Kimura, TakashiGuigo Corominas, Roderic2021-10-182021-10-182021https://hdl.handle.net/2144/43178The goal of the present thesis is to study new examples, applications and com- putational aspects of the topological recursion formalism introduced by Eynard and Orantin. We develop efficient methods for the calculation of non-perturbative wave functions associated to spectral curves of genus one. Our results are used to test two conjectures. The first one relates perturbative knot invariants obtained from the AJ Conjecture and a state integral model to the wave function obtained from topological recursion. The second conjecture describes the structure of the quantum curve for the Weierstrass spectral curve. We are able to verify the conjectures up to some order in a formal parameter h and we state a stronger version of the conjecture in the case of the Weierstrass curve. Some of these results are based upon joint work with Greyson Potter.en-USMathematicsThesis/Dissertation2021-10-07