Hyperbolic Structure Arising from a Knot Invariant

TL;DR

This paper introduces a new knot invariant derived from the quantum dilogarithm, linking quantum topology with hyperbolic geometry through classical limit analysis.

Contribution

It presents a non-compact analogue of known knot invariants, connecting quantum dilogarithm properties with hyperbolic structures in a novel way.

Findings

The invariant is expressed as an integral form based on the quantum dilogarithm.

Hyperbolicity conditions emerge naturally from the classical limit.

The invariant relates to the geometric structure of knot complements.

Abstract

We study the knot invariant based on the quantum dilogarithm function. This invariant can be regarded as a non-compact analogue of Kashaev's invariant, or the colored Jones invariant, and is defined by an integral form. The 3-dimensional picture of our invariant originates from the pentagon identity of the quantum dilogarithm function, and we show that the hyperbolicity consistency conditions in gluing polyhedra arise naturally in the classical limit as the saddle point equation of our invariant.