Equivalences of racks, Legendrian racks, and symmetric racks

Abstract: Racks and Legendrian racks are nonassociative algebraic structures based on the framed and Legendrian Reidemeister moves, respectively. Motivated by the classification problem for Legendrian knots, we construct an equivalence of categories between racks and Legendrian racks (and, hence, GL-quandles). We deduce equivalences between kink-involutory racks and Legendrian quandles, involutory racks and Legendrian kei, and the respective pairs of full subcategories whose objects are medial. As applications, we classify objects in these categories up to order 8 and classify several families of symmetric racks; these results are likely to be of independent interest. In particular, the categories of kei with good involutions, Legendrian kei, and involutory racks are all equivalent.