On the tail of Jones polynomials of closed braids with a full twist

TL;DR

This paper analyzes the initial terms of Jones polynomials for closed braids with full twists, revealing their structure and constraints, and extends findings to colored Jones polynomials for positive braids.

Contribution

It explicitly determines the leading terms of Jones polynomials for certain closed braids and establishes a braid index constraint, extending results to colored Jones polynomials.

Findings

First n-k+1 terms of Jones polynomial are determined

Jones polynomial satisfies a braid index gap constraint

Extended results to colored Jones polynomials for positive braids

Abstract

For a closed n-braid L with a full positive twist and with k negative crossings, 0\leq k \leq n, we determine the first n-k+1 terms of the Jones polynomial V_L(t). We show that V_L(t) satisfies a braid index constraint, which is a gap of length at least n-k between the first two nonzero coefficients of (1-t^2)V_L(t). For a closed positive n-braid with a full positive twist, we extend our results to the colored Jones polynomials. For N>n-1, we determine the first n(N-1)+1 terms of the normalized N-th colored Jones polynomial.