Turaev-Viro invariants and cabling operations

TL;DR

This paper investigates how Turaev-Viro invariants of 3-manifolds change under cabling operations, demonstrating stability of the Chen-Yang volume conjecture for coprime parameters using TQFT operator analysis.

Contribution

It establishes the stability of the Chen-Yang volume conjecture under (p,q)-cabling for coprime p and q by analyzing the invertibility of associated linear operators in TQFT.

Findings

Proves the stability of the Chen-Yang volume conjecture under cabling.

Characterizes the invertibility conditions of the linear operator RT_r.

Connects Turaev-Viro invariants with Reshetikhin-Turaev TQFT operators.

Abstract

In this paper, we study the variation of the Turaev--Viro invariants for $3$-manifolds with toroidal boundary under the operation of attaching a $(p,q)$-cable space. We apply our results to a conjecture of Chen and Yang which relates the asymptotics of the Turaev--Viro invariants to the simplicial volume of a compact oriented $3$-manifold. For $p$ and $q$ coprime, we show that the Chen--Yang volume conjecture is stable under $\left(p,q\right)$-cabling. We achieve our results by studying the linear operator $RT_r$ associated to the torus knot cable spaces by the Reshetikhin--Turaev $SO_3$-Topological Quantum Field Theory (TQFT), where the TQFT is well-known to be closely related to the desired Turaev--Viro invariants. In particular, our utilized method relies on the invertibility of the linear operator for which we provide necessary and sufficient conditions.