Kimura, TakashiPotter, Greyson2025-11-102025-11-102023https://hdl.handle.net/2144/517702023The goal of the present thesis is to develop a computational approach to analyzing the generalized volume conjecture, which relates topological recursion and Chern-Simons theory for hyperbolic 3-manifolds with torus boundary. The conjecture states that there is an equality between two series in a formal parameter h: the non-perturbative wave function from topological recursion and a state-integral model for SL(2,C) Chern-Simons theory. Both series are expressible as sums over graphs. We develop algorithms for efficiently computing the graph sum for the non-perturbative wave function via quantum Airy structures and a software package, called qairy, which implements our algorithms. We further develop tools and techniques which are widely applicable to the calculation and analysis of the non-perturbative wave functions associated to genus one spectral curves. Using these tools, we are able to verify the conjecture in several cases up to much higher order in h than was previously accessible as well as analyze the arithmetic aspects of the conjecture.en-USMathematicsChern-Simons theoryGeneralized volume conjectureHyperbolic knot invariantsTopological recursionThesis/Dissertation2025-11-09