Asymptotic behaviors of the colored Jones polynomials of a torus knot

TL;DR

This paper investigates the asymptotic properties of colored Jones polynomials for torus knots, revealing that their limits do not correspond to volumes or Chern-Simons invariants but can relate to the inverse of the Alexander polynomial in some cases.

Contribution

It provides new insights into the asymptotic behavior of colored Jones polynomials of torus knots, contrasting with previous work linking them to 3-manifold invariants.

Findings

Limits do not give volumes or Chern-Simons invariants.

In some cases, limits relate to the inverse of the Alexander polynomial.

Provides conditions under which the limits correspond to the Alexander polynomial.

Abstract

We study the asymptotic behaviors of the colored Jones polynomials of torus knots. Contrary to the works by R. Kashaev, O. Tirkkonen, Y. Yokota, and the author, they do not seem to give the volumes or the Chern-Simons invariants of the three-manifolds obtained by Dehn surgeries. On the other hand it is proved that in some cases the limits give the inverse of the Alexander polynomial.