Quantum invariants and volume of links on surfaces and knotted graphs

Since its discovery by Kashaev, the influential Volume Conjecture has attracted much attention towards the ongoing effort of relating diagrammatic and quantum invariants of links in the 3-sphere to the geometry of the link complement. The conjecture has inspired similar open problems which extend it in multiple directions, either to links in more general 3-manifolds or to quantum invariants of other topological objects. This thesis presents contributions to these efforts on both fronts. First, for D a reduced alternating link diagram on a surface Sigma, we bound the twist number of D in terms of the coefficients of a polynomial invariant. To this end, we introduce a generalization of the homological Kauffman bracket defined by Krushkal. Combined with work of Futer, Kalfagianni, and Purcell, this yields a bound for the hyperbolic volume of a class of alternating surface links in terms of these coefficients. Second, we prove an instance of Yang's volume conjecture for the relative Turaev–Viro invariants, which are defined for a compact, orientable 3-manifold M via a partially ideal triangulation. We consider the case where M is a punctured 3-sphere obtained from a knotted trivalent graph Gamma belonging to a particular family. We evaluate the limit of relative Turaev–Viro invariants by utilizing techniques relating them to the Reshetikhin–Turaev TQFT and colored Jones polynomials, and we show that this limit equals the volume of O, where O is the outside of Gamma, as defined by van der Veen.