Turaev-Viro invariants and cabling operations

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.