Asymptotically multiplicative quantum invariants

TL;DR

This paper demonstrates that a specific perturbative quantum invariant of cusped hyperbolic 3-manifolds becomes asymptotically multiplicative under cyclic covers, revealing new structural insights and polynomial relations involving twisted Neumann--Zagier data.

Contribution

It introduces a new t-deformation of quantum invariants that are asymptotically multiplicative under cyclic covers, expanding understanding of quantum invariants' behavior.

Findings

The perturbative power series is asymptotically multiplicative under cyclic covers.

Coefficients are determined by polynomials from twisted Neumann--Zagier data.

Illustrated with examples involving hyperbolic knots.

Abstract

The Euler characteristic and the volume are two best-known multiplicative invariants of manifolds under finite covers. On the other hand, quantum invariants of 3-manifolds are not multiplicative. We show that a perturbative power series, introduced by Dimofte and the first author and shown to be a topological invariant of cusped hyperbolic 3-manifolds by Storzer--Wheeler and the first author, and conjectured to agree with the asymptotics of the Kashaev invariant to all orders in perturbation theory, is asymptotically multiplicative under cyclic covers. Moreover, its coefficients are determined by polynomials constructed out of twisted Neumann--Zagier data. This gives a new $t$-deformation of the perturbative quantum invariants, different than the $x$-deformation obtained by deforming the geometric representation. We illustrate our results with several hyperbolic knots.