Optimistic limits of the colored Jones polynomials

TL;DR

This paper demonstrates that for hyperbolic knots, the optimistic limits of colored Jones polynomials align with those of Kashaev invariants modulo 4π², revealing a deep connection between these quantum invariants.

Contribution

It establishes the equivalence of optimistic limits of colored Jones polynomials and Kashaev invariants modulo 4π² for hyperbolic knots, advancing understanding of quantum knot invariants.

Findings

Optimistic limits of colored Jones polynomials match Kashaev invariants modulo 4π².

Provides evidence for the conjectured relationship between these quantum invariants.

Enhances the theoretical framework connecting knot invariants and hyperbolic geometry.

Abstract

We show that the optimistic limits of the colored Jones polynomials of the hyperbolic knots coincide with the optimistic limits of the Kashaev invariants modulo $4\pi^2$.