The hyperbolic volume of knots from quantum dilogarithm

TL;DR

This work investigates a knot invariant defined via the cyclic quantum dilogarithm, showing that its large-$N$ limit encodes the hyperbolic volume of a knot complement through $2\pi \log|\langle L\rangle| \sim N V(L)$. By formulating the invariant at a primitive $N$-th root of unity and applying analytic continuation to convert sums into contour integrals, the author performs saddle-point analyses for three hyperbolic knots ($4_1$, $5_2$, $6_1$) and finds $|\langle L\rangle| \sim \exp(V(L)/(2\gamma))$ with $\gamma=\pi/N$, where the resulting $V(L)$ matches the known hyperbolic volumes. These results support the view that this cyclic quantum dilogarithm-based construction provides a quantum generalization of hyperbolic volume and suggests a link to quantum 2+1 dimensional gravity via a simplicial TQFT. The work thus connects topological quantum field theory, hyperbolic geometry, and quantum gravity in a concrete asymptotic framework.

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.