Braid representatives minimizing the number of simple walks - Boden, Hans U. (Ars mathematica contemporanea, 2023)
Given a knot, we find the braid representative that minimizes the number of simple walks. Such braids lead to an efficient method for computing the colored Jones polynomial of the knot, 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.
Keywords: barvni Jonesov polinom; braids; colored Jones polynomial; enostaven sprehod; knots; pletenice; simple walk; vozli