The Bubble Skein Element and Applications

TL;DR

This paper introduces a skein element in the Kauffman bracket skein module, uses it to analyze the colored Jones polynomial's head and tail, and provides new q-series and theta coefficient computations.

Contribution

It presents a novel skein element and demonstrates its applications in computing the tail of the colored Jones polynomial and theta coefficients.

Findings

Derived a simple q-series for the tail of knot 8_5

Expanded skein element in terms of independent basis elements

Facilitated easier determination of theta coefficients

Abstract

We study a certain skein element in the relative Kauffman bracket skein module of the disk with some marked points, and expand this element in terms linearly independent elements of this module. This expansion is used to compute and study the head and the tail of the colored Jones polynomial and in particular we give a simple $q-$series for the tail of the knot $8_5$. Furthermore, we use this expansion to obtain an easy determination of the theta coefficients.