The hyperbolic volume of knots from quantum dilogarithm

TL;DR

This paper explores how the quantum dilogarithm-based invariant of hyperbolic knots grows exponentially with the parameter N, revealing a connection between the invariant's growth rate and the knot's hyperbolic volume.

Contribution

It proposes a novel link between the asymptotic behavior of quantum invariants and hyperbolic volume for knots, supported by analysis of specific examples.

Findings

Invariant grows exponentially with N for hyperbolic knots

Growth rate of invariant equals hyperbolic volume

Supports conjecture relating quantum invariants to geometric topology

Abstract

The invariant of a link in three-sphere, associated with the cyclic quantum dilogarithm, depends on a natural number $N$. By the analysis of particular examples it is argued that for a hyperbolic knot (link) the absolute value of this invariant grows exponentially at large $N$, the hyperbolic volume of the knot (link) complement being the growth rate.