  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.
