Tait coloring and a moduli space

TL;DR

This paper introduces a moduli space linked to planar trivalent graphs, explores its properties, and connects it to Tait colorings and SU(3) representations, revealing new insights into graph colorings and topological structures.

Contribution

It constructs a novel moduli space for planar trivalent graphs and establishes its relationship with Tait colorings and fundamental group representations.

Findings

Euler characteristic of M(G) equals the number of Tait colorings for bipartite graphs

Decomposition properties of M(G) are established

M(G) can be interpreted as a representation space of the fundamental group into SU(3)

Abstract

We associate a moduli space M(G) to a planar trivalent graph G. We proved several decomposition properties of M(G), which implies that the Euler characteristic of M(G) equals to the number of Tait colorings of G when G is bipartite. Then we interpret M(G) as a representation space of the fundamental group of G to SU(3).