Asymptotic Behavior of Colored Jones polynomial and Turaev-Viro Invariant of figure eight knot

TL;DR

This paper studies the asymptotic behavior of colored Jones polynomials and Turaev-Viro invariants for the figure eight knot, revealing how their growth rates relate and providing formulas for their asymptotic expansions.

Contribution

It derives the asymptotic expansion formulas for the colored Jones polynomials and Turaev-Viro invariants of the figure eight knot, especially near specific ratios, and suggests a generalization to other hyperbolic knots.

Findings

Asymptotic expansion formula for colored Jones polynomials near s=1

Exponential growth rate varies with s, being less near s=1/2

Turaev-Viro invariant sum dominated by terms with s close to 1

Abstract

In this paper we investigate the asymptotic behavior of the colored Jones polynomials and the Turaev-Viro invariants for the figure eight knot. More precisely, we consider the $M$-th colored Jones polynomials evaluated at $(N+1/2)$-th root of unity with a fixed limiting ratio, $s$, of $M$ and $(N+1/2)$. We find out the asymptotic expansion formula (AEF) of the colored Jones polynomials of the figure eight knot with $s$ close to $1$. Nonetheless, we show that the exponential growth rate of the colored Jones polynomials of the figure eight knot with $s$ close to $1/2$ is strictly less than those with $s$ close to $1$. It is known that the Turaev Viro invariant of the figure eight knot can be expressed in terms of a sum of its colored Jones polynomials. Our results show that this sum is asymptotically equal to the sum of the terms with $s$ close to 1. As an application of the asymptotic behavior of the colored Jones polynomials, we obtain the asymptotic expansion formula for the Turaev-Viro invariants of the figure eight knot. Finally, we suggest a possible generalization of our approach so as to relate the AEF for the colored Jones polynomials and the AEF for the Turaev-Viro invariants for general hyperbolic knots.