The Chen-Yang volume conjecture for knots in handlebodies

Abstract

In 2015, Chen and Yang proposed a volume conjecture that stated that certain Turaev-Viro invariants of an hyperbolic 3-manifold should grow exponentially with a rate equal to the hyperbolic volume. Since then, this conjecture has been proven or numerically tested for several hyperbolic 3-manifolds, either closed or with boundary, the boundary being either a family of tori or a family of higher genus surfaces. The current paper now provides new numerical checks of this volume conjecture for 3-manifolds with one toroidal boundary component and one geodesic boundary component. More precisely, we study a family of hyperbolic 3-manifolds $M_g$ introduced by Frigerio. Each $M_g$ can be seen as the complement of a knot in an handlebody of genus $g$. We provide an explicit code that computes the Turaev-Viro invariants of these manifolds $M_g$, and we then numerically check the Chen-Yang volume conjecture for the first six members of this family. Furthermore, we propose an extension of the volume conjecture, where the second coefficient of the asymptotic expansion only depends on the topology of the boundary of the manifold. We numerically check this property for the manifolds $M_2$ to $M_7$ and we also observe that the second coefficient grows linearly in the Euler characteristic $\chi(\partial M_g)$.