Knot Colorings: Coloring and Goeritz matrices

TL;DR

This paper explores knot colorings and their associated matrices, providing elementary methods to compute invariants like knot determinant and nullity, with applications to pretzel knots, without relying on algebraic topology.

Contribution

It offers an elementary approach to relate coloring matrices and Goeritz matrices, simplifying the computation of knot invariants such as determinant and nullity.

Findings

Elementary proof of equivalence between coloring and Goeritz matrices

Explicit computation of knot determinants for pretzel knots

Simplified methods for calculating knot invariants

Abstract

Knot colorings are one of the simplest ways to distinguish knots, dating back to Reidemeister, and popularized by Fox. In this mostly expository article, we discuss knot invariants like colorability, knot determinant and number of colorings, and how these can be computed from either the coloring matrix or the Goeritz matrix. We give an elementary approach to this equivalence, without using any algebraic topology. We also compute knot determinant, nullity of pretzel knots with arbitrarily many twist regions.