Twist Regions and Coefficients Stability of the Colored Jones Polynomial

TL;DR

This paper demonstrates that the coefficients of the colored Jones polynomial for alternating links stabilize as the number of twists in certain regions increases, leading to a family of stable q-power series.

Contribution

It introduces a stability result for the coefficients of the colored Jones polynomial in relation to twist regions in alternating links, expanding understanding of their asymptotic behavior.

Findings

Coefficients stabilize with increasing twists

Infinite family of q-power series constructed

Provides new insights into link invariants stability

Abstract

We prove that the coefficients of the colored Jones polynomial of alternating links stabilize under increasing the number of twists in the twist regions of the link diagram. This gives us an infinite family of $q$-power series derived from the colored Jones polynomial parametrized by the color and the twist regions of the alternating link diagram.