Walks Along Braids and the Colored Jones Polynomial

TL;DR

This paper introduces a new combinatorial method to compute the colored Jones polynomial using walks along braids, and demonstrates that for positive braid closures, the initial coefficients of the polynomial are trivial.

Contribution

It provides a novel combinatorial framework for the colored Jones polynomial based on braid walks, extending previous quantum determinant approaches.

Findings

First N coefficients are trivial for positive braid closures

New combinatorial description simplifies polynomial computation

Connects braid walks with quantum invariants

Abstract

Using the Huynh and Le quantum determinant description of the colored Jones polynomial, we construct a new combinatorial description of the colored Jones polynomial in terms of walks along a braid. We then use this description to show that for a knot which is the closure of a positive braid, the first N coefficients of the N-th colored Jones polynomial are trivial.