  We use Dehn surgery methods to construct infinite families of hyperbolic
knots in the 3-sphere satisfying a weak form of the Turaev--Viro invariants
volume conjecture. The results have applications to a conjecture of Andersen,
Masbaum, and Ueno about quantum representations of surface mapping class
groups. We obtain an explicit family of pseudo-Anosov mapping classes acting on
surfaces of any genus and with one boundary component that satisfy the
conjecture.
