Quandles are algebraic structures designed to mesh with the Reidemeister moves of knot theory. Joyce and Matveev showed that quandles give rise to a complete invariant of oriented knots. Since the Yang-Baxter equation resembles the third Reidemeister move, it is not surprising that quandles also form a class of set-theoretic solutions of the Yang-Baxter equation. In this talk I will explain how quandles and connected quandles can be enumerated up to isomorphism and list a few open problems. I will also present two additional classes (involutive and idempotent) of set-theoretic solutions of the Yang-Baxter equation with rich algebraic theory.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
