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Colorings, determinants and Alexander polynomials for spatial graphs
Journal of Knot Theory and Its Ramifications
  • Terry Kong
  • Alec Lewald
  • Blake Mellor, Loyola Marymount University
  • Vadim Pigrish
Document Type
Article - post-print
Publication Date
1-1-2016
Disciplines
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 Γ(p,m,k). We finish by proving some properties of the Alexander polynomial.

Comments

This is a post-print version of the article.

Citation Information
Kong, T., A. Lewald, B. Mellor, and V. Pigrish, 2016: Colorings, determinants and Alexander polynomials for spatial graphs. J. Knot Theory Ramif., 25.4, arXiv:1506.06083.