Braid representatives minimizing the number of simple walks

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.