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, Elhamdadi, and Saito 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 non-trivial $n$-cocycles for Alexander biquandles. For a biquandle $X,$ its geometric realization $BX$ is discussed, which has the potential to build invariants of links and knotted surfaces. In particular, we demonstrate that the second homotopy group of $BX$ is finitely generated if the biquandle $X$ is finite.