On uniconnected solutions of the Yang-Baxter equation and Dehornoy's class

Abstract

In the first part, we focus on indecomposable involutive solutions of the Yang-Baxter equation whose permutation group forces them to be uniconnected. Indecomposable involutive solutions with a permutation group isomorphic to a dihedral group or a minimal non-cyclic group are studied in detail. In the last part, we study the Dehornoy class of involutive solutions (not necessarily indecomposable) and its link with left braces. As an application, we give an upper bound for several families of indecomposable involutive solutions and we compute the precise value in some other cases.