Colorings, determinants and Alexander polynomials for spatial graphs

TL;DR

This paper extends Alexander invariants to balanced spatial graphs, exploring their properties, colorings, and determinants, and linking these to graph colorability and fundamental group representations.

Contribution

It introduces the Alexander module and polynomial for balanced spatial graphs, analyzing their behavior and applications in graph colorings and group representations.

Findings

The determinant determines p-colorability of the graph.

p-colorings correspond to representations into metacyclic groups.

Properties of the Alexander polynomial are established.

Abstract

A {\em balanced} spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to Kinoshita \cite{ki}), and examine their behavior under some common operations on the graph. We use the Alexander module to define the determinant and $p$-colorings of a balanced spatial graph, and provide examples. We show that the determinant of a spatial graph determines for which $p$ the graph is $p$-colorable, and that a $p$-coloring of a graph corresponds to a representation of the fundamental group of its complement into a metacyclic group $\Gamma(p,m,k)$. We finish by proving some properties of the Alexander polynomial.