Generalized Legendrian racks: Classification, tensors, and knot coloring invariants

Abstract: Generalized Legendrian racks are nonassociative algebraic structures based on the Legendrian Reidemeister moves. We study algebraic aspects of GL-racks and coloring invariants of Legendrian links. We answer an open question characterizing the group of GL-structures on a given rack. As applications, we classify several infinite families of GL-racks. We also compute automorphism groups of dihedral GL-quandles and the categorical center of GL-racks. Then we construct an equivalence of categories between racks and GL-quandles. We also study tensor products of racks and GL-racks coming from universal algebra. Surprisingly, the categories of racks and GL-racks have tensor units. The induced symmetric monoidal structure on medial racks is closed, and similarly for medial GL-racks. Answering another open question, we use GL-racks to distinguish Legendrian knots whose classical invariants are identical. In particular, we complete the classification of Legendrian $8_{13}$ knots. Finally, we use exhaustive search algorithms to classify GL-racks up to order 8.