Some limits of the colored Jones polynomials of the figure-eight knot

TL;DR

This paper investigates the asymptotic behavior of the colored Jones polynomials for the figure-eight knot, revealing connections to the volumes of cone manifolds with singularities in specific limits.

Contribution

It demonstrates how certain limits of the colored Jones polynomials relate to the geometric volumes of cone manifolds with singularities along the knot.

Findings

Limits of colored Jones polynomials correspond to cone manifold volumes.

Asymptotic analysis links quantum invariants to geometric structures.

Results highlight constraints in the relationship between polynomials and manifold volumes.

Abstract

We will study the asymptotic behaviors of the colored Jones polynomials of the figure-eight knot. In particular we will show that for certain limits we obtain the volumes of the cone manifolds with singularities along the knot.