Asymptotically multiplicative quantum invariants

The Euler characteristic and the volume are two best-known multiplicative invariants of manifolds under finite covers. On the other hand, quantum invariants of 3-manifolds are not multiplicative. We show that a perturbative power series, introduced by Dimofte and the first author and shown to be a topological invariant of cusped hyperbolic 3-manifolds by Storzer--Wheeler and the first author, and conjectured to agree with the asymptotics of the Kashaev invariant to all orders in perturbation theory, is asymptotically multiplicative under cyclic covers. Moreover, its coefficients are determined by polynomials constructed out of twisted Neumann--Zagier data. This gives a new $t$-deformation of the perturbative quantum invariants, different than the $x$-deformation obtained by deforming the geometric representation. We illustrate our results with several hyperbolic knots.