The Four Color Theorem meets Shapes of Polyhedra

TL;DR

This paper explores the classification of solutions to the 4-color problem on sphere triangulations, linking combinatorial types to lattice points in rational polyhedral cones and relating these to Thurston's moduli space.

Contribution

It introduces a novel parametrization of solutions via lattice points in convex cones and connects these to geometric structures in moduli space.

Findings

Solutions are parametrized by lattice points in 4D rational polyhedral cones.

Each combinatorial type corresponds to a specific cone and quadratic form.

The structure relates to the octahedral stratum of Thurston's moduli space.

Abstract

We consider solutions to the $4$-color problem for the vertices of sphere triangulations with degree sequence $6,...,6,4,4,4,4,4,4$. We sort these solutions into combinatorial types and show that each generic type $\tau$ is parametrized by the set of integer lattice points inside a $4$-dimensional rational polyhedral convex cone ${\cal C\/}_{\tau}$. There is an integral quadratic form $Q_{\tau}$ on ${\cal C\/}_{\tau}$ whose diagonal part, evaluated on a lattice point, is $3$ times the number of triangles in the corresponding triangulation. We relate this structure to the octahedral stratum of Thurston's moduli space of flat cone structures on the sphere.