  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.
