Colourings and the Alexander Polynomial

TL;DR

This paper develops methods to compute the number of colourings of prime knots using linear Alexander quandles, simplifying the process through matrix triangularization and exploring the distinguishing power of colourings related to Alexander polynomials.

Contribution

It introduces a technique to triangularize colouring matrices for prime knots, enabling closed-form calculations and analysis of the relationship between colourings and Alexander polynomials.

Findings

Triangularization simplifies the calculation of colourings.

Some colouring matrices resist triangularization.

Colourings can distinguish knots with different Alexander polynomials.

Abstract

In this paper we look for closed expressions to calculate the number of colourings of prime knots for given linear Alexander quandles. For this purpose the colouring matrices are simplified to a triangular form, when possible. The operations used to perform this triangularization preserve the property that the entries in each row add up to zero, thereby simplifying the solution of the equations giving the number of colourings. When the colouring matrices (of prime knots up to ten crossings) can be triangularized, closed expressions giving the number of colourings can be obtained in a straightforward way. We use these results to show that there are colouring matrices that cannot be triangularized. In the case of knots with triangularizable colouring matrices we present a way to find linear Alexander quandles that distinguish by colourings knots with different Alexander polynomials. The colourings of knots with the same Alexander polynomial are also studied as regards when they can and cannot be distinguished by colourings.