Work

Colorings, determinants and Alexander polynomials for spatial graphs

Year: 2015

Type: preprint

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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.1506.06083

Authors: Kong, Terry, Lewald, Alec, Mellor, Blake, Pigrish, Vadim

Institutions: Stanford University, California State Polytechnic University, Loyola Marymount University, University of Pennsylvania

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: National Science Foundation

Award: Topological symmetries and intrinsic properties of graphs embedded in 3-space

OpenAlex ID: W1575223769

Cites: 7

Cited by:

Related to: 10