We study the Turaev-Viro invariants of 3-manifolds as well as their relationship to invariants arising from hyperbolic geometry. We first construct a closed formula for the Turaev-Viro invariants for hyperbolic once-punctured torus bundles. We then examine a conjecture by Chen and Yang which states that the asymptotics of the Turaev-Viro invariants recover the hyperbolic volume of the manifold. Using topological tools, we are able to construct infinite families of new examples of hyperbolic links in the 3-sphere which satisfy the conjecture. Additionally, we show a general method for augmenting a link such that the resulting link has hyperbolic complement which satisfies the conjecture. As an application of the constructed links in the 3-sphere, we extend the class of known examples which satisfy a conjecture by Andersen, Masbaum, and Ueno relating the quantum representations of surface mapping class groups to its Nielsen-Thurston classification. Most notably, we provide explicit elements in the mapping class group for a genus zero surface with n boundary components, for n > 3, as well as elements in the mapping class group for any genus g surface with four boundary components such that the obtained elements satisfy the conjecture. This is accomplished by utilizing the intrinsic relationship between the Turaev-Viro invariants and the quantum representations shown by Detcherry and Kalfagianni.
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