diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.pdf b/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.pdf
index fb88766..fb2610f 100644
Binary files a/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.pdf and b/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.pdf differ
diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.tex b/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.tex
index 18b17b2..e72901d 100644
--- a/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.tex
+++ b/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.tex
@@ -46,7 +46,7 @@ A proper $3$-edge-colouring of $G'$ exists iff some $\sigma$ is
 achievable as both $\sigma_0$ and $\sigma_1$, i.e.\
 $\mathcal{R}_0 \cap \mathcal{R}_1 \neq \emptyset$ where
 $\mathcal{R}_i := \{\sigma_i \mid \chi_i \text{ proper}\}$.  $G'$ a
-counterexample ⇒ $\mathcal{R}_0 \cap \mathcal{R}_1 = \emptyset$.
+counterexample $\Rightarrow$ $\mathcal{R}_0 \cap \mathcal{R}_1 = \emptyset$.
 
 \subsection*{Layered decomposition via cut tires}
 
@@ -82,7 +82,7 @@ spokes}|}$ corresponds, via the bijection $\{\text{out spokes of }
 T_d\} \to \{\text{specific edges on face boundaries of } T_{d-1}
 \text{'s}\}$, to a projection of $T_{d-1}$'s face-boundary colouring.
 
-Symmetrically: in spokes of $T_d$ ↔ specific face-boundary edges of
+Symmetrically: in spokes of $T_d$ $\leftrightarrow$ specific face-boundary edges of
 some $T_{d+1}$.
 
 \subsection*{Result inheritance via the partial-tire-dual identification}