Colored Jones polynomials with polynomial growth

TL;DR

This paper investigates the behavior of colored Jones polynomials of hyperbolic knots, focusing on cases where their growth is polynomial rather than exponential, and explores the transition between these growth regimes.

Contribution

It identifies conditions under which colored Jones polynomials exhibit polynomial growth, extending understanding beyond the exponential growth predicted by the volume conjecture.

Findings

Colored Jones polynomials grow polynomially for certain parameter values.

The paper delineates the boundary between exponential and polynomial growth regimes.

Provides new insights into the asymptotic behavior of knot invariants.

Abstract

The volume conjecture and its generalizations say that the colored Jones polynomial corresponding to the N-dimensional irreducible representation of sl(2;C) of a (hyperbolic) knot evaluated at exp(c/N) grows exponentially with respect to N if one fixes a complex number c near 2*Pi*I. On the other hand if the absolute value of c is small enough, it converges to the inverse of the Alexander polynomial evaluated at exp(c). In this paper we study cases where it grows polynomially.