Colored Jones polynomials without tails

TL;DR

This paper demonstrates that the stability and tail behavior of colored Jones polynomials observed in alternating knots do not extend straightforwardly to all knots, by providing an infinite family with linearly growing first coefficients.

Contribution

It introduces an infinite family of knots where the first coefficient of the n-colored Jones polynomial grows linearly, challenging previous assumptions about polynomial stability.

Findings

First coefficient grows linearly with n for the constructed knots.

Stability and tail phenomena are not universal for all knots.

Challenges the generalization of tail behavior beyond alternating knots.

Abstract

We exhibit an infinite family of knots with the property that the first coefficient of the n-colored Jones polynomial grows linearly with n. This shows that the concept of stability and tail seen in the colored Jones polynomials of alternating knots does not generalize naively.