Constructing Goeritz matrix from Dehn coloring matrix

TL;DR

This paper presents a method to construct a Goeritz matrix directly from a Dehn coloring matrix, establishing a new algebraic link between knot invariants and coloring solutions.

Contribution

It introduces a novel algebraic construction of the Goeritz matrix from Dehn coloring matrices, especially for prime knot diagrams.

Findings

Construction of Goeritz matrix from Dehn coloring matrix

Isomorphism between solution space and Dehn colorings

Purely algebraic method for prime knot diagrams

Abstract

Associated to a knot diagram, Goeritz introduced an integral matrix, which is now called a Goeritz matrix. It was shown by Traldi that the solution space of the equations with Goeritz matrix (precisely, unreduced Goeritz matrix called in his paper) as a coefficient matrix is isomorphic to the linear space consisting of the Dehn colorings for a knot. In this paper, we give a construction of a Goeritz matrix from a Dehn coloring matrix, from which Dehn colorings are induced. Moreover, if the knot diagram is prime, we give a purely algebraic construction of a Goeritz matrix from a Dehn coloring matrix.