Dehn coloring and the dimer model for knots

TL;DR

This paper explores the relationship between Dehn coloring and the dimer model for knots, introducing graph-theoretic methods to compute knot invariants like the determinant and Smith normal form using Kasteleyn theory.

Contribution

It establishes a novel connection between Dehn coloring and the dimer model, enabling new computational techniques for knot invariants via graph theory.

Findings

Dehn coloring data is encoded by a weighted balanced overlaid Tait graph.

Kasteleyn theory provides methods to compute the determinant and Smith normal form of a knot.

The constructions relate to Kauffman's state sum for the Alexander polynomial.

Abstract

Fox coloring provides a combinatorial framework for studying dihedral representations of the knot group. The less well-known concept of Dehn coloring captures the same data. Recent work of Carter-Silver-Williams clarifies the relationship between the two focusing on how one transitions between Fox and Dehn colorings. In our work, we relate Dehn coloring to the dimer model for knots showing that Dehn coloring data is encoded by a certain weighted balanced overlaid Tait graph. Using Kasteleyn theory, we provide graph theoretic methods for computing the determinant and Smith normal form of a knot. These constructions are closely related to Kauffman's work on a state sum for the Alexander polynomial.