Constructions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> -hyperbolic knots

We use Dehn surgery methods to construct infinite families of hyperbolic knots in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>3</mml:mn> </mml:math> -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.