UC Davis

The topics of this dissertation fall under the purview of quantum topology, which seeks tobuild connections between the insights and constructions of quantum physics and classical topol- ogy. A pivotal theme will be the appearance of topologically interesting q-holonomic systems in quantum invariants. These manifest in the quasiperiodic behavior of Witten-Reshetikhin-Turaev (WRT) invariants, and as certain modules associated to lagrangians in quantized character vari- eties. This work was motivated by the AJ conjecture [Gar04, Guk05], which predicts that these two manifestations are the two sides of a single coin. The main result of this dissertation is that the ADO invariant is q-holonomic, meaning it exhibits strong recursive behavior. Some subtlety is involved in the definition of q-holonomicity in this setting, as the ADO invariant exhibits a topologically uninteresting quasi-periodicity because of the appearance of roots of unity. This invariant is closely related to the colored Jones polynomial of the AJ conjecture, and acts as its analytic continuation.