Fundamental shadow links realized as links in $S^3$

TL;DR

This paper constructs infinite families of hyperbolic links in the 3-sphere that satisfy the Turaev-Viro invariant volume conjecture, extending known cases and linking to quantum representations of surface mapping class groups.

Contribution

It introduces topological methods to generate links satisfying the volume conjecture and connects these to the AMU conjecture on quantum representations.

Findings

Constructed infinite families of hyperbolic links satisfying the volume conjecture.

Proved that all links in $S^3$ are sublinks of links satisfying the conjecture.

Extended the class of links known to satisfy the AMU conjecture.

Abstract

We use purely topological tools to construct several infinite families of hyperbolic links in the 3-sphere that satisfy the Turaev-Viro invariant volume conjecture posed by Chen and Yang. To show that our links satisfy the volume conjecture, we prove that each has complement homeomorphic to the complement of a fundamental shadow link. These are links in connected sums of copies of $S^2 \times S^1$ for which the conjecture is known due to Belletti, Detcherry, Kalfagianni, and Yang. Our methods also verify the conjecture for several hyperbolic links with crossing number less than twelve. In addition, we show that every link in the $3$-sphere is a sublink of a link that satisfies the conjecture. As an application of our results, we extend the class of known examples that satisfy the AMU conjecture on quantum representations of surface mapping class groups. For example, we give explicit elements in the mapping class group of a genus $g$ surface with four boundary components for any $g$. For this, we use techniques developed by Detcherry and Kalfagianni which relate the Turaev-Viro invariant volume conjecture to the AMU conjecture.