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

Xiao Wang; Seung Yeop Yang

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arXiv · 2002.04567

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

Xiao Wang·Seung Yeop Yang

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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.

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Xiao Wang, Seung Yeop Yang. 2020-02-11. The geometric realization of a normalized set-theoretic Yang-Baxter homology of biquandles. https://arxiv.org/abs/2002.04567

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