Classification and structure of generalized Legendrian racks

Abstract

We study algebraic aspects of generalized Legendrian racks, which are nonassociative structures based on the Legendrian Reidemeister moves. 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. Then we compute the centers of the category of GL-racks and several of its full subcategories. We also 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.

29 pages, 4 tables; comments welcome. Section 7 and the end of Appendix A have been omitted due to computational errors identified by Zhiyi He. The end of Section 4 has been merged with Section 5. The title, abstract, and introduction are updated to reflect these changes