Turaev-Viro invariants and cabling operations

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.