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Given a knot, we develop methods for finding the braid representative thatminimizes the number of simple walks. Such braids lead to an efficient methodfor computing the colored Jones polynomial of $K$, following an approachdeveloped by Armond and implemented by Hajij and Levitt. We use this method tocompute 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, andcyclic permutation of the braid, and we prove an invariance property whichrelates the simple walks of a braid to those of its reflection under cyclicpermutation. We study the growth rate of the number of simple walks forfamilies of torus knots. Finally, we present a table of braid words thatminimize the number of simple walks for knots up to 13 crossings.

Braid representatives minimizing the number of simple walks