We use purely topological tools to construct several infinite families of hyperbolic links in
the
–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
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
–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
surface with four boundary
components for any
.
For this, we use techniques developed by Detcherry and Kalfagianni which relate the
Turaev–Viro invariant volume conjecture to the AMU conjecture.
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