Braid representatives minimizing the number of simple walks

TL;DR

This paper introduces methods to find braid representatives with minimal simple walks, enabling efficient computation of the colored Jones polynomial for various knots, and explores invariance properties and growth rates of simple walks.

Contribution

It develops a new approach to minimize simple walks in braid representatives, improving the computation of colored Jones polynomials and analyzing their invariance and growth.

Findings

Computed colored Jones polynomials for specific knots in closed form.

Proved invariance properties of simple walks under certain braid transformations.

Provided a table of minimal simple walk braid words for knots up to 13 crossings.

Abstract

Given a knot, we develop methods for finding the braid representative that minimizes the number of simple walks. Such braids lead to an efficient method for computing the colored Jones polynomial of $K$, following an approach developed by Armond and implemented by Hajij and Levitt. We use this method to compute the colored Jones polynomial in closed form for the knots $5_2, 6_1,$ and $7_2$. The set of simple walks can change under reflection, rotation, and cyclic permutation of the braid, and we prove an invariance property which relates the simple walks of a braid to those of its reflection under cyclic permutation. We study the growth rate of the number of simple walks for families of torus knots. Finally, we present a table of braid words that minimize the number of simple walks for knots up to 13 crossings.