Set-theoretic Yang–Baxter (co)homology theory of involutive non-degenerate solutions

Abstract

W. Rump showed that there exists a one-to-one correspondence between involutive right non-degenerate solutions of the Yang–Baxter equation and cycle sets. J. S. Carter, M. Elhamdadi, and M. Saito, meanwhile, introduced a homology theory of set-theoretic solutions of the Yang–Baxter equation in order to define cocycle invariants of classical knots. In this paper, we introduce the normalized homology theory of an involutive right non-degenerate solution of the Yang–Baxter equation and compute the normalized set-theoretic Yang–Baxter homology of cyclic racks. Moreover, we explicitly calculate some two-cocycles, which can be used to classify certain families of torus links.