Simply connected latin quandles

Abstract

A (left) quandle is connected if its left translations generate a group that acts transitively on the underlying set. In 2014, Eisermann introduced the concept of quandle coverings, corresponding to constant quandle cocycles of Andruskiewitsch and Graña. A connected quandle is simply connected if it has no nontrivial coverings, or, equivalently, if all its second constant cohomology sets with coefficients in symmetric groups are trivial.In this paper we develop a combinatorial approach to constant cohomology. We prove that connected quandles that are affine over cyclic groups are simply connected (extending a result of Graña for quandles of prime size) and that finite doubly transitive quandles of order different from 4 are simply connected. We also consider constant cohomology with coefficients in arbitrary groups.

Embeddings of quandles into groups

“…A quandle Q is called latin if for every pair of elements x, y ∈ Q there exists a unique element z ∈ Q such that x = y * z. Such quandles were studied, for example, in [6,19,21]. If Q is a finite latin quandle then the multiplication table of Q is a latin square.…”

A Universal algebraic approach to rack coverings

“…Connected quandles with a cyclic displacement group and connected quandles with doubly transitive displacement group are simply connected [BV18] and simply connected quandles of size p 2 have been classified in [GIV17]. Table 1 collects all the other simply connected quandles up to size 47 which do not fall in these families (the data have been computed using Theorem 3.2 and the labels are taken from the RIG library of GAP [RIG]).…”