:{coloring[f'u{i}']}" for i in graph.up_edges)
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parts.append("U[" + up + "]")
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downs = []
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for i in graph.singleton_down_edges:
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downs.append(f"d{i}:{coloring[f'd{i}']}")
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for i, j in sorted(graph.bites):
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downs.append(f"p{i}_{j}:{coloring[f'p{i}_{j}']}")
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if downs:
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parts.append("D[" + " ".join(downs) + "]")
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return " ".join(parts)
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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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# group entries by graph
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by_graph: dict[int, list[Coloring]] = defaultdict(list)
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for gidx, coloring in exp.by_sequence[seq]:
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by_graph[gidx].append(coloring)
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colour_multiset = {}
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for c in seq:
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colour_multiset[c] = colour_multiset.get(c, 0) + 1
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counts = ", ".join(f"{n}×colour{c}" for c, n in sorted(colour_multiset.items()))
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lines = []
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lines.append(f"# Up-tooth apex sequence `{s}`")
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lines.append("")
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lines.append(
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f"Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic "
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f"order around A(T), reduced modulo the six colour permutations) for "
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f"Kempe-balanced 3-colourings of M(T) with **n = {exp.n}**, "
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f"**m = {exp.m} up teeth**."
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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_graph)}** of {len(exp.graphs)} M(T) "
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f"(dihedral classes).")
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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 up-tooth 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 are 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 gidx in sorted(by_graph):
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g = exp.graphs[gidx]
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cols = by_graph[gidx]
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lines.append(f"## {graph_id(gidx)} — {describe_graph(g)}")
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lines.append("")
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lines.append(f"{len(cols)} colouring(s) with up-tooth sequence `{s}`:")
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lines.append("")
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for coloring in cols:
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raw = seq_str(up_tooth_sequence(g, coloring))
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lines.append(f"- up apexes (raw labels) `{raw}` → canonical `{s}` · "
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f"`{compact_coloring(g, 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, notes_dir_name):
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lines = []
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lines.append(f"# Up-tooth apex sequences of Kempe-balanced colourings "
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f"(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} and exactly "
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f"{exp.m} up teeth, one representative per dihedral class: "
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f"**{len(exp.graphs)} M(T)**. For each we enumerate the Kempe-balanced "
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f"(valid) proper 3-colourings (modulo colour permutation), read the "
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f"up-tooth apex colour sequence u0<u1<... in cyclic order, and reduce it "
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f"modulo colour permutation (NOT 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 up-tooth 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 up-tooth sequences")
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lines.append("")
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lines.append("| sequence | colour multiset | #M(T) realising | #colourings |")
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lines.append("|---|---|---|---|")
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for seq in exp.sequences():
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cm = {}
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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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ngraphs = len({g for g, _ in exp.by_sequence[seq]})
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lines.append(f"| `{seq_str(seq)}` | {cms} | {ngraphs} | "
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f"{len(exp.by_sequence[seq])} |")
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lines.append("")
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# The outer-face Kempe-parity rule says every colour pair meets the up-tooth
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# apexes an even number of times; equivalently the three colour-counts share
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# one parity. With m apexes that parity is forced by m mod 2.
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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 outer-face Kempe-parity "
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"constraint (each colour pair meets the up-tooth apexes an even "
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f"number of times). With m = {exp.m} up teeth (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 M(T) by their set of unique up-tooth "
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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.graphs)} M(T) fall into **{len(groups)}** groups "
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f"by the set of canonical up-tooth sequences they realise:")
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lines.append("")
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lines.append("| #M(T) | set of up-tooth sequences | M(T) ids |")
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lines.append("|---|---|---|")
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for sset, gidxs in groups:
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seqs = ", ".join(f"`{seq_str(s)}`" for s in sorted(sset))
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ids = ", ".join(graph_id(i) for i in gidxs)
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lines.append(f"| {len(gidxs)} | {{ {seqs} }} | {ids} |")
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lines.append("")
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lines.append("## M(T) atlas (ids)")
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lines.append("")
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lines.append("| id | word / bites | #Kempe-balanced | up-tooth sequence set |")
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lines.append("|---|---|---|---|")
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for gidx, g in enumerate(exp.graphs):
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sset = exp.graph_seq_sets[gidx]
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seqs = ", ".join(f"`{seq_str(s)}`" for s in sorted(sset))
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lines.append(f"| {graph_id(gidx)} | {describe_graph(g)} | "
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f"{len(exp.colorings[gidx])} | {{ {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.graphs)} M(T) (dihedral classes)")
|
||||
print(f"distinct canonical up-tooth sequences: {len(exp.by_sequence)}")
|
||||
for seq in exp.sequences():
|
||||
ng = len({g for g, _ in exp.by_sequence[seq]})
|
||||
print(f" {seq_str(seq)}: {ng} M(T), {len(exp.by_sequence[seq])} colourings")
|
||||
print(f"groups by sequence-set: {len(exp.groups())}")
|
||||
|
||||
notes_dir = os.path.join(HERE, f"kempe_up_tooth_sequences_n{args.n}_m{args.m}")
|
||||
os.makedirs(notes_dir, exist_ok=True)
|
||||
|
||||
write_summary(exp, os.path.join(notes_dir, "summary.md"), notes_dir)
|
||||
for seq in exp.sequences():
|
||||
s = seq_str(seq)
|
||||
fig_name = f"seq_{s}.png"
|
||||
write_sequence_note(exp, seq, os.path.join(notes_dir, f"seq_{s}.md"), fig_name)
|
||||
if not args.no_figures:
|
||||
draw_sequence(exp, seq,
|
||||
os.path.join(notes_dir, fig_name),
|
||||
os.path.join(notes_dir, f"seq_{s}.pdf"))
|
||||
|
||||
|
||||
def main():
|
||||
parser = argparse.ArgumentParser(description=__doc__)
|
||||
parser.add_argument("--n", type=int, default=9)
|
||||
parser.add_argument("--m", type=int, default=4, help="number of up teeth")
|
||||
parser.add_argument("--no-figures", action="store_true")
|
||||
run(parser.parse_args())
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
main()
|
||||
+325
@@ -0,0 +1,325 @@
|
||||
# Up-tooth apex sequence `012`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 3 up teeth**.
|
||||
|
||||
- Colour multiset: 1×colour0, 1×colour1, 1×colour2.
|
||||
- Realised by **23** of 23 M(T) (dihedral classes).
|
||||
- **199** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_012.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G00 — word=UUUDDDDDD bites=-
|
||||
|
||||
22 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012010101 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012010102 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012010121 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012010201 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012010202 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012010212 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012012012 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012012021 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012012101 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012012102 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012012121 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020101 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020102 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020121 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020201 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020202 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020212 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012021012 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:2 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012021021 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:2 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012021201 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012021202 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012021212 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G01 — word=UUUDDDDDD bites=(3,8)
|
||||
|
||||
12 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012010101 U[u0:2 u1:0 u2:1] D[d4:2 d5:2 d6:2 d7:2 p3_8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012010121 U[u0:2 u1:0 u2:1] D[d4:2 d5:2 d6:0 d7:0 p3_8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012010201 U[u0:2 u1:0 u2:1] D[d4:2 d5:1 d6:1 d7:2 p3_8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012012021 U[u0:2 u1:0 u2:1] D[d4:0 d5:1 d6:1 d7:0 p3_8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012012101 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:2 d7:2 p3_8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012012121 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:0 d7:0 p3_8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020102 U[u0:2 u1:0 u2:1] D[d4:1 d5:2 d6:2 d7:1 p3_8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020202 U[u0:2 u1:0 u2:1] D[d4:1 d5:1 d6:1 d7:1 p3_8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020212 U[u0:2 u1:0 u2:1] D[d4:1 d5:1 d6:0 d7:0 p3_8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012021012 U[u0:2 u1:0 u2:1] D[d4:0 d5:2 d6:2 d7:0 p3_8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012021202 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:1 d7:1 p3_8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012021212 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:0 d7:0 p3_8:1]`
|
||||
|
||||
## G02 — word=UUDUDDDDD bites=-
|
||||
|
||||
11 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020101 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020102 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020121 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020201 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020202 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020212 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012021012 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:2 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012021021 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:2 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012021201 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012021202 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012021212 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G03 — word=UUDUDDDDD bites=(2,4)
|
||||
|
||||
6 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020101 U[u0:2 u1:0 u3:1] D[d5:2 d6:2 d7:2 d8:2 p2_4:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020102 U[u0:2 u1:0 u3:1] D[d5:2 d6:2 d7:1 d8:1 p2_4:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020121 U[u0:2 u1:0 u3:1] D[d5:2 d6:0 d7:0 d8:2 p2_4:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020201 U[u0:2 u1:0 u3:1] D[d5:1 d6:1 d7:2 d8:2 p2_4:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020202 U[u0:2 u1:0 u3:1] D[d5:1 d6:1 d7:1 d8:1 p2_4:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020212 U[u0:2 u1:0 u3:1] D[d5:1 d6:0 d7:0 d8:1 p2_4:1]`
|
||||
|
||||
## G04 — word=UUDUDDDDD bites=(2,8)
|
||||
|
||||
6 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020102 U[u0:2 u1:0 u3:1] D[d4:1 d5:2 d6:2 d7:1 p2_8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020202 U[u0:2 u1:0 u3:1] D[d4:1 d5:1 d6:1 d7:1 p2_8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020212 U[u0:2 u1:0 u3:1] D[d4:1 d5:1 d6:0 d7:0 p2_8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012021012 U[u0:2 u1:0 u3:1] D[d4:0 d5:2 d6:2 d7:0 p2_8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012021202 U[u0:2 u1:0 u3:1] D[d4:0 d5:0 d6:1 d7:1 p2_8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012021212 U[u0:2 u1:0 u3:1] D[d4:0 d5:0 d6:0 d7:0 p2_8:1]`
|
||||
|
||||
## G05 — word=UUDDUDDDD bites=-
|
||||
|
||||
17 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020101 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020102 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020121 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020202 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012102012 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012102021 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012102101 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012102102 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012102121 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120101 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120102 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120121 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120201 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120202 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120212 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G06 — word=UUDDUDDDD bites=(3,5)
|
||||
|
||||
3 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d6:1 d7:2 d8:2 p3_5:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020202 U[u0:2 u1:0 u4:1] D[d2:1 d6:1 d7:1 d8:1 p3_5:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d6:0 d7:0 d8:1 p3_5:1]`
|
||||
|
||||
## G07 — word=UUDDUDDDD bites=(2,8)
|
||||
|
||||
3 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020102 U[u0:2 u1:0 u4:1] D[d3:1 d5:2 d6:2 d7:1 p2_8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020202 U[u0:2 u1:0 u4:1] D[d3:1 d5:1 d6:1 d7:1 p2_8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020212 U[u0:2 u1:0 u4:1] D[d3:1 d5:1 d6:0 d7:0 p2_8:1]`
|
||||
|
||||
## G08 — word=UUDDDUDDD bites=-
|
||||
|
||||
13 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012010201 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012010202 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012010212 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012012012 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:0 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012012021 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:0 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020202 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020212 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012102012 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d4:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012102021 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d4:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120201 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120212 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G09 — word=UUDDDUDDD bites=(4,6)
|
||||
|
||||
5 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d7:2 d8:2 p4_6:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020202 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d7:1 d8:1 p4_6:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012102021 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d7:0 d8:2 p4_6:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120201 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d7:2 d8:2 p4_6:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d7:1 d8:1 p4_6:1]`
|
||||
|
||||
## G10 — word=UUDDDUDDD bites=(2,8)
|
||||
|
||||
5 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012010202 U[u0:2 u1:0 u5:1] D[d3:2 d4:2 d6:1 d7:1 p2_8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012010212 U[u0:2 u1:0 u5:1] D[d3:2 d4:2 d6:0 d7:0 p2_8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012012012 U[u0:2 u1:0 u5:1] D[d3:2 d4:0 d6:2 d7:0 p2_8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020202 U[u0:2 u1:0 u5:1] D[d3:1 d4:1 d6:1 d7:1 p2_8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020212 U[u0:2 u1:0 u5:1] D[d3:1 d4:1 d6:0 d7:0 p2_8:1]`
|
||||
|
||||
## G11 — word=UUDDDUDDD bites=(2,8),(3,7),(4,6)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012020202 U[u0:2 u1:0 u5:1] D[p2_8:1 p3_7:1 p4_6:1]`
|
||||
|
||||
## G12 — word=UDUDUDDDD bites=-
|
||||
|
||||
26 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010212012 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010212021 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010212101 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010212102 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010212121 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012012012 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012012021 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012012101 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012012102 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012012121 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012021012 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:2 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012021021 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:2 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012021201 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012021202 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012021212 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012102012 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012102021 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012102101 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012102102 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012102121 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120101 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120102 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120121 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120201 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120202 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120212 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G13 — word=UDUDUDDDD bites=(3,5)
|
||||
|
||||
3 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010212101 U[u0:2 u2:1 u4:0] D[d1:2 d6:2 d7:2 d8:2 p3_5:0]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010212102 U[u0:2 u2:1 u4:0] D[d1:2 d6:2 d7:1 d8:1 p3_5:0]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010212121 U[u0:2 u2:1 u4:0] D[d1:2 d6:0 d7:0 d8:2 p3_5:0]`
|
||||
|
||||
## G14 — word=UDUDUDDDD bites=(1,3)
|
||||
|
||||
6 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120101 U[u0:2 u2:0 u4:1] D[d5:2 d6:2 d7:2 d8:2 p1_3:0]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120102 U[u0:2 u2:0 u4:1] D[d5:2 d6:2 d7:1 d8:1 p1_3:0]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120121 U[u0:2 u2:0 u4:1] D[d5:2 d6:0 d7:0 d8:2 p1_3:0]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120201 U[u0:2 u2:0 u4:1] D[d5:1 d6:1 d7:2 d8:2 p1_3:0]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120202 U[u0:2 u2:0 u4:1] D[d5:1 d6:1 d7:1 d8:1 p1_3:0]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120212 U[u0:2 u2:0 u4:1] D[d5:1 d6:0 d7:0 d8:1 p1_3:0]`
|
||||
|
||||
## G15 — word=UDUDDUDDD bites=-
|
||||
|
||||
20 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010201201 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010201202 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010201212 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010202101 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010202102 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010202121 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010212101 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010212102 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010212121 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012012101 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012012102 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012012121 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012021201 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012021202 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012021212 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012102012 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d4:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012102021 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d4:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120201 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120212 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G16 — word=UDUDDUDDD bites=(4,6)
|
||||
|
||||
6 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010212121 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d7:0 d8:2 p4_6:0]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012012121 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d7:0 d8:2 p4_6:0]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012021212 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d7:0 d8:1 p4_6:0]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012102021 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d7:0 d8:2 p4_6:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120201 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d7:2 d8:2 p4_6:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d7:1 d8:1 p4_6:1]`
|
||||
|
||||
## G17 — word=UDUDDUDDD bites=(1,3)
|
||||
|
||||
3 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120201 U[u0:2 u2:0 u5:1] D[d4:1 d6:1 d7:2 d8:2 p1_3:0]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120202 U[u0:2 u2:0 u5:1] D[d4:1 d6:1 d7:1 d8:1 p1_3:0]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120212 U[u0:2 u2:0 u5:1] D[d4:1 d6:0 d7:0 d8:1 p1_3:0]`
|
||||
|
||||
## G18 — word=UDUDDUDDD bites=(1,8)
|
||||
|
||||
5 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010201201 U[u0:2 u2:1 u5:0] D[d3:1 d4:2 d6:1 d7:2 p1_8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010202101 U[u0:2 u2:1 u5:0] D[d3:1 d4:1 d6:2 d7:2 p1_8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010202121 U[u0:2 u2:1 u5:0] D[d3:1 d4:1 d6:0 d7:0 p1_8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010212101 U[u0:2 u2:1 u5:0] D[d3:0 d4:0 d6:2 d7:2 p1_8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010212121 U[u0:2 u2:1 u5:0] D[d3:0 d4:0 d6:0 d7:0 p1_8:2]`
|
||||
|
||||
## G19 — word=UDUDDUDDD bites=(1,8),(3,7),(4,6)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010212121 U[u0:2 u2:1 u5:0] D[p1_8:2 p3_7:0 p4_6:0]`
|
||||
|
||||
## G20 — word=UDDUDDUDD bites=-
|
||||
|
||||
18 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=010120201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=010121021 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=010121201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=010121202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:0 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010201212 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:2 d5:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010202121 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:1 d5:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=010210201 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=010210202 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=010212021 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:0 d5:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012020121 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012020212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:1 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012021212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:0 d5:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120201 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d5:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d5:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012121021 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012121201 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012121202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:1 d8:1]`
|
||||
|
||||
## G21 — word=UDDUDDUDD bites=(5,7)
|
||||
|
||||
6 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d8:1 p5_7:1]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010201212 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:2 d8:1 p5_7:0]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010202121 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:1 d8:2 p5_7:0]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=010210202 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d8:1 p5_7:1]`
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=012021212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:0 d8:1 p5_7:0]`
|
||||
- up apexes (raw labels) `201` → canonical `012` · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d8:1 p5_7:1]`
|
||||
|
||||
## G22 — word=UDDUDDUDD bites=(1,8),(2,4),(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `012`:
|
||||
|
||||
- up apexes (raw labels) `210` → canonical `012` · `A=010202121 U[u0:2 u3:1 u6:0] D[p1_8:2 p2_4:1 p5_7:0]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequences of Kempe-balanced colourings (n=9, m=3)
|
||||
|
||||
Every full medial tire graph M(T) with |A(T)| = 9 and exactly 3 up teeth, one representative per dihedral class: **23 M(T)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the up-tooth apex colour sequence u0<u1<... in cyclic order, and reduce it modulo colour permutation (NOT dihedral symmetry).
|
||||
|
||||
- Total Kempe-balanced colourings (mod colour permutation): **199**.
|
||||
- Distinct canonical up-tooth sequences overall: **1**.
|
||||
|
||||
## Distinct canonical up-tooth sequences
|
||||
|
||||
| sequence | colour multiset | #M(T) realising | #colourings |
|
||||
|---|---|---|---|
|
||||
| `012` | 1+1+1 | 23 | 199 |
|
||||
|
||||
Note: every realised sequence has its three colour-counts of **equal parity** — exactly the outer-face Kempe-parity constraint (each colour pair meets the up-tooth apexes an even number of times). With m = 3 up teeth (m is odd) every count must be **odd**, so the only admissible colour multisets are 1+1+1.
|
||||
|
||||
## Step 4 — grouping M(T) by their set of unique up-tooth sequences
|
||||
|
||||
The 23 M(T) fall into **1** groups by the set of canonical up-tooth sequences they realise:
|
||||
|
||||
| #M(T) | set of up-tooth sequences | M(T) ids |
|
||||
|---|---|---|
|
||||
| 23 | { `012` } | G00, G01, G02, G03, G04, G05, G06, G07, G08, G09, G10, G11, G12, G13, G14, G15, G16, G17, G18, G19, G20, G21, G22 |
|
||||
|
||||
## M(T) atlas (ids)
|
||||
|
||||
| id | word / bites | #Kempe-balanced | up-tooth sequence set |
|
||||
|---|---|---|---|
|
||||
| G00 | word=UUUDDDDDD bites=- | 22 | { `012` } |
|
||||
| G01 | word=UUUDDDDDD bites=(3,8) | 12 | { `012` } |
|
||||
| G02 | word=UUDUDDDDD bites=- | 11 | { `012` } |
|
||||
| G03 | word=UUDUDDDDD bites=(2,4) | 6 | { `012` } |
|
||||
| G04 | word=UUDUDDDDD bites=(2,8) | 6 | { `012` } |
|
||||
| G05 | word=UUDDUDDDD bites=- | 17 | { `012` } |
|
||||
| G06 | word=UUDDUDDDD bites=(3,5) | 3 | { `012` } |
|
||||
| G07 | word=UUDDUDDDD bites=(2,8) | 3 | { `012` } |
|
||||
| G08 | word=UUDDDUDDD bites=- | 13 | { `012` } |
|
||||
| G09 | word=UUDDDUDDD bites=(4,6) | 5 | { `012` } |
|
||||
| G10 | word=UUDDDUDDD bites=(2,8) | 5 | { `012` } |
|
||||
| G11 | word=UUDDDUDDD bites=(2,8),(3,7),(4,6) | 1 | { `012` } |
|
||||
| G12 | word=UDUDUDDDD bites=- | 26 | { `012` } |
|
||||
| G13 | word=UDUDUDDDD bites=(3,5) | 3 | { `012` } |
|
||||
| G14 | word=UDUDUDDDD bites=(1,3) | 6 | { `012` } |
|
||||
| G15 | word=UDUDDUDDD bites=- | 20 | { `012` } |
|
||||
| G16 | word=UDUDDUDDD bites=(4,6) | 6 | { `012` } |
|
||||
| G17 | word=UDUDDUDDD bites=(1,3) | 3 | { `012` } |
|
||||
| G18 | word=UDUDDUDDD bites=(1,8) | 5 | { `012` } |
|
||||
| G19 | word=UDUDDUDDD bites=(1,8),(3,7),(4,6) | 1 | { `012` } |
|
||||
| G20 | word=UDDUDDUDD bites=- | 18 | { `012` } |
|
||||
| G21 | word=UDDUDDUDD bites=(5,7) | 6 | { `012` } |
|
||||
| G22 | word=UDDUDDUDD bites=(1,8),(2,4),(5,7) | 1 | { `012` } |
|
||||
|
||||
## Per-sequence notes
|
||||
|
||||
- [`012`](seq_012.md) — figure `seq_012.png`
|
||||
|
||||
+221
@@ -0,0 +1,221 @@
|
||||
# Up-tooth apex sequence `0000`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 up teeth**.
|
||||
|
||||
- Colour multiset: 4×colour0.
|
||||
- Realised by **27** of 29 M(T) (dihedral classes).
|
||||
- **75** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_0000.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G00 — word=UUUUDDDDD bites=-
|
||||
|
||||
10 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101012 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101202 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101212 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010102021 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010102101 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010102121 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
|
||||
|
||||
## G01 — word=UUUUDDDDD bites=(4,8)
|
||||
|
||||
4 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d5:2 d6:1 d7:0 p4_8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d5:0 d6:1 d7:2 p4_8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d5:1 d6:2 d7:0 p4_8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d5:0 d6:2 d7:1 p4_8:1]`
|
||||
|
||||
## G02 — word=UUUDUDDDD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101202 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101212 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G03 — word=UUUDUDDDD bites=(3,5)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d6:2 d7:0 d8:1 p3_5:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d6:1 d7:0 d8:2 p3_5:2]`
|
||||
|
||||
## G04 — word=UUUDUDDDD bites=(3,8)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d5:2 d6:1 d7:0 p3_8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d5:0 d6:1 d7:2 p3_8:2]`
|
||||
|
||||
## G05 — word=UUUDDUDDD bites=-
|
||||
|
||||
7 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010120101 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010120102 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010120121 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010121012 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010121021 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
|
||||
|
||||
## G06 — word=UUUDDUDDD bites=(4,6)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d7:0 d8:1 p4_6:2]`
|
||||
|
||||
## G07 — word=UUUDDUDDD bites=(3,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d4:2 d6:1 d7:0 p3_8:2]`
|
||||
|
||||
## G08 — word=UUDUUDDDD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101201 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101202 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101212 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G09 — word=UUDUUDDDD bites=(2,5)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d6:2 d7:0 d8:1 p2_5:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d6:1 d7:0 d8:2 p2_5:2]`
|
||||
|
||||
## G10 — word=UUDUDUDDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:1 d7:0 d8:2]`
|
||||
|
||||
## G11 — word=UUDUDUDDD bites=(4,6)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d7:0 d8:1 p4_6:2]`
|
||||
|
||||
## G12 — word=UUDUDUDDD bites=(2,4)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d6:2 d7:0 d8:1 p2_4:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d6:1 d7:0 d8:2 p2_4:2]`
|
||||
|
||||
## G13 — word=UUDUDUDDD bites=(2,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d4:2 d6:1 d7:0 p2_8:2]`
|
||||
|
||||
## G14 — word=UUDUDDUDD bites=-
|
||||
|
||||
4 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:2 d5:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010102012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:1 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010102102 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:1 d8:1]`
|
||||
|
||||
## G16 — word=UUDUDDUDD bites=(2,4)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d5:2 d7:0 d8:1 p2_4:2]`
|
||||
|
||||
## G17 — word=UUDUDDUDD bites=(2,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d4:1 d5:0 d7:2 p2_8:2]`
|
||||
|
||||
## G18 — word=UUDUDDDUD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:2 d5:0 d6:1 d8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 d6:2 d8:2]`
|
||||
|
||||
## G19 — word=UUDUDDDUD bites=(6,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 p6_8:2]`
|
||||
|
||||
## G20 — word=UUDUDDDUD bites=(2,8)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d4:2 d5:0 d6:1 p2_8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d4:1 d5:0 d6:2 p2_8:2]`
|
||||
|
||||
## G21 — word=UUDDUUDDD bites=-
|
||||
|
||||
7 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101021 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010201012 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010210101 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010210102 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:0 d7:0 d8:2]`
|
||||
|
||||
## G22 — word=UUDDUUDDD bites=(3,6)
|
||||
|
||||
3 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d7:0 d8:1 p3_6:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d7:0 d8:2 p3_6:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d7:0 d8:2 p3_6:0]`
|
||||
|
||||
## G23 — word=UUDDUDUDD bites=-
|
||||
|
||||
4 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101012 U[u0:2 u1:2 u4:2 u6:2] D[d2:2 d3:2 d5:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:1 d8:1]`
|
||||
|
||||
## G24 — word=UUDDUDUDD bites=(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d8:2 p5_7:2]`
|
||||
|
||||
## G25 — word=UUDDUDUDD bites=(2,8)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d3:1 d5:2 d7:0 p2_8:1]`
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d3:0 d5:2 d7:1 p2_8:1]`
|
||||
|
||||
## G26 — word=UDUDUDUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d3:2 d5:2 d7:0 d8:1]`
|
||||
|
||||
## G28 — word=UDUDUDUDD bites=(3,5)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0000`:
|
||||
|
||||
- up apexes (raw labels) `2222` → canonical `0000` · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d7:0 d8:1 p3_5:2]`
|
||||
|
||||
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|
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# Up-tooth apex sequence `0011`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 up teeth**.
|
||||
|
||||
- Colour multiset: 2×colour0, 2×colour1.
|
||||
- Realised by **29** of 29 M(T) (dihedral classes).
|
||||
- **140** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_0011.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G00 — word=UUUUDDDDD bites=-
|
||||
|
||||
10 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010201012 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010201202 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010201212 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202021 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202101 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202121 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
|
||||
|
||||
## G01 — word=UUUUDDDDD bites=(4,8)
|
||||
|
||||
4 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d5:2 d6:1 d7:0 p4_8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d5:0 d6:1 d7:2 p4_8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d5:1 d6:2 d7:0 p4_8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d5:0 d6:2 d7:1 p4_8:1]`
|
||||
|
||||
## G02 — word=UUUDUDDDD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202101 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202121 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:0 d7:0 d8:2]`
|
||||
|
||||
## G03 — word=UUUDUDDDD bites=(3,5)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d6:2 d7:0 d8:1 p3_5:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d6:1 d7:0 d8:2 p3_5:1]`
|
||||
|
||||
## G04 — word=UUUDUDDDD bites=(3,8)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d5:1 d6:2 d7:0 p3_8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d5:0 d6:2 d7:1 p3_8:1]`
|
||||
|
||||
## G05 — word=UUUDDUDDD bites=-
|
||||
|
||||
7 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010210201 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010210202 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010210212 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010212012 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010212021 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
|
||||
|
||||
## G06 — word=UUUDDUDDD bites=(4,6)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d7:0 d8:2 p4_6:1]`
|
||||
|
||||
## G07 — word=UUUDDUDDD bites=(3,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d4:1 d6:2 d7:0 p3_8:1]`
|
||||
|
||||
## G08 — word=UUDUUDDDD bites=-
|
||||
|
||||
15 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121201 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121202 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121212 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202101 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202102 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202121 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212101 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212102 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212121 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
|
||||
|
||||
## G09 — word=UUDUUDDDD bites=(2,5)
|
||||
|
||||
6 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d6:2 d7:0 d8:1 p2_5:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d6:1 d7:0 d8:2 p2_5:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d6:2 d7:0 d8:1 p2_5:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d6:1 d7:0 d8:2 p2_5:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d6:2 d7:0 d8:1 p2_5:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d6:1 d7:0 d8:2 p2_5:1]`
|
||||
|
||||
## G10 — word=UUDUDUDDD bites=-
|
||||
|
||||
8 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121202 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212101 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]`
|
||||
|
||||
## G11 — word=UUDUDUDDD bites=(4,6)
|
||||
|
||||
3 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d7:0 d8:1 p4_6:0]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d7:0 d8:2 p4_6:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d7:0 d8:2 p4_6:0]`
|
||||
|
||||
## G12 — word=UUDUDUDDD bites=(2,4)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d6:2 d7:0 d8:1 p2_4:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d6:1 d7:0 d8:2 p2_4:1]`
|
||||
|
||||
## G13 — word=UUDUDUDDD bites=(2,8)
|
||||
|
||||
3 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d4:0 d6:1 d7:2 p2_8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d4:1 d6:2 d7:0 p2_8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d4:0 d6:2 d7:1 p2_8:1]`
|
||||
|
||||
## G14 — word=UUDUDDUDD bites=-
|
||||
|
||||
10 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010120212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:1 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d5:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010201021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010201201 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:1 d5:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010210121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]`
|
||||
|
||||
## G15 — word=UUDUDDUDD bites=(5,7)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d8:1 p5_7:0]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d8:2 p5_7:0]`
|
||||
|
||||
## G16 — word=UUDUDDUDD bites=(2,4)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d5:1 d7:0 d8:2 p2_4:1]`
|
||||
|
||||
## G17 — word=UUDUDDUDD bites=(2,8)
|
||||
|
||||
3 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d4:1 d5:2 d7:0 p2_8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d4:2 d5:0 d7:1 p2_8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d4:2 d5:1 d7:0 p2_8:1]`
|
||||
|
||||
## G18 — word=UUDUDDDUD bites=-
|
||||
|
||||
12 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:2 d6:0 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010120212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:1 d6:0 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121012 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:1 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:0 d6:0 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 d6:1 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:1 d5:0 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010210121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212021 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]`
|
||||
|
||||
## G19 — word=UUDUDDDUD bites=(6,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 p6_8:1]`
|
||||
|
||||
## G20 — word=UUDUDDDUD bites=(2,8)
|
||||
|
||||
6 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d4:1 d5:2 d6:0 p2_8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d4:0 d5:2 d6:1 p2_8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d4:2 d5:0 d6:1 p2_8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d4:1 d5:0 d6:2 p2_8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d4:2 d5:1 d6:0 p2_8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d4:0 d5:1 d6:2 p2_8:1]`
|
||||
|
||||
## G21 — word=UUDDUUDDD bites=-
|
||||
|
||||
13 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010102021 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010120201 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010120202 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121201 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121202 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202012 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212101 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212102 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:0 d7:0 d8:2]`
|
||||
|
||||
## G22 — word=UUDDUUDDD bites=(3,6)
|
||||
|
||||
5 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d7:0 d8:1 p3_6:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d7:0 d8:1 p3_6:0]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d7:0 d8:1 p3_6:0]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d7:0 d8:2 p3_6:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d7:0 d8:2 p3_6:0]`
|
||||
|
||||
## G23 — word=UUDDUDUDD bites=-
|
||||
|
||||
6 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:2 d5:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d5:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010202021 U[u0:2 u1:2 u4:1 u6:1] D[d2:1 d3:1 d5:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d5:0 d7:0 d8:2]`
|
||||
|
||||
## G24 — word=UUDDUDUDD bites=(5,7)
|
||||
|
||||
3 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d8:1 p5_7:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d8:1 p5_7:0]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d8:2 p5_7:0]`
|
||||
|
||||
## G25 — word=UUDDUDUDD bites=(2,8)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d3:2 d5:1 d7:0 p2_8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d3:0 d5:1 d7:2 p2_8:2]`
|
||||
|
||||
## G26 — word=UDUDUDUDD bites=-
|
||||
|
||||
4 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010102021 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:2 d5:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010120201 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d5:0 d7:0 d8:1]`
|
||||
|
||||
## G27 — word=UDUDUDUDD bites=(5,7)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2211` → canonical `0011` · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d8:1 p5_7:1]`
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d8:1 p5_7:0]`
|
||||
|
||||
## G28 — word=UDUDUDUDD bites=(3,5)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0011`:
|
||||
|
||||
- up apexes (raw labels) `2200` → canonical `0011` · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d7:0 d8:1 p3_5:0]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `0101`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 up teeth**.
|
||||
|
||||
- Colour multiset: 2×colour0, 2×colour1.
|
||||
- Realised by **16** of 29 M(T) (dihedral classes).
|
||||
- **59** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_0101.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G08 — word=UUDUUDDDD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `0101`:
|
||||
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012101 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012102 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012121 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
|
||||
|
||||
## G09 — word=UUDUUDDDD bites=(2,5)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0101`:
|
||||
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d6:2 d7:0 d8:1 p2_5:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d6:1 d7:0 d8:2 p2_5:1]`
|
||||
|
||||
## G10 — word=UUDUDUDDD bites=-
|
||||
|
||||
9 colouring(s) with up-tooth sequence `0101`:
|
||||
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012101 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012101201 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012101202 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012101212 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012102101 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012102102 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012102121 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:0 d7:0 d8:2]`
|
||||
|
||||
## G11 — word=UUDUDUDDD bites=(4,6)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0101`:
|
||||
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d7:0 d8:2 p4_6:0]`
|
||||
|
||||
## G13 — word=UUDUDUDDD bites=(2,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0101`:
|
||||
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d4:0 d6:2 d7:1 p2_8:1]`
|
||||
|
||||
## G14 — word=UUDUDDUDD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `0101`:
|
||||
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012010121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d5:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d5:0 d7:0 d8:2]`
|
||||
|
||||
## G15 — word=UUDUDDUDD bites=(5,7)
|
||||
|
||||
3 colouring(s) with up-tooth sequence `0101`:
|
||||
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d8:2 p5_7:0]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d8:1 p5_7:0]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d8:2 p5_7:0]`
|
||||
|
||||
## G17 — word=UUDUDDUDD bites=(2,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0101`:
|
||||
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d4:2 d5:1 d7:0 p2_8:1]`
|
||||
|
||||
## G18 — word=UUDUDDDUD bites=-
|
||||
|
||||
11 colouring(s) with up-tooth sequence `0101`:
|
||||
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012010121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012021 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012101012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012101021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:1 d8:2]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012101212 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:0 d6:0 d8:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012102012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012102021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:1 d8:2]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012102121 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:0 d6:0 d8:2]`
|
||||
|
||||
## G20 — word=UUDUDDDUD bites=(2,8)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0101`:
|
||||
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d4:2 d5:1 d6:0 p2_8:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d4:0 d5:1 d6:2 p2_8:1]`
|
||||
|
||||
## G21 — word=UUDDUUDDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `0101`:
|
||||
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012101201 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012101202 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012101212 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G23 — word=UUDDUDUDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `0101`:
|
||||
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012010121 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:1 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d5:0 d7:0 d8:1]`
|
||||
|
||||
## G24 — word=UUDDUDUDD bites=(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0101`:
|
||||
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d8:1 p5_7:0]`
|
||||
|
||||
## G25 — word=UUDDUDUDD bites=(2,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0101`:
|
||||
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d3:2 d5:1 d7:0 p2_8:1]`
|
||||
|
||||
## G26 — word=UDUDUDUDD bites=-
|
||||
|
||||
8 colouring(s) with up-tooth sequence `0101`:
|
||||
|
||||
- up apexes (raw labels) `2121` → canonical `0101` · `A=010201021 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2121` → canonical `0101` · `A=010201201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2121` → canonical `0101` · `A=010201202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2121` → canonical `0101` · `A=010210201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2121` → canonical `0101` · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2121` → canonical `0101` · `A=012010201 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2121` → canonical `0101` · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d5:0 d7:0 d8:1]`
|
||||
|
||||
## G27 — word=UDUDUDUDD bites=(5,7)
|
||||
|
||||
3 colouring(s) with up-tooth sequence `0101`:
|
||||
|
||||
- up apexes (raw labels) `2121` → canonical `0101` · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d8:1 p5_7:1]`
|
||||
- up apexes (raw labels) `2121` → canonical `0101` · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d8:1 p5_7:1]`
|
||||
- up apexes (raw labels) `2020` → canonical `0101` · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d8:1 p5_7:0]`
|
||||
|
||||
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|
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# Up-tooth apex sequence `0110`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 up teeth**.
|
||||
|
||||
- Colour multiset: 2×colour0, 2×colour1.
|
||||
- Realised by **22** of 29 M(T) (dihedral classes).
|
||||
- **69** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_0110.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G00 — word=UUUUDDDDD bites=-
|
||||
|
||||
10 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101012 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101202 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101212 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012102021 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012102101 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012102121 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
|
||||
|
||||
## G01 — word=UUUUDDDDD bites=(4,8)
|
||||
|
||||
4 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d5:2 d6:1 d7:0 p4_8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d5:0 d6:1 d7:2 p4_8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d5:1 d6:2 d7:0 p4_8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d5:0 d6:2 d7:1 p4_8:1]`
|
||||
|
||||
## G02 — word=UUUDUDDDD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101202 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101212 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G03 — word=UUUDUDDDD bites=(3,5)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d6:2 d7:0 d8:1 p3_5:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d6:1 d7:0 d8:2 p3_5:2]`
|
||||
|
||||
## G04 — word=UUUDUDDDD bites=(3,8)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d5:2 d6:1 d7:0 p3_8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d5:0 d6:1 d7:2 p3_8:2]`
|
||||
|
||||
## G05 — word=UUUDDUDDD bites=-
|
||||
|
||||
7 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012120101 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012120102 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012120121 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012121012 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012121021 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
|
||||
|
||||
## G06 — word=UUUDDUDDD bites=(4,6)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d7:0 d8:1 p4_6:2]`
|
||||
|
||||
## G07 — word=UUUDDUDDD bites=(3,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d4:2 d6:1 d7:0 p3_8:2]`
|
||||
|
||||
## G10 — word=UUDUDUDDD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012120101 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012120102 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012120121 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:1 d7:0 d8:2]`
|
||||
|
||||
## G12 — word=UUDUDUDDD bites=(2,4)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d6:2 d7:0 d8:1 p2_4:0]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d6:1 d7:0 d8:2 p2_4:0]`
|
||||
|
||||
## G14 — word=UUDUDDUDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012120102 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:0 d5:2 d7:0 d8:1]`
|
||||
|
||||
## G15 — word=UUDUDDUDD bites=(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d8:2 p5_7:2]`
|
||||
|
||||
## G16 — word=UUDUDDUDD bites=(2,4)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d5:2 d7:0 d8:1 p2_4:0]`
|
||||
|
||||
## G18 — word=UUDUDDDUD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 d6:2 d8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012120201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:1 d6:1 d8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012121201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:0 d5:0 d6:1 d8:2]`
|
||||
|
||||
## G19 — word=UUDUDDDUD bites=(6,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 p6_8:2]`
|
||||
|
||||
## G21 — word=UUDDUUDDD bites=-
|
||||
|
||||
4 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012021012 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012121012 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012121021 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:1 d7:0 d8:2]`
|
||||
|
||||
## G22 — word=UUDDUUDDD bites=(3,6)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d7:0 d8:2 p3_6:1]`
|
||||
|
||||
## G23 — word=UUDDUDUDD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:1 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012012101 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012121012 U[u0:2 u1:0 u4:0 u6:2] D[d2:0 d3:0 d5:2 d7:0 d8:1]`
|
||||
|
||||
## G25 — word=UUDDUDUDD bites=(2,8)
|
||||
|
||||
3 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d3:2 d5:1 d7:0 p2_8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d3:2 d5:0 d7:1 p2_8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d3:1 d5:2 d7:0 p2_8:1]`
|
||||
|
||||
## G26 — word=UDUDUDUDD bites=-
|
||||
|
||||
6 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2112` → canonical `0110` · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:1 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `2112` → canonical `0110` · `A=010202101 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2112` → canonical `0110` · `A=010202102 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2112` → canonical `0110` · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `2112` → canonical `0110` · `A=012020102 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `2002` → canonical `0110` · `A=012121012 U[u0:2 u2:0 u4:0 u6:2] D[d1:0 d3:0 d5:2 d7:0 d8:1]`
|
||||
|
||||
## G27 — word=UDUDUDUDD bites=(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2112` → canonical `0110` · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d8:2 p5_7:2]`
|
||||
|
||||
## G28 — word=UDUDUDUDD bites=(3,5)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `0110`:
|
||||
|
||||
- up apexes (raw labels) `2112` → canonical `0110` · `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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|
||||
# Up-tooth apex sequences of Kempe-balanced colourings (n=9, m=4)
|
||||
|
||||
Every full medial tire graph M(T) with |A(T)| = 9 and exactly 4 up teeth, one representative per dihedral class: **29 M(T)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the up-tooth apex colour sequence u0<u1<... in cyclic order, and reduce it modulo colour permutation (NOT dihedral symmetry).
|
||||
|
||||
- Total Kempe-balanced colourings (mod colour permutation): **343**.
|
||||
- Distinct canonical up-tooth sequences overall: **4**.
|
||||
|
||||
## Distinct canonical up-tooth sequences
|
||||
|
||||
| sequence | colour multiset | #M(T) realising | #colourings |
|
||||
|---|---|---|---|
|
||||
| `0000` | 4 | 27 | 75 |
|
||||
| `0011` | 2+2 | 29 | 140 |
|
||||
| `0101` | 2+2 | 16 | 59 |
|
||||
| `0110` | 2+2 | 22 | 69 |
|
||||
|
||||
Note: every realised sequence has its three colour-counts of **equal parity** — exactly the outer-face Kempe-parity constraint (each colour pair meets the up-tooth apexes an even number of times). With m = 4 up teeth (m is even) every count must be **even**, so the only admissible colour multisets are 2+2, 4.
|
||||
|
||||
## Step 4 — grouping M(T) by their set of unique up-tooth sequences
|
||||
|
||||
The 29 M(T) fall into **4** groups by the set of canonical up-tooth sequences they realise:
|
||||
|
||||
| #M(T) | set of up-tooth sequences | M(T) ids |
|
||||
|---|---|---|
|
||||
| 13 | { `0000`, `0011`, `0110` } | G00, G01, G02, G03, G04, G05, G06, G07, G12, G16, G19, G22, G28 |
|
||||
| 7 | { `0000`, `0011`, `0101` } | G08, G09, G11, G13, G17, G20, G24 |
|
||||
| 7 | { `0000`, `0011`, `0101`, `0110` } | G10, G14, G18, G21, G23, G25, G26 |
|
||||
| 2 | { `0011`, `0101`, `0110` } | G15, G27 |
|
||||
|
||||
## M(T) atlas (ids)
|
||||
|
||||
| id | word / bites | #Kempe-balanced | up-tooth sequence set |
|
||||
|---|---|---|---|
|
||||
| G00 | word=UUUUDDDDD bites=- | 30 | { `0000`, `0011`, `0110` } |
|
||||
| G01 | word=UUUUDDDDD bites=(4,8) | 12 | { `0000`, `0011`, `0110` } |
|
||||
| G02 | word=UUUDUDDDD bites=- | 15 | { `0000`, `0011`, `0110` } |
|
||||
| G03 | word=UUUDUDDDD bites=(3,5) | 6 | { `0000`, `0011`, `0110` } |
|
||||
| G04 | word=UUUDUDDDD bites=(3,8) | 6 | { `0000`, `0011`, `0110` } |
|
||||
| G05 | word=UUUDDUDDD bites=- | 21 | { `0000`, `0011`, `0110` } |
|
||||
| G06 | word=UUUDDUDDD bites=(4,6) | 3 | { `0000`, `0011`, `0110` } |
|
||||
| G07 | word=UUUDDUDDD bites=(3,8) | 3 | { `0000`, `0011`, `0110` } |
|
||||
| G08 | word=UUDUUDDDD bites=- | 25 | { `0000`, `0011`, `0101` } |
|
||||
| G09 | word=UUDUUDDDD bites=(2,5) | 10 | { `0000`, `0011`, `0101` } |
|
||||
| G10 | word=UUDUDUDDD bites=- | 24 | { `0000`, `0011`, `0101`, `0110` } |
|
||||
| G11 | word=UUDUDUDDD bites=(4,6) | 5 | { `0000`, `0011`, `0101` } |
|
||||
| G12 | word=UUDUDUDDD bites=(2,4) | 6 | { `0000`, `0011`, `0110` } |
|
||||
| G13 | word=UUDUDUDDD bites=(2,8) | 5 | { `0000`, `0011`, `0101` } |
|
||||
| G14 | word=UUDUDDUDD bites=- | 22 | { `0000`, `0011`, `0101`, `0110` } |
|
||||
| G15 | word=UUDUDDUDD bites=(5,7) | 6 | { `0011`, `0101`, `0110` } |
|
||||
| G16 | word=UUDUDDUDD bites=(2,4) | 3 | { `0000`, `0011`, `0110` } |
|
||||
| G17 | word=UUDUDDUDD bites=(2,8) | 5 | { `0000`, `0011`, `0101` } |
|
||||
| G18 | word=UUDUDDDUD bites=- | 28 | { `0000`, `0011`, `0101`, `0110` } |
|
||||
| G19 | word=UUDUDDDUD bites=(6,8) | 3 | { `0000`, `0011`, `0110` } |
|
||||
| G20 | word=UUDUDDDUD bites=(2,8) | 10 | { `0000`, `0011`, `0101` } |
|
||||
| G21 | word=UUDDUUDDD bites=- | 27 | { `0000`, `0011`, `0101`, `0110` } |
|
||||
| G22 | word=UUDDUUDDD bites=(3,6) | 9 | { `0000`, `0011`, `0110` } |
|
||||
| G23 | word=UUDDUDUDD bites=- | 18 | { `0000`, `0011`, `0101`, `0110` } |
|
||||
| G24 | word=UUDDUDUDD bites=(5,7) | 5 | { `0000`, `0011`, `0101` } |
|
||||
| G25 | word=UUDDUDUDD bites=(2,8) | 8 | { `0000`, `0011`, `0101`, `0110` } |
|
||||
| G26 | word=UDUDUDUDD bites=- | 19 | { `0000`, `0011`, `0101`, `0110` } |
|
||||
| G27 | word=UDUDUDUDD bites=(5,7) | 6 | { `0011`, `0101`, `0110` } |
|
||||
| G28 | word=UDUDUDUDD bites=(3,5) | 3 | { `0000`, `0011`, `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`
|
||||
|
||||
+150
@@ -0,0 +1,150 @@
|
||||
# Up-tooth apex sequence `00012`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 up teeth**.
|
||||
|
||||
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
|
||||
- Realised by **18** of 18 M(T) (dihedral classes).
|
||||
- **49** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_00012.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G00 — word=UUUUUDDDD bites=-
|
||||
|
||||
6 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010120101 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010120102 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010120202 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010120212 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G01 — word=UUUUDUDDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010120202 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010120212 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G02 — word=UUUUDDUDD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:1 d5:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010120202 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:1 d5:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010121021 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010121201 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010121202 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:0 d7:1 d8:1]`
|
||||
|
||||
## G03 — word=UUUUDDUDD bites=(4,8),(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010120202 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[p4_8:1 p5_7:1]`
|
||||
|
||||
## G04 — word=UUUDUUDDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010102101 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010102102 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:0 d7:0 d8:2]`
|
||||
|
||||
## G05 — word=UUUDUDUDD bites=-
|
||||
|
||||
6 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[d3:2 d5:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010120121 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[d3:0 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010120212 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[d3:0 d5:1 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010121021 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010121201 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010121202 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:0 d7:1 d8:1]`
|
||||
|
||||
## G06 — word=UUUDUDUDD bites=(3,8),(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[p3_8:2 p5_7:0]`
|
||||
|
||||
## G07 — word=UUUDUDDUD bites=-
|
||||
|
||||
6 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010102012 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:1 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010102021 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:1 d6:1 d8:2]`
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:0 d6:0 d8:2]`
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010120121 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:0 d5:2 d6:0 d8:2]`
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010120212 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:0 d5:1 d6:0 d8:1]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010121202 U[u0:2 u1:2 u2:2 u4:0 u7:1] D[d3:0 d5:0 d6:1 d8:1]`
|
||||
|
||||
## G08 — word=UUUDUDDUD bites=(3,5),(6,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010121202 U[u0:2 u1:2 u2:2 u4:0 u7:1] D[p3_5:0 p6_8:1]`
|
||||
|
||||
## G09 — word=UUUDDUUDD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010101201 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:2 d4:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010101202 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:2 d4:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010120212 U[u0:2 u1:2 u2:2 u5:1 u6:0] D[d3:0 d4:1 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010121201 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:0 d4:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010121202 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:0 d4:0 d7:1 d8:1]`
|
||||
|
||||
## G10 — word=UUUDDUUDD bites=(3,8),(4,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010101201 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[p3_8:2 p4_7:2]`
|
||||
|
||||
## G11 — word=UUDUUDUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010102121 U[u0:2 u1:2 u3:2 u4:1 u6:0] D[d2:2 d5:0 d7:0 d8:2]`
|
||||
|
||||
## G12 — word=UUDUUDUDD bites=(2,8),(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010102121 U[u0:2 u1:2 u3:2 u4:1 u6:0] D[p2_8:2 p5_7:0]`
|
||||
|
||||
## G13 — word=UUDUDUUDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010101201 U[u0:2 u1:2 u3:2 u5:0 u6:1] D[d2:2 d4:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010101202 U[u0:2 u1:2 u3:2 u5:0 u6:1] D[d2:2 d4:2 d7:1 d8:1]`
|
||||
|
||||
## G14 — word=UUDUDUUDD bites=(2,8),(4,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010101201 U[u0:2 u1:2 u3:2 u5:0 u6:1] D[p2_8:2 p4_7:2]`
|
||||
|
||||
## G15 — word=UUDUDUDUD bites=-
|
||||
|
||||
4 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010101202 U[u0:2 u1:2 u3:2 u5:0 u7:1] D[d2:2 d4:2 d6:1 d8:1]`
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010102012 U[u0:2 u1:2 u3:2 u5:1 u7:0] D[d2:2 d4:1 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010102021 U[u0:2 u1:2 u3:2 u5:1 u7:0] D[d2:2 d4:1 d6:1 d8:2]`
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010102102 U[u0:2 u1:2 u3:2 u5:0 u7:1] D[d2:2 d4:1 d6:2 d8:1]`
|
||||
|
||||
## G16 — word=UUDUDUDUD bites=(2,4),(6,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22201` → canonical `00012` · `A=010101202 U[u0:2 u1:2 u3:2 u5:0 u7:1] D[p2_4:2 p6_8:1]`
|
||||
|
||||
## G17 — word=UUDUDUDUD bites=(2,8),(4,6)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00012`:
|
||||
|
||||
- up apexes (raw labels) `22210` → canonical `00012` · `A=010102021 U[u0:2 u1:2 u3:2 u5:1 u7:0] D[p2_8:2 p4_6:1]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `00102`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 up teeth**.
|
||||
|
||||
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
|
||||
- Realised by **11** of 18 M(T) (dihedral classes).
|
||||
- **26** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_00102.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G04 — word=UUUDUUDDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `00102`:
|
||||
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010201201 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010201202 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G05 — word=UUUDUDUDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `00102`:
|
||||
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[d3:1 d5:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010210121 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[d3:0 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010210212 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[d3:0 d5:1 d7:0 d8:1]`
|
||||
|
||||
## G06 — word=UUUDUDUDD bites=(3,8),(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00102`:
|
||||
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[p3_8:1 p5_7:0]`
|
||||
|
||||
## G07 — word=UUUDUDDUD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `00102`:
|
||||
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010201012 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:2 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010201021 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:2 d6:1 d8:2]`
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:0 d6:0 d8:1]`
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010210121 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:0 d5:2 d6:0 d8:2]`
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010210212 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:0 d5:1 d6:0 d8:1]`
|
||||
|
||||
## G09 — word=UUUDDUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00102`:
|
||||
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010210121 U[u0:2 u1:2 u2:1 u5:2 u6:0] D[d3:0 d4:2 d7:0 d8:2]`
|
||||
|
||||
## G11 — word=UUDUUDUDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `00102`:
|
||||
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010201212 U[u0:2 u1:2 u3:1 u4:2 u6:0] D[d2:1 d5:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `22021` → canonical `00102` · `A=010210201 U[u0:2 u1:2 u3:0 u4:2 u6:1] D[d2:1 d5:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22021` → canonical `00102` · `A=010210202 U[u0:2 u1:2 u3:0 u4:2 u6:1] D[d2:1 d5:1 d7:1 d8:1]`
|
||||
|
||||
## G12 — word=UUDUUDUDD bites=(2,8),(5,7)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `00102`:
|
||||
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010201212 U[u0:2 u1:2 u3:1 u4:2 u6:0] D[p2_8:1 p5_7:0]`
|
||||
- up apexes (raw labels) `22021` → canonical `00102` · `A=010210202 U[u0:2 u1:2 u3:0 u4:2 u6:1] D[p2_8:1 p5_7:1]`
|
||||
|
||||
## G13 — word=UUDUDUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00102`:
|
||||
|
||||
- up apexes (raw labels) `22021` → canonical `00102` · `A=010121021 U[u0:2 u1:2 u3:0 u5:2 u6:1] D[d2:2 d4:0 d7:0 d8:2]`
|
||||
|
||||
## G14 — word=UUDUDUUDD bites=(2,8),(4,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00102`:
|
||||
|
||||
- up apexes (raw labels) `22021` → canonical `00102` · `A=010121021 U[u0:2 u1:2 u3:0 u5:2 u6:1] D[p2_8:2 p4_7:0]`
|
||||
|
||||
## G15 — word=UUDUDUDUD bites=-
|
||||
|
||||
4 colouring(s) with up-tooth sequence `00102`:
|
||||
|
||||
- up apexes (raw labels) `22021` → canonical `00102` · `A=010120102 U[u0:2 u1:2 u3:0 u5:2 u7:1] D[d2:2 d4:1 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010201012 U[u0:2 u1:2 u3:1 u5:2 u7:0] D[d2:1 d4:2 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010201021 U[u0:2 u1:2 u3:1 u5:2 u7:0] D[d2:1 d4:2 d6:1 d8:2]`
|
||||
- up apexes (raw labels) `22021` → canonical `00102` · `A=010210102 U[u0:2 u1:2 u3:0 u5:2 u7:1] D[d2:1 d4:2 d6:2 d8:1]`
|
||||
|
||||
## G17 — word=UUDUDUDUD bites=(2,8),(4,6)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `00102`:
|
||||
|
||||
- up apexes (raw labels) `22120` → canonical `00102` · `A=010201012 U[u0:2 u1:2 u3:1 u5:2 u7:0] D[p2_8:1 p4_6:2]`
|
||||
- up apexes (raw labels) `22021` → canonical `00102` · `A=010210102 U[u0:2 u1:2 u3:0 u5:2 u7:1] D[p2_8:1 p4_6:2]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `00120`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 up teeth**.
|
||||
|
||||
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
|
||||
- Realised by **16** of 18 M(T) (dihedral classes).
|
||||
- **39** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_00120.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G00 — word=UUUUUDDDD bites=-
|
||||
|
||||
6 colouring(s) with up-tooth sequence `00120`:
|
||||
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010210101 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010210121 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010210201 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010210202 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010210212 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G01 — word=UUUUDUDDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `00120`:
|
||||
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010210101 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010210121 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:0 d7:0 d8:2]`
|
||||
|
||||
## G02 — word=UUUUDDUDD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `00120`:
|
||||
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010210101 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:2 d5:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:2 d5:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010212012 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:1 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010212101 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010212102 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:0 d7:1 d8:1]`
|
||||
|
||||
## G03 — word=UUUUDDUDD bites=(4,8),(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00120`:
|
||||
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010210101 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[p4_8:2 p5_7:2]`
|
||||
|
||||
## G05 — word=UUUDUDUDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `00120`:
|
||||
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010212012 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:1 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010212101 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010212102 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:0 d7:1 d8:1]`
|
||||
|
||||
## G07 — word=UUUDUDDUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00120`:
|
||||
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010212101 U[u0:2 u1:2 u2:1 u4:0 u7:2] D[d3:0 d5:0 d6:2 d8:2]`
|
||||
|
||||
## G08 — word=UUUDUDDUD bites=(3,5),(6,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00120`:
|
||||
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010212101 U[u0:2 u1:2 u2:1 u4:0 u7:2] D[p3_5:0 p6_8:2]`
|
||||
|
||||
## G09 — word=UUUDDUUDD bites=-
|
||||
|
||||
4 colouring(s) with up-tooth sequence `00120`:
|
||||
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010202101 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:1 d4:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010202102 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:1 d4:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010212101 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:0 d4:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010212102 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:0 d4:0 d7:1 d8:1]`
|
||||
|
||||
## G10 — word=UUUDDUUDD bites=(3,8),(4,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00120`:
|
||||
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010202102 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[p3_8:1 p4_7:1]`
|
||||
|
||||
## G11 — word=UUDUUDUDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `00120`:
|
||||
|
||||
- up apexes (raw labels) `22012` → canonical `00120` · `A=010120101 U[u0:2 u1:2 u3:0 u4:1 u6:2] D[d2:2 d5:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22012` → canonical `00120` · `A=010120102 U[u0:2 u1:2 u3:0 u4:1 u6:2] D[d2:2 d5:2 d7:1 d8:1]`
|
||||
|
||||
## G12 — word=UUDUUDUDD bites=(2,8),(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00120`:
|
||||
|
||||
- up apexes (raw labels) `22012` → canonical `00120` · `A=010120101 U[u0:2 u1:2 u3:0 u4:1 u6:2] D[p2_8:2 p5_7:2]`
|
||||
|
||||
## G13 — word=UUDUDUUDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `00120`:
|
||||
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010202101 U[u0:2 u1:2 u3:1 u5:0 u6:2] D[d2:1 d4:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010202102 U[u0:2 u1:2 u3:1 u5:0 u6:2] D[d2:1 d4:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `22012` → canonical `00120` · `A=010212012 U[u0:2 u1:2 u3:0 u5:1 u6:2] D[d2:1 d4:0 d7:0 d8:1]`
|
||||
|
||||
## G14 — word=UUDUDUUDD bites=(2,8),(4,7)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `00120`:
|
||||
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010202102 U[u0:2 u1:2 u3:1 u5:0 u6:2] D[p2_8:1 p4_7:1]`
|
||||
- up apexes (raw labels) `22012` → canonical `00120` · `A=010212012 U[u0:2 u1:2 u3:0 u5:1 u6:2] D[p2_8:1 p4_7:0]`
|
||||
|
||||
## G15 — word=UUDUDUDUD bites=-
|
||||
|
||||
4 colouring(s) with up-tooth sequence `00120`:
|
||||
|
||||
- up apexes (raw labels) `22012` → canonical `00120` · `A=010120201 U[u0:2 u1:2 u3:0 u5:1 u7:2] D[d2:2 d4:1 d6:1 d8:2]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010201201 U[u0:2 u1:2 u3:1 u5:0 u7:2] D[d2:1 d4:2 d6:1 d8:2]`
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010202101 U[u0:2 u1:2 u3:1 u5:0 u7:2] D[d2:1 d4:1 d6:2 d8:2]`
|
||||
- up apexes (raw labels) `22012` → canonical `00120` · `A=010210201 U[u0:2 u1:2 u3:0 u5:1 u7:2] D[d2:1 d4:2 d6:1 d8:2]`
|
||||
|
||||
## G16 — word=UUDUDUDUD bites=(2,4),(6,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00120`:
|
||||
|
||||
- up apexes (raw labels) `22102` → canonical `00120` · `A=010202101 U[u0:2 u1:2 u3:1 u5:0 u7:2] D[p2_4:1 p6_8:2]`
|
||||
|
||||
## G17 — word=UUDUDUDUD bites=(2,8),(4,6)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `00120`:
|
||||
|
||||
- up apexes (raw labels) `22012` → canonical `00120` · `A=010120201 U[u0:2 u1:2 u3:0 u5:1 u7:2] D[p2_8:2 p4_6:1]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `01002`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 up teeth**.
|
||||
|
||||
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
|
||||
- Realised by **5** of 18 M(T) (dihedral classes).
|
||||
- **9** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_01002.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G11 — word=UUDUUDUDD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `01002`:
|
||||
|
||||
- up apexes (raw labels) `20221` → canonical `01002` · `A=012010201 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:1 d5:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20221` → canonical `01002` · `A=012010202 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:1 d5:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20221` → canonical `01002` · `A=012101021 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `20221` → canonical `01002` · `A=012101201 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20221` → canonical `01002` · `A=012101202 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:0 d7:1 d8:1]`
|
||||
|
||||
## G12 — word=UUDUUDUDD bites=(2,8),(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01002`:
|
||||
|
||||
- up apexes (raw labels) `20221` → canonical `01002` · `A=012010202 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[p2_8:1 p5_7:1]`
|
||||
|
||||
## G13 — word=UUDUDUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01002`:
|
||||
|
||||
- up apexes (raw labels) `20221` → canonical `01002` · `A=012101021 U[u0:2 u1:0 u3:2 u5:2 u6:1] D[d2:0 d4:2 d7:0 d8:2]`
|
||||
|
||||
## G15 — word=UUDUDUDUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01002`:
|
||||
|
||||
- up apexes (raw labels) `20221` → canonical `01002` · `A=012010102 U[u0:2 u1:0 u3:2 u5:2 u7:1] D[d2:1 d4:2 d6:2 d8:1]`
|
||||
|
||||
## G17 — word=UUDUDUDUD bites=(2,8),(4,6)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01002`:
|
||||
|
||||
- up apexes (raw labels) `20221` → canonical `01002` · `A=012010102 U[u0:2 u1:0 u3:2 u5:2 u7:1] D[p2_8:1 p4_6:2]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `01020`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 up teeth**.
|
||||
|
||||
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
|
||||
- Realised by **4** of 18 M(T) (dihedral classes).
|
||||
- **7** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_01020.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G11 — word=UUDUUDUDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `01020`:
|
||||
|
||||
- up apexes (raw labels) `20212` → canonical `01020` · `A=012102012 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:1 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `20212` → canonical `01020` · `A=012102101 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20212` → canonical `01020` · `A=012102102 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:0 d7:1 d8:1]`
|
||||
|
||||
## G13 — word=UUDUDUUDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `01020`:
|
||||
|
||||
- up apexes (raw labels) `20212` → canonical `01020` · `A=012012012 U[u0:2 u1:0 u3:2 u5:1 u6:2] D[d2:1 d4:0 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `20212` → canonical `01020` · `A=012102012 U[u0:2 u1:0 u3:2 u5:1 u6:2] D[d2:0 d4:1 d7:0 d8:1]`
|
||||
|
||||
## G14 — word=UUDUDUUDD bites=(2,8),(4,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01020`:
|
||||
|
||||
- up apexes (raw labels) `20212` → canonical `01020` · `A=012012012 U[u0:2 u1:0 u3:2 u5:1 u6:2] D[p2_8:1 p4_7:0]`
|
||||
|
||||
## G15 — word=UUDUDUDUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01020`:
|
||||
|
||||
- up apexes (raw labels) `20212` → canonical `01020` · `A=012010201 U[u0:2 u1:0 u3:2 u5:1 u7:2] D[d2:1 d4:2 d6:1 d8:2]`
|
||||
|
||||
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+108
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|
||||
# Up-tooth apex sequence `01112`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 up teeth**.
|
||||
|
||||
- Colour multiset: 1×colour0, 3×colour1, 1×colour2.
|
||||
- Realised by **13** of 18 M(T) (dihedral classes).
|
||||
- **32** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_01112.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G00 — word=UUUUUDDDD bites=-
|
||||
|
||||
6 colouring(s) with up-tooth sequence `01112`:
|
||||
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012120101 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012120102 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012120121 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012120202 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012120212 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G01 — word=UUUUDUDDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `01112`:
|
||||
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012120202 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012120212 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G02 — word=UUUUDDUDD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `01112`:
|
||||
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:1 d5:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012120202 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:1 d5:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012121021 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012121201 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012121202 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:0 d7:1 d8:1]`
|
||||
|
||||
## G03 — word=UUUUDDUDD bites=(4,8),(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01112`:
|
||||
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012120202 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[p4_8:1 p5_7:1]`
|
||||
|
||||
## G05 — word=UUUDUDUDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `01112`:
|
||||
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012121021 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012121201 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012121202 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:0 d7:1 d8:1]`
|
||||
|
||||
## G07 — word=UUUDUDDUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01112`:
|
||||
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012121202 U[u0:2 u1:0 u2:0 u4:0 u7:1] D[d3:0 d5:0 d6:1 d8:1]`
|
||||
|
||||
## G08 — word=UUUDUDDUD bites=(3,5),(6,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01112`:
|
||||
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012121202 U[u0:2 u1:0 u2:0 u4:0 u7:1] D[p3_5:0 p6_8:1]`
|
||||
|
||||
## G09 — word=UUUDDUUDD bites=-
|
||||
|
||||
4 colouring(s) with up-tooth sequence `01112`:
|
||||
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012101201 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:2 d4:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012101202 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:2 d4:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012121201 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:0 d4:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012121202 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:0 d4:0 d7:1 d8:1]`
|
||||
|
||||
## G10 — word=UUUDDUUDD bites=(3,8),(4,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01112`:
|
||||
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012101201 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[p3_8:2 p4_7:2]`
|
||||
|
||||
## G11 — word=UUDUUDUDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `01112`:
|
||||
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012121021 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012121201 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012121202 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:0 d7:1 d8:1]`
|
||||
|
||||
## G13 — word=UUDUDUUDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `01112`:
|
||||
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012121201 U[u0:2 u1:0 u3:0 u5:0 u6:1] D[d2:0 d4:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012121202 U[u0:2 u1:0 u3:0 u5:0 u6:1] D[d2:0 d4:0 d7:1 d8:1]`
|
||||
|
||||
## G15 — word=UUDUDUDUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01112`:
|
||||
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012121202 U[u0:2 u1:0 u3:0 u5:0 u7:1] D[d2:0 d4:0 d6:1 d8:1]`
|
||||
|
||||
## G16 — word=UUDUDUDUD bites=(2,4),(6,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01112`:
|
||||
|
||||
- up apexes (raw labels) `20001` → canonical `01112` · `A=012121202 U[u0:2 u1:0 u3:0 u5:0 u7:1] D[p2_4:0 p6_8:1]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `01121`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 up teeth**.
|
||||
|
||||
- Colour multiset: 1×colour0, 3×colour1, 1×colour2.
|
||||
- Realised by **8** of 18 M(T) (dihedral classes).
|
||||
- **17** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_01121.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G04 — word=UUUDUUDDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `01121`:
|
||||
|
||||
- up apexes (raw labels) `20010` → canonical `01121` · `A=012102101 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20010` → canonical `01121` · `A=012102102 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20010` → canonical `01121` · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:0 d7:0 d8:2]`
|
||||
|
||||
## G05 — word=UUUDUDUDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `01121`:
|
||||
|
||||
- up apexes (raw labels) `20010` → canonical `01121` · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:2 d5:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `20010` → canonical `01121` · `A=012120121 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:0 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `20010` → canonical `01121` · `A=012120212 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:0 d5:1 d7:0 d8:1]`
|
||||
|
||||
## G06 — word=UUUDUDUDD bites=(3,8),(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01121`:
|
||||
|
||||
- up apexes (raw labels) `20010` → canonical `01121` · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[p3_8:2 p5_7:0]`
|
||||
|
||||
## G07 — word=UUUDUDDUD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `01121`:
|
||||
|
||||
- up apexes (raw labels) `20010` → canonical `01121` · `A=012102012 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:1 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `20010` → canonical `01121` · `A=012102021 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:1 d6:1 d8:2]`
|
||||
- up apexes (raw labels) `20010` → canonical `01121` · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:0 d6:0 d8:2]`
|
||||
- up apexes (raw labels) `20010` → canonical `01121` · `A=012120121 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:0 d5:2 d6:0 d8:2]`
|
||||
- up apexes (raw labels) `20010` → canonical `01121` · `A=012120212 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:0 d5:1 d6:0 d8:1]`
|
||||
|
||||
## G09 — word=UUUDDUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01121`:
|
||||
|
||||
- up apexes (raw labels) `20010` → canonical `01121` · `A=012120212 U[u0:2 u1:0 u2:0 u5:1 u6:0] D[d3:0 d4:1 d7:0 d8:1]`
|
||||
|
||||
## G11 — word=UUDUUDUDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `01121`:
|
||||
|
||||
- up apexes (raw labels) `20010` → canonical `01121` · `A=012120121 U[u0:2 u1:0 u3:0 u4:1 u6:0] D[d2:0 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `20010` → canonical `01121` · `A=012120212 U[u0:2 u1:0 u3:0 u4:1 u6:0] D[d2:0 d5:1 d7:0 d8:1]`
|
||||
|
||||
## G13 — word=UUDUDUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01121`:
|
||||
|
||||
- up apexes (raw labels) `20010` → canonical `01121` · `A=012120212 U[u0:2 u1:0 u3:0 u5:1 u6:0] D[d2:0 d4:1 d7:0 d8:1]`
|
||||
|
||||
## G15 — word=UUDUDUDUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01121`:
|
||||
|
||||
- up apexes (raw labels) `20010` → canonical `01121` · `A=012120212 U[u0:2 u1:0 u3:0 u5:1 u7:0] D[d2:0 d4:1 d6:0 d8:1]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `01200`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 up teeth**.
|
||||
|
||||
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
|
||||
- Realised by **15** of 18 M(T) (dihedral classes).
|
||||
- **39** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_01200.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G00 — word=UUUUUDDDD bites=-
|
||||
|
||||
6 colouring(s) with up-tooth sequence `01200`:
|
||||
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010121 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010201 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010202 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010212 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G01 — word=UUUUDUDDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `01200`:
|
||||
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010121 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:0 d7:0 d8:2]`
|
||||
|
||||
## G02 — word=UUUUDDUDD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `01200`:
|
||||
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:2 d5:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:2 d5:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012012012 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:1 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012012101 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012012102 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:0 d7:1 d8:1]`
|
||||
|
||||
## G03 — word=UUUUDDUDD bites=(4,8),(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01200`:
|
||||
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[p4_8:2 p5_7:2]`
|
||||
|
||||
## G04 — word=UUUDUUDDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `01200`:
|
||||
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010102 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010121 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:0 d7:0 d8:2]`
|
||||
|
||||
## G05 — word=UUUDUDUDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `01200`:
|
||||
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u6:2] D[d3:2 d5:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010102 U[u0:2 u1:0 u2:1 u4:2 u6:2] D[d3:2 d5:2 d7:1 d8:1]`
|
||||
|
||||
## G06 — word=UUUDUDUDD bites=(3,8),(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01200`:
|
||||
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u6:2] D[p3_8:2 p5_7:2]`
|
||||
|
||||
## G07 — word=UUUDUDDUD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `01200`:
|
||||
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u7:2] D[d3:2 d5:2 d6:2 d8:2]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010201 U[u0:2 u1:0 u2:1 u4:2 u7:2] D[d3:2 d5:1 d6:1 d8:2]`
|
||||
|
||||
## G08 — word=UUUDUDDUD bites=(3,5),(6,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01200`:
|
||||
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u7:2] D[p3_5:2 p6_8:2]`
|
||||
|
||||
## G09 — word=UUUDDUUDD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `01200`:
|
||||
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:2 d4:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010102 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:2 d4:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012020101 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012020102 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012021012 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:0 d7:0 d8:1]`
|
||||
|
||||
## G10 — word=UUUDDUUDD bites=(3,8),(4,7)
|
||||
|
||||
3 colouring(s) with up-tooth sequence `01200`:
|
||||
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[p3_8:2 p4_7:2]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012020102 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[p3_8:1 p4_7:1]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012021012 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[p3_8:1 p4_7:0]`
|
||||
|
||||
## G13 — word=UUDUDUUDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `01200`:
|
||||
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012020101 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012020102 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012021012 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:0 d7:0 d8:1]`
|
||||
|
||||
## G14 — word=UUDUDUUDD bites=(2,8),(4,7)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `01200`:
|
||||
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012020102 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[p2_8:1 p4_7:1]`
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012021012 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[p2_8:1 p4_7:0]`
|
||||
|
||||
## G15 — word=UUDUDUDUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01200`:
|
||||
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012020101 U[u0:2 u1:0 u3:1 u5:2 u7:2] D[d2:1 d4:1 d6:2 d8:2]`
|
||||
|
||||
## G16 — word=UUDUDUDUD bites=(2,4),(6,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01200`:
|
||||
|
||||
- up apexes (raw labels) `20122` → canonical `01200` · `A=012020101 U[u0:2 u1:0 u3:1 u5:2 u7:2] D[p2_4:1 p6_8:2]`
|
||||
|
||||
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|
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+97
@@ -0,0 +1,97 @@
|
||||
# Up-tooth apex sequence `01211`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 up teeth**.
|
||||
|
||||
- Colour multiset: 1×colour0, 3×colour1, 1×colour2.
|
||||
- Realised by **12** of 18 M(T) (dihedral classes).
|
||||
- **26** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_01211.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G04 — word=UUUDUUDDD bites=-
|
||||
|
||||
6 colouring(s) with up-tooth sequence `01211`:
|
||||
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012012101 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012012102 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012021201 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012021202 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G05 — word=UUUDUDUDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `01211`:
|
||||
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[d3:2 d5:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[d3:1 d5:0 d7:0 d8:1]`
|
||||
|
||||
## G06 — word=UUUDUDUDD bites=(3,8),(5,7)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `01211`:
|
||||
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[p3_8:2 p5_7:0]`
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[p3_8:1 p5_7:0]`
|
||||
|
||||
## G07 — word=UUUDUDDUD bites=-
|
||||
|
||||
6 colouring(s) with up-tooth sequence `01211`:
|
||||
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012012012 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:1 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012012021 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:1 d6:1 d8:2]`
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:0 d6:0 d8:2]`
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012021012 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:2 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012021021 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:2 d6:1 d8:2]`
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:0 d6:0 d8:1]`
|
||||
|
||||
## G09 — word=UUUDDUUDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `01211`:
|
||||
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012012121 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[d3:2 d4:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[d3:1 d4:0 d7:0 d8:1]`
|
||||
|
||||
## G10 — word=UUUDDUUDD bites=(3,8),(4,7)
|
||||
|
||||
2 colouring(s) with up-tooth sequence `01211`:
|
||||
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012012121 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[p3_8:2 p4_7:0]`
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[p3_8:1 p4_7:0]`
|
||||
|
||||
## G11 — word=UUDUUDUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01211`:
|
||||
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u3:1 u4:0 u6:0] D[d2:1 d5:0 d7:0 d8:1]`
|
||||
|
||||
## G12 — word=UUDUUDUDD bites=(2,8),(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01211`:
|
||||
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u3:1 u4:0 u6:0] D[p2_8:1 p5_7:0]`
|
||||
|
||||
## G13 — word=UUDUDUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01211`:
|
||||
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u6:0] D[d2:1 d4:0 d7:0 d8:1]`
|
||||
|
||||
## G14 — word=UUDUDUUDD bites=(2,8),(4,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01211`:
|
||||
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u6:0] D[p2_8:1 p4_7:0]`
|
||||
|
||||
## G15 — word=UUDUDUDUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01211`:
|
||||
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u7:0] D[d2:1 d4:0 d6:0 d8:1]`
|
||||
|
||||
## G17 — word=UUDUDUDUD bites=(2,8),(4,6)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01211`:
|
||||
|
||||
- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u7:0] D[p2_8:1 p4_6:0]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `01222`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 up teeth**.
|
||||
|
||||
- Colour multiset: 1×colour0, 1×colour1, 3×colour2.
|
||||
- Realised by **18** of 18 M(T) (dihedral classes).
|
||||
- **41** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_01222.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G00 — word=UUUUUDDDD bites=-
|
||||
|
||||
6 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020101 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020102 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020121 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020212 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G01 — word=UUUUDUDDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020212 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G02 — word=UUUUDDUDD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:1 d5:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:1 d5:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012021021 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:2 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012021201 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:0 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012021202 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:0 d7:1 d8:1]`
|
||||
|
||||
## G03 — word=UUUUDDUDD bites=(4,8),(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[p4_8:1 p5_7:1]`
|
||||
|
||||
## G04 — word=UUUDUUDDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020201 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:1 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020212 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:0 d7:0 d8:1]`
|
||||
|
||||
## G05 — word=UUUDUDUDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020201 U[u0:2 u1:0 u2:1 u4:1 u6:1] D[d3:1 d5:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u2:1 u4:1 u6:1] D[d3:1 d5:1 d7:1 d8:1]`
|
||||
|
||||
## G06 — word=UUUDUDUDD bites=(3,8),(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u2:1 u4:1 u6:1] D[p3_8:1 p5_7:1]`
|
||||
|
||||
## G07 — word=UUUDUDDUD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020102 U[u0:2 u1:0 u2:1 u4:1 u7:1] D[d3:1 d5:2 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u2:1 u4:1 u7:1] D[d3:1 d5:1 d6:1 d8:1]`
|
||||
|
||||
## G08 — word=UUUDUDDUD bites=(3,5),(6,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u2:1 u4:1 u7:1] D[p3_5:1 p6_8:1]`
|
||||
|
||||
## G09 — word=UUUDDUUDD bites=-
|
||||
|
||||
5 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012010201 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:2 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012010202 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:2 d7:1 d8:1]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012012021 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:0 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020201 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:1 d4:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:1 d4:1 d7:1 d8:1]`
|
||||
|
||||
## G10 — word=UUUDDUUDD bites=(3,8),(4,7)
|
||||
|
||||
3 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012010201 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[p3_8:2 p4_7:2]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012012021 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[p3_8:2 p4_7:0]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[p3_8:1 p4_7:1]`
|
||||
|
||||
## G11 — word=UUDUUDUDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020201 U[u0:2 u1:0 u3:1 u4:1 u6:1] D[d2:1 d5:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u3:1 u4:1 u6:1] D[d2:1 d5:1 d7:1 d8:1]`
|
||||
|
||||
## G12 — word=UUDUUDUDD bites=(2,8),(5,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u3:1 u4:1 u6:1] D[p2_8:1 p5_7:1]`
|
||||
|
||||
## G13 — word=UUDUDUUDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020201 U[u0:2 u1:0 u3:1 u5:1 u6:1] D[d2:1 d4:1 d7:2 d8:2]`
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u3:1 u5:1 u6:1] D[d2:1 d4:1 d7:1 d8:1]`
|
||||
|
||||
## G14 — word=UUDUDUUDD bites=(2,8),(4,7)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u3:1 u5:1 u6:1] D[p2_8:1 p4_7:1]`
|
||||
|
||||
## G15 — word=UUDUDUDUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u3:1 u5:1 u7:1] D[d2:1 d4:1 d6:1 d8:1]`
|
||||
|
||||
## G16 — word=UUDUDUDUD bites=(2,4),(6,8)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u3:1 u5:1 u7:1] D[p2_4:1 p6_8:1]`
|
||||
|
||||
## G17 — word=UUDUDUDUD bites=(2,8),(4,6)
|
||||
|
||||
1 colouring(s) with up-tooth sequence `01222`:
|
||||
|
||||
- up apexes (raw labels) `20111` → canonical `01222` · `A=012020202 U[u0:2 u1:0 u3:1 u5:1 u7:1] D[p2_8:1 p4_6:1]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequences of Kempe-balanced colourings (n=9, m=5)
|
||||
|
||||
Every full medial tire graph M(T) with |A(T)| = 9 and exactly 5 up teeth, one representative per dihedral class: **18 M(T)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the up-tooth apex colour sequence u0<u1<... in cyclic order, and reduce it modulo colour permutation (NOT dihedral symmetry).
|
||||
|
||||
- Total Kempe-balanced colourings (mod colour permutation): **285**.
|
||||
- Distinct canonical up-tooth sequences overall: **10**.
|
||||
|
||||
## Distinct canonical up-tooth sequences
|
||||
|
||||
| sequence | colour multiset | #M(T) realising | #colourings |
|
||||
|---|---|---|---|
|
||||
| `00012` | 3+1+1 | 18 | 49 |
|
||||
| `00102` | 3+1+1 | 11 | 26 |
|
||||
| `00120` | 3+1+1 | 16 | 39 |
|
||||
| `01002` | 3+1+1 | 5 | 9 |
|
||||
| `01020` | 3+1+1 | 4 | 7 |
|
||||
| `01112` | 3+1+1 | 13 | 32 |
|
||||
| `01121` | 3+1+1 | 8 | 17 |
|
||||
| `01200` | 3+1+1 | 15 | 39 |
|
||||
| `01211` | 3+1+1 | 12 | 26 |
|
||||
| `01222` | 3+1+1 | 18 | 41 |
|
||||
|
||||
Note: every realised sequence has its three colour-counts of **equal parity** — exactly the outer-face Kempe-parity constraint (each colour pair meets the up-tooth apexes an even number of times). With m = 5 up teeth (m is odd) every count must be **odd**, so the only admissible colour multisets are 3+1+1.
|
||||
|
||||
## Step 4 — grouping M(T) by their set of unique up-tooth sequences
|
||||
|
||||
The 18 M(T) fall into **8** groups by the set of canonical up-tooth sequences they realise:
|
||||
|
||||
| #M(T) | set of up-tooth sequences | M(T) ids |
|
||||
|---|---|---|
|
||||
| 6 | { `00012`, `00120`, `01112`, `01200`, `01222` } | G00, G01, G02, G03, G08, G16 |
|
||||
| 3 | { `00012`, `00102`, `00120`, `01112`, `01121`, `01200`, `01211`, `01222` } | G05, G07, G09 |
|
||||
| 2 | { `00012`, `00102`, `01121`, `01200`, `01211`, `01222` } | G04, G06 |
|
||||
| 2 | { `00012`, `00102`, `00120`, `01002`, `01211`, `01222` } | G12, G17 |
|
||||
| 2 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } | G13, G15 |
|
||||
| 1 | { `00012`, `00120`, `01112`, `01200`, `01211`, `01222` } | G10 |
|
||||
| 1 | { `00012`, `00102`, `00120`, `01020`, `01200`, `01211`, `01222` } | G14 |
|
||||
| 1 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01211`, `01222` } | G11 |
|
||||
|
||||
## M(T) atlas (ids)
|
||||
|
||||
| id | word / bites | #Kempe-balanced | up-tooth sequence set |
|
||||
|---|---|---|---|
|
||||
| G00 | word=UUUUUDDDD bites=- | 30 | { `00012`, `00120`, `01112`, `01200`, `01222` } |
|
||||
| G01 | word=UUUUDUDDD bites=- | 15 | { `00012`, `00120`, `01112`, `01200`, `01222` } |
|
||||
| G02 | word=UUUUDDUDD bites=- | 25 | { `00012`, `00120`, `01112`, `01200`, `01222` } |
|
||||
| G03 | word=UUUUDDUDD bites=(4,8),(5,7) | 5 | { `00012`, `00120`, `01112`, `01200`, `01222` } |
|
||||
| G04 | word=UUUDUUDDD bites=- | 21 | { `00012`, `00102`, `01121`, `01200`, `01211`, `01222` } |
|
||||
| G05 | word=UUUDUDUDD bites=- | 24 | { `00012`, `00102`, `00120`, `01112`, `01121`, `01200`, `01211`, `01222` } |
|
||||
| G06 | word=UUUDUDUDD bites=(3,8),(5,7) | 7 | { `00012`, `00102`, `01121`, `01200`, `01211`, `01222` } |
|
||||
| G07 | word=UUUDUDDUD bites=- | 28 | { `00012`, `00102`, `00120`, `01112`, `01121`, `01200`, `01211`, `01222` } |
|
||||
| G08 | word=UUUDUDDUD bites=(3,5),(6,8) | 5 | { `00012`, `00120`, `01112`, `01200`, `01222` } |
|
||||
| G09 | word=UUUDDUUDD bites=- | 27 | { `00012`, `00102`, `00120`, `01112`, `01121`, `01200`, `01211`, `01222` } |
|
||||
| G10 | word=UUUDDUUDD bites=(3,8),(4,7) | 11 | { `00012`, `00120`, `01112`, `01200`, `01211`, `01222` } |
|
||||
| G11 | word=UUDUUDUDD bites=- | 22 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01211`, `01222` } |
|
||||
| G12 | word=UUDUUDUDD bites=(2,8),(5,7) | 7 | { `00012`, `00102`, `00120`, `01002`, `01211`, `01222` } |
|
||||
| G13 | word=UUDUDUUDD bites=- | 18 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } |
|
||||
| G14 | word=UUDUDUUDD bites=(2,8),(4,7) | 9 | { `00012`, `00102`, `00120`, `01020`, `01200`, `01211`, `01222` } |
|
||||
| G15 | word=UUDUDUDUD bites=- | 19 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } |
|
||||
| G16 | word=UUDUDUDUD bites=(2,4),(6,8) | 5 | { `00012`, `00120`, `01112`, `01200`, `01222` } |
|
||||
| G17 | word=UUDUDUDUD bites=(2,8),(4,6) | 7 | { `00012`, `00102`, `00120`, `01002`, `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`
|
||||
|
||||
+36
@@ -0,0 +1,36 @@
|
||||
# Up-tooth apex sequence `000000`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**.
|
||||
|
||||
- Colour multiset: 6×colour0.
|
||||
- Realised by **4** of 7 M(T) (dihedral classes).
|
||||
- **5** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_000000.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G00 — word=UUUUUUDDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `000000`:
|
||||
|
||||
- up apexes (raw labels) `222222` → canonical `000000` · `A=010101012 U[u0:2 u1:2 u2:2 u3:2 u4:2 u5:2] D[d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `222222` → canonical `000000` · `A=010101021 U[u0:2 u1:2 u2:2 u3:2 u4:2 u5:2] D[d6:1 d7:0 d8:2]`
|
||||
|
||||
## G01 — word=UUUUUDUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `000000`:
|
||||
|
||||
- up apexes (raw labels) `222222` → canonical `000000` · `A=010101012 U[u0:2 u1:2 u2:2 u3:2 u4:2 u6:2] D[d5:2 d7:0 d8:1]`
|
||||
|
||||
## G02 — word=UUUUDUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `000000`:
|
||||
|
||||
- up apexes (raw labels) `222222` → canonical `000000` · `A=010101012 U[u0:2 u1:2 u2:2 u3:2 u5:2 u6:2] D[d4:2 d7:0 d8:1]`
|
||||
|
||||
## G04 — word=UUUDUUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `000000`:
|
||||
|
||||
- up apexes (raw labels) `222222` → canonical `000000` · `A=010101012 U[u0:2 u1:2 u2:2 u4:2 u5:2 u6:2] D[d3:2 d7:0 d8:1]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `000011`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **7** of 7 M(T) (dihedral classes).
|
||||
- **12** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_000011.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G00 — word=UUUUUUDDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `000011`:
|
||||
|
||||
- up apexes (raw labels) `222211` → canonical `000011` · `A=010102012 U[u0:2 u1:2 u2:2 u3:2 u4:1 u5:1] D[d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `222211` → canonical `000011` · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u4:1 u5:1] D[d6:1 d7:0 d8:2]`
|
||||
|
||||
## G01 — word=UUUUUDUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `000011`:
|
||||
|
||||
- up apexes (raw labels) `222211` → canonical `000011` · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u4:1 u6:1] D[d5:1 d7:0 d8:2]`
|
||||
|
||||
## G02 — word=UUUUDUUDD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `000011`:
|
||||
|
||||
- up apexes (raw labels) `222200` → canonical `000011` · `A=010101212 U[u0:2 u1:2 u2:2 u3:2 u5:0 u6:0] D[d4:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `222211` → canonical `000011` · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u5:1 u6:1] D[d4:1 d7:0 d8:2]`
|
||||
- up apexes (raw labels) `222200` → canonical `000011` · `A=010102121 U[u0:2 u1:2 u2:2 u3:2 u5:0 u6:0] D[d4:1 d7:0 d8:2]`
|
||||
|
||||
## G03 — word=UUUUDUDUD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `000011`:
|
||||
|
||||
- up apexes (raw labels) `222200` → canonical `000011` · `A=010101212 U[u0:2 u1:2 u2:2 u3:2 u5:0 u7:0] D[d4:2 d6:0 d8:1]`
|
||||
- up apexes (raw labels) `222200` → canonical `000011` · `A=010102121 U[u0:2 u1:2 u2:2 u3:2 u5:0 u7:0] D[d4:1 d6:0 d8:2]`
|
||||
|
||||
## G04 — word=UUUDUUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `000011`:
|
||||
|
||||
- up apexes (raw labels) `222200` → canonical `000011` · `A=010101212 U[u0:2 u1:2 u2:2 u4:2 u5:0 u6:0] D[d3:2 d7:0 d8:1]`
|
||||
|
||||
## G05 — word=UUUDUUDUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `000011`:
|
||||
|
||||
- up apexes (raw labels) `222200` → canonical `000011` · `A=010101212 U[u0:2 u1:2 u2:2 u4:2 u5:0 u7:0] D[d3:2 d6:0 d8:1]`
|
||||
|
||||
## G06 — word=UUDUUDUUD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `000011`:
|
||||
|
||||
- up apexes (raw labels) `222211` → canonical `000011` · `A=010101202 U[u0:2 u1:2 u3:2 u4:2 u6:1 u7:1] D[d2:2 d5:0 d8:1]`
|
||||
- up apexes (raw labels) `222200` → canonical `000011` · `A=010101212 U[u0:2 u1:2 u3:2 u4:2 u6:0 u7:0] D[d2:2 d5:0 d8:1]`
|
||||
|
||||
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+39
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|
||||
# Up-tooth apex sequence `000101`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **4** of 7 M(T) (dihedral classes).
|
||||
- **8** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_000101.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G02 — word=UUUUDUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `000101`:
|
||||
|
||||
- up apexes (raw labels) `222020` → canonical `000101` · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u5:2 u6:0] D[d4:1 d7:0 d8:2]`
|
||||
|
||||
## G03 — word=UUUUDUDUD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `000101`:
|
||||
|
||||
- up apexes (raw labels) `222020` → canonical `000101` · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:1 d6:0 d8:2]`
|
||||
- up apexes (raw labels) `222020` → canonical `000101` · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:0 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `222020` → canonical `000101` · `A=010121021 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:0 d6:1 d8:2]`
|
||||
|
||||
## G05 — word=UUUDUUDUD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `000101`:
|
||||
|
||||
- up apexes (raw labels) `222121` → canonical `000101` · `A=010120102 U[u0:2 u1:2 u2:2 u4:1 u5:2 u7:1] D[d3:0 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `222020` → canonical `000101` · `A=010121012 U[u0:2 u1:2 u2:2 u4:0 u5:2 u7:0] D[d3:0 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `222020` → canonical `000101` · `A=010121021 U[u0:2 u1:2 u2:2 u4:0 u5:2 u7:0] D[d3:0 d6:1 d8:2]`
|
||||
|
||||
## G06 — word=UUDUUDUUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `000101`:
|
||||
|
||||
- up apexes (raw labels) `222121` → canonical `000101` · `A=010102102 U[u0:2 u1:2 u3:2 u4:1 u6:2 u7:1] D[d2:2 d5:0 d8:1]`
|
||||
|
||||
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+43
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|
||||
# Up-tooth apex sequence `000110`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **5** of 7 M(T) (dihedral classes).
|
||||
- **7** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_000110.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G00 — word=UUUUUUDDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `000110`:
|
||||
|
||||
- up apexes (raw labels) `222002` → canonical `000110` · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u4:0 u5:2] D[d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `222002` → canonical `000110` · `A=010121021 U[u0:2 u1:2 u2:2 u3:0 u4:0 u5:2] D[d6:1 d7:0 d8:2]`
|
||||
|
||||
## G01 — word=UUUUUDUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `000110`:
|
||||
|
||||
- up apexes (raw labels) `222002` → canonical `000110` · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u4:0 u6:2] D[d5:2 d7:0 d8:1]`
|
||||
|
||||
## G03 — word=UUUUDUDUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `000110`:
|
||||
|
||||
- up apexes (raw labels) `222002` → canonical `000110` · `A=010121201 U[u0:2 u1:2 u2:2 u3:0 u5:0 u7:2] D[d4:0 d6:1 d8:2]`
|
||||
|
||||
## G04 — word=UUUDUUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `000110`:
|
||||
|
||||
- up apexes (raw labels) `222112` → canonical `000110` · `A=010102012 U[u0:2 u1:2 u2:2 u4:1 u5:1 u6:2] D[d3:2 d7:0 d8:1]`
|
||||
|
||||
## G05 — word=UUUDUUDUD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `000110`:
|
||||
|
||||
- up apexes (raw labels) `222112` → canonical `000110` · `A=010120201 U[u0:2 u1:2 u2:2 u4:1 u5:1 u7:2] D[d3:0 d6:1 d8:2]`
|
||||
- up apexes (raw labels) `222002` → canonical `000110` · `A=010121201 U[u0:2 u1:2 u2:2 u4:0 u5:0 u7:2] D[d3:0 d6:1 d8:2]`
|
||||
|
||||
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+23
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|
||||
# Up-tooth apex sequence `001001`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **2** of 7 M(T) (dihedral classes).
|
||||
- **2** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_001001.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G04 — word=UUUDUUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `001001`:
|
||||
|
||||
- up apexes (raw labels) `221221` → canonical `001001` · `A=010201021 U[u0:2 u1:2 u2:1 u4:2 u5:2 u6:1] D[d3:1 d7:0 d8:2]`
|
||||
|
||||
## G05 — word=UUUDUUDUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `001001`:
|
||||
|
||||
- up apexes (raw labels) `221221` → canonical `001001` · `A=010210102 U[u0:2 u1:2 u2:1 u4:2 u5:2 u7:1] D[d3:0 d6:2 d8:1]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `001010`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **2** of 7 M(T) (dihedral classes).
|
||||
- **2** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_001010.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G05 — word=UUUDUUDUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `001010`:
|
||||
|
||||
- up apexes (raw labels) `221212` → canonical `001010` · `A=010210201 U[u0:2 u1:2 u2:1 u4:2 u5:1 u7:2] D[d3:0 d6:1 d8:2]`
|
||||
|
||||
## G06 — word=UUDUUDUUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `001010`:
|
||||
|
||||
- up apexes (raw labels) `221212` → canonical `001010` · `A=010201201 U[u0:2 u1:2 u3:1 u4:2 u6:1 u7:2] D[d2:1 d5:0 d8:2]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `001100`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **4** of 7 M(T) (dihedral classes).
|
||||
- **6** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_001100.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G00 — word=UUUUUUDDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `001100`:
|
||||
|
||||
- up apexes (raw labels) `221122` → canonical `001100` · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u4:2 u5:2] D[d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `221122` → canonical `001100` · `A=010201021 U[u0:2 u1:2 u2:1 u3:1 u4:2 u5:2] D[d6:1 d7:0 d8:2]`
|
||||
|
||||
## G01 — word=UUUUUDUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `001100`:
|
||||
|
||||
- up apexes (raw labels) `221122` → canonical `001100` · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u4:2 u6:2] D[d5:2 d7:0 d8:1]`
|
||||
|
||||
## G02 — word=UUUUDUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `001100`:
|
||||
|
||||
- up apexes (raw labels) `221122` → canonical `001100` · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u5:2 u6:2] D[d4:2 d7:0 d8:1]`
|
||||
|
||||
## G06 — word=UUDUUDUUD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `001100`:
|
||||
|
||||
- up apexes (raw labels) `221122` → canonical `001100` · `A=010202101 U[u0:2 u1:2 u3:1 u4:1 u6:2 u7:2] D[d2:1 d5:0 d8:2]`
|
||||
- up apexes (raw labels) `220022` → canonical `001100` · `A=010212101 U[u0:2 u1:2 u3:0 u4:0 u6:2 u7:2] D[d2:1 d5:0 d8:2]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `001111`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**.
|
||||
|
||||
- Colour multiset: 2×colour0, 4×colour1.
|
||||
- Realised by **5** of 7 M(T) (dihedral classes).
|
||||
- **7** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_001111.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G00 — word=UUUUUUDDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `001111`:
|
||||
|
||||
- up apexes (raw labels) `221111` → canonical `001111` · `A=010202012 U[u0:2 u1:2 u2:1 u3:1 u4:1 u5:1] D[d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `221111` → canonical `001111` · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u4:1 u5:1] D[d6:1 d7:0 d8:2]`
|
||||
|
||||
## G01 — word=UUUUUDUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `001111`:
|
||||
|
||||
- up apexes (raw labels) `221111` → canonical `001111` · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u4:1 u6:1] D[d5:1 d7:0 d8:2]`
|
||||
|
||||
## G02 — word=UUUUDUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `001111`:
|
||||
|
||||
- up apexes (raw labels) `221111` → canonical `001111` · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u5:1 u6:1] D[d4:1 d7:0 d8:2]`
|
||||
|
||||
## G04 — word=UUUDUUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `001111`:
|
||||
|
||||
- up apexes (raw labels) `221111` → canonical `001111` · `A=010202021 U[u0:2 u1:2 u2:1 u4:1 u5:1 u6:1] D[d3:1 d7:0 d8:2]`
|
||||
|
||||
## G06 — word=UUDUUDUUD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `001111`:
|
||||
|
||||
- up apexes (raw labels) `220000` → canonical `001111` · `A=010121212 U[u0:2 u1:2 u3:0 u4:0 u6:0 u7:0] D[d2:2 d5:0 d8:1]`
|
||||
- up apexes (raw labels) `220000` → canonical `001111` · `A=010212121 U[u0:2 u1:2 u3:0 u4:0 u6:0 u7:0] D[d2:1 d5:0 d8:2]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `001122`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**.
|
||||
|
||||
- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
|
||||
- Realised by **5** of 7 M(T) (dihedral classes).
|
||||
- **8** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_001122.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G02 — word=UUUUDUUDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `001122`:
|
||||
|
||||
- up apexes (raw labels) `221100` → canonical `001122` · `A=010201212 U[u0:2 u1:2 u2:1 u3:1 u5:0 u6:0] D[d4:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `221100` → canonical `001122` · `A=010202121 U[u0:2 u1:2 u2:1 u3:1 u5:0 u6:0] D[d4:1 d7:0 d8:2]`
|
||||
|
||||
## G03 — word=UUUUDUDUD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `001122`:
|
||||
|
||||
- up apexes (raw labels) `221100` → canonical `001122` · `A=010201212 U[u0:2 u1:2 u2:1 u3:1 u5:0 u7:0] D[d4:2 d6:0 d8:1]`
|
||||
- up apexes (raw labels) `221100` → canonical `001122` · `A=010202121 U[u0:2 u1:2 u2:1 u3:1 u5:0 u7:0] D[d4:1 d6:0 d8:2]`
|
||||
|
||||
## G04 — word=UUUDUUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `001122`:
|
||||
|
||||
- up apexes (raw labels) `221100` → canonical `001122` · `A=010202121 U[u0:2 u1:2 u2:1 u4:1 u5:0 u6:0] D[d3:1 d7:0 d8:2]`
|
||||
|
||||
## G05 — word=UUUDUUDUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `001122`:
|
||||
|
||||
- up apexes (raw labels) `221100` → canonical `001122` · `A=010202121 U[u0:2 u1:2 u2:1 u4:1 u5:0 u7:0] D[d3:1 d6:0 d8:2]`
|
||||
|
||||
## G06 — word=UUDUUDUUD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `001122`:
|
||||
|
||||
- up apexes (raw labels) `220011` → canonical `001122` · `A=010121202 U[u0:2 u1:2 u3:0 u4:0 u6:1 u7:1] D[d2:2 d5:0 d8:1]`
|
||||
- up apexes (raw labels) `221100` → canonical `001122` · `A=010202121 U[u0:2 u1:2 u3:1 u4:1 u6:0 u7:0] D[d2:1 d5:0 d8:2]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `001212`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**.
|
||||
|
||||
- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
|
||||
- Realised by **3** of 7 M(T) (dihedral classes).
|
||||
- **6** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_001212.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G02 — word=UUUUDUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `001212`:
|
||||
|
||||
- up apexes (raw labels) `221010` → canonical `001212` · `A=010210212 U[u0:2 u1:2 u2:1 u3:0 u5:1 u6:0] D[d4:2 d7:0 d8:1]`
|
||||
|
||||
## G03 — word=UUUUDUDUD bites=-
|
||||
|
||||
3 colouring(s) with up-tooth sequence `001212`:
|
||||
|
||||
- up apexes (raw labels) `221010` → canonical `001212` · `A=010210212 U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0] D[d4:2 d6:0 d8:1]`
|
||||
- up apexes (raw labels) `221010` → canonical `001212` · `A=010212012 U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0] D[d4:0 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `221010` → canonical `001212` · `A=010212021 U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0] D[d4:0 d6:1 d8:2]`
|
||||
|
||||
## G05 — word=UUUDUUDUD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `001212`:
|
||||
|
||||
- up apexes (raw labels) `221010` → canonical `001212` · `A=010212012 U[u0:2 u1:2 u2:1 u4:0 u5:1 u7:0] D[d3:0 d6:2 d8:1]`
|
||||
- up apexes (raw labels) `221010` → canonical `001212` · `A=010212021 U[u0:2 u1:2 u2:1 u4:0 u5:1 u7:0] D[d3:0 d6:1 d8:2]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `001221`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**.
|
||||
|
||||
- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
|
||||
- Realised by **4** of 7 M(T) (dihedral classes).
|
||||
- **5** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_001221.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G00 — word=UUUUUUDDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `001221`:
|
||||
|
||||
- up apexes (raw labels) `221001` → canonical `001221` · `A=010212012 U[u0:2 u1:2 u2:1 u3:0 u4:0 u5:1] D[d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `221001` → canonical `001221` · `A=010212021 U[u0:2 u1:2 u2:1 u3:0 u4:0 u5:1] D[d6:1 d7:0 d8:2]`
|
||||
|
||||
## G01 — word=UUUUUDUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `001221`:
|
||||
|
||||
- up apexes (raw labels) `221001` → canonical `001221` · `A=010212021 U[u0:2 u1:2 u2:1 u3:0 u4:0 u6:1] D[d5:1 d7:0 d8:2]`
|
||||
|
||||
## G03 — word=UUUUDUDUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `001221`:
|
||||
|
||||
- up apexes (raw labels) `221001` → canonical `001221` · `A=010212102 U[u0:2 u1:2 u2:1 u3:0 u5:0 u7:1] D[d4:0 d6:2 d8:1]`
|
||||
|
||||
## G05 — word=UUUDUUDUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `001221`:
|
||||
|
||||
- up apexes (raw labels) `221001` → canonical `001221` · `A=010212102 U[u0:2 u1:2 u2:1 u4:0 u5:0 u7:1] D[d3:0 d6:2 d8:1]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `010001`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **1** of 7 M(T) (dihedral classes).
|
||||
- **1** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_010001.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G06 — word=UUDUUDUUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `010001`:
|
||||
|
||||
- up apexes (raw labels) `202220` → canonical `010001` · `A=012101012 U[u0:2 u1:0 u3:2 u4:2 u6:2 u7:0] D[d2:0 d5:2 d8:1]`
|
||||
|
||||
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|
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+17
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|
||||
# Up-tooth apex sequence `010100`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **1** of 7 M(T) (dihedral classes).
|
||||
- **1** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_010100.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G06 — word=UUDUUDUUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `010100`:
|
||||
|
||||
- up apexes (raw labels) `202022` → canonical `010100` · `A=012012101 U[u0:2 u1:0 u3:2 u4:0 u6:2 u7:2] D[d2:1 d5:0 d8:2]`
|
||||
|
||||
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|
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|
||||
# Up-tooth apex sequence `010111`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**.
|
||||
|
||||
- Colour multiset: 2×colour0, 4×colour1.
|
||||
- Realised by **1** of 7 M(T) (dihedral classes).
|
||||
- **1** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_010111.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G06 — word=UUDUUDUUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `010111`:
|
||||
|
||||
- up apexes (raw labels) `202000` → canonical `010111` · `A=012012121 U[u0:2 u1:0 u3:2 u4:0 u6:0 u7:0] D[d2:1 d5:0 d8:2]`
|
||||
|
||||
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|
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|
||||
# Up-tooth apex sequence `010221`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**.
|
||||
|
||||
- Colour multiset: 2×colour0, 2×colour1, 2×colour2.
|
||||
- Realised by **1** of 7 M(T) (dihedral classes).
|
||||
- **1** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_010221.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G06 — word=UUDUUDUUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `010221`:
|
||||
|
||||
- up apexes (raw labels) `202110` → canonical `010221` · `A=012102021 U[u0:2 u1:0 u3:2 u4:1 u6:1 u7:0] D[d2:0 d5:1 d8:2]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `011000`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**.
|
||||
|
||||
- Colour multiset: 4×colour0, 2×colour1.
|
||||
- Realised by **4** of 7 M(T) (dihedral classes).
|
||||
- **5** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_011000.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G00 — word=UUUUUUDDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `011000`:
|
||||
|
||||
- up apexes (raw labels) `200222` → canonical `011000` · `A=012101012 U[u0:2 u1:0 u2:0 u3:2 u4:2 u5:2] D[d6:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `200222` → canonical `011000` · `A=012101021 U[u0:2 u1:0 u2:0 u3:2 u4:2 u5:2] D[d6:1 d7:0 d8:2]`
|
||||
|
||||
## G01 — word=UUUUUDUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `011000`:
|
||||
|
||||
- up apexes (raw labels) `200222` → canonical `011000` · `A=012101012 U[u0:2 u1:0 u2:0 u3:2 u4:2 u6:2] D[d5:2 d7:0 d8:1]`
|
||||
|
||||
## G02 — word=UUUUDUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `011000`:
|
||||
|
||||
- up apexes (raw labels) `200222` → canonical `011000` · `A=012101012 U[u0:2 u1:0 u2:0 u3:2 u5:2 u6:2] D[d4:2 d7:0 d8:1]`
|
||||
|
||||
## G04 — word=UUUDUUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `011000`:
|
||||
|
||||
- up apexes (raw labels) `200222` → canonical `011000` · `A=012101012 U[u0:2 u1:0 u2:0 u4:2 u5:2 u6:2] D[d3:2 d7:0 d8:1]`
|
||||
|
||||
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|
||||
# Up-tooth apex sequence `011011`
|
||||
|
||||
Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**.
|
||||
|
||||
- Colour multiset: 2×colour0, 4×colour1.
|
||||
- Realised by **4** of 7 M(T) (dihedral classes).
|
||||
- **6** Kempe-balanced colourings (mod colour permutation) produce it.
|
||||
- Figure: `seq_011011.png` (black rings mark the up-tooth 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 are 0/1/2 = 0:orange, 1:blue, 2:green.
|
||||
|
||||
## G02 — word=UUUUDUUDD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `011011`:
|
||||
|
||||
- up apexes (raw labels) `200200` → canonical `011011` · `A=012101212 U[u0:2 u1:0 u2:0 u3:2 u5:0 u6:0] D[d4:2 d7:0 d8:1]`
|
||||
- up apexes (raw labels) `200200` → canonical `011011` · `A=012102121 U[u0:2 u1:0 u2:0 u3:2 u5:0 u6:0] D[d4:1 d7:0 d8:2]`
|
||||
|
||||
## G03 — word=UUUUDUDUD bites=-
|
||||
|
||||
2 colouring(s) with up-tooth sequence `011011`:
|
||||
|
||||
- up apexes (raw labels) `200200` → canonical `011011` · `A=012101212 U[u0:2 u1:0 u2:0 u3:2 u5:0 u7:0] D[d4:2 d6:0 d8:1]`
|
||||
- up apexes (raw labels) `200200` → canonical `011011` · `A=012102121 U[u0:2 u1:0 u2:0 u3:2 u5:0 u7:0] D[d4:1 d6:0 d8:2]`
|
||||
|
||||
## G04 — word=UUUDUUUDD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `011011`:
|
||||
|
||||
- up apexes (raw labels) `200200` → canonical `011011` · `A=012101212 U[u0:2 u1:0 u2:0 u4:2 u5:0 u6:0] D[d3:2 d7:0 d8:1]`
|
||||
|
||||
## G05 — word=UUUDUUDUD bites=-
|
||||
|
||||
1 colouring(s) with up-tooth sequence `011011`:
|
||||
|
||||
- up apexes (raw labels) `200200` → canonical `011011` · `A=012101212 U[u0:2 u1:0 u2:0 u4:2 u5:0 u7:0] D[d3:2 d6:0 d8:1]`
|
||||
|
||||
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Some files were not shown because too many files have changed in this diff Show More
Reference in New Issue
Block a user