Braid representatives minimizing the number of simple walks
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abstract

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 $K$, 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.