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 $R{T}_r$ associated to the torus knot cable spaces by the Reshetikhin--Turaev $S{O}_3$-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.

22 pages. Significant revision of the technical results of Section 4 and filled a gap in the proof of Proposition 4.2 using referee suggestions. Relaxed the assumptions of the main results of the paper following revisions. Updated abstract. To appear in the International Journal of Mathematics