The geometric realization of a normalized set-theoretic Yang–Baxter homology of biquandles

Abstract

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.