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.

Biracks and biquandles, which are useful for studying the knot theory, are special families of solutions of the set-theoretic Yang–Baxter equation. A homology theory for the set-theoretic Yang–Baxter equation was developed by Carter et al. in order to construct knot invariants. In this paper, we construct a normalized (co)homology theory of a set-theoretic solution of the Yang–Baxter equation. We obtain some concrete examples of nontrivial [Formula: see text]-cocycles for Alexander biquandles. For a biquandle [Formula: see text] its geometric realization [Formula: see text] is discussed, which has the potential to build invariants of links and knotted surfaces. In particular, we demonstrate that the second homotopy group of [Formula: see text] is finitely generated if the biquandle [Formula: see text] is finite.