Knot Colorings: Coloring and Goeritz Matrices

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 the knot determinant and the nullity of pretzel knots with arbitrarily many twist regions.