Volume 21, issue 6 (2021)

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

Sanjay Kumar

Algebraic & Geometric Topology 21 (2021) 3153–3198

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 S2 × S1 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.

fundamental shadow links, Turaev–Viro invariant volume conjecture, AMU conjecture
Mathematical Subject Classification
Primary: 57K31, 57K32, 57M50
Received: 15 June 2020
Revised: 11 October 2020
Accepted: 26 October 2020
Published: 22 November 2021
Sanjay Kumar
Department of Mathematics
Michigan State University
East Lansing, MI
United States