Stability properties of the colored Jones polynomial

TL;DR

This paper extends the concept of the tail of the colored Jones polynomial, originally defined for +adequate links, to all links, showing it is trivial precisely when the link is non +adequate, revealing new topological insights.

Contribution

It introduces a generalized tail for the colored Jones polynomial applicable to all links, not just +adequate ones, and characterizes non +adequate links through its triviality.

Findings

The generalized tail exists for all links.

The tail is trivial if and only if the link is non +adequate.

Provides a new topological invariant related to link adequacy.

Abstract

It is known that the colored Jones polynomial of a $+$-adequate link has a well-defined tail consisting of stable coefficients, and that the coefficients of the tail carry geometric and topological information on the $+$-adequate link complement. We show that a power series similar to the tail of the colored Jones polynomial for $+$-adequate links can be defined for $all$ links, and that it is trivial if and only if the link is non $+$-adequate.