Volume Conjecture and quantum hyperbolic invariants: the figure eight knot complement

TL;DR

This paper analyzes the quantum hyperbolic invariants of the figure-eight knot complement, revealing their real part's dependence on holonomy representations and its relation to hyperbolic volume.

Contribution

It demonstrates the rigidity of the semi-classical limit of QHI and establishes a parity condition linking it to hyperbolic volume.

Findings

The real part of the semi-classical limit is independent of holonomy representation.

It is either zero or proportional to the hyperbolic volume, depending on a parity condition.

The paper surveys features of quantum hyperbolic invariants.

Abstract

We compute the real part of the semi-classical limit of the sequence of quantum hyperbolic invariants (QHI) of the figure-eight knot complement $M$. We show that it is rigid, in the sense that it does not depend on the choice of holonomy representation of $M$, and it is either $0$ or equal to the hyperbolic volume of $M$ divided by $2\pi$, depending on a parity condition satisfied by logarithms of the holonomy eigenvalues on the canonical longitude, where the logarithms are parameters of the QHI of $M$. Along the way we also survey some relevant general features of the QHI.