Skein-valued Gromov-Witten theory and Hofer geometry in contact manifolds

The thesis consists of an introduction and five research articles in the fields of Hofer contact Geometry and skein-valued open Gromov-Witten theory. In Paper I, we refine Murphy's h-principle for loose Legendrians by obtaining upper bounds on the Shelukhin-Chekanov-Hofer distance of loose Legendrians, generalizing an earlier result of Dimitroglou Rizell and Sullivan. In Paper II, we give first examples of closed Legendrian submanifolds with vanishing Shelukhin-Chekanov-Hofer metric, thereby providing counterexamples to a conjecture of Rosen and Zhang. In Paper III, we introduce a pseudo-metric on the contactomorphism group and on isotopy classes of Legendrian submanifolds whose topology agrees with the interval topology of Chernov and Nemirovski. We prove a dichotomy for its non-degeneracy which resolves a question of Chernov and Nemirovski. In Paper IV, we introduce a family of partition functions in the skein of a disjoint union of solid tori, one for each compact, oriented surface with boundary, which reduce to the BPS partition functions in the case without boundary. We prove gluing formulas and a version of the unknot skein relation for all partition functions and a crossing formula in case of disks which generalizes the pentagon relation for disk partition functions. In Paper V, we conjecture in joint work with T. Ekholm and P. Longhi the existence of a skein-valued D-module for links in the three-sphere and exemplify this general conjecture in the case of the Hopf link. We use this to obtain quiver-like expressions for different fillings of the Hopf link unit conormal covering the augmentation variety.