Turaev–Viro invariants and cabling operations

Abstract

In this paper, we study the variation of the Turaev–Viro invariants for [Formula: see text]-manifolds with toroidal boundary under the operation of attaching a [Formula: see text]-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 [Formula: see text]-manifold. For [Formula: see text] and [Formula: see text] coprime, we show that the Chen–Yang volume conjecture is stable under [Formula: see text]-cabling. We achieve our results by studying the linear operator [Formula: see text] associated to the torus knot cable spaces by the Reshetikhin–Turaev [Formula: see text]-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.