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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 n -cocycles for Alexander biquandles. For a biquandle X , its geometric realization B X is discussed, which has the potential to build invariants of links and knotted surfaces. In particular, we demonstrate that the second homotopy group of B X is finitely generated if the biquandle X is finite. [ABSTRACT FROM AUTHOR]

We introduce a modified homology and cohomology theory for involutory biquandles (also known as bikei). We use bikei 2-cocycles to enhance the bikei counting invariant for unoriented knots and links as well as unoriented and non-orientable knotted surfaces in R4. [ABSTRACT FROM AUTHOR]

We compute the third cohomology group of a dihedral quandle of odd order with finite field coefficient, generalizing our previous work. In particular, we obtain many new examples of non-trivial 3-cocycles. We check non-triviality of the associated invariant for knots. [ABSTRACT FROM AUTHOR]

We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virtual knots and links. In particular, they provide a method for detecting non-classicality of virtual knots and links. [ABSTRACT FROM AUTHOR]

Fox's shadow p-colorings of a knot K define two kinds of Laurent polynomials, Φp(K) and $\Phi_{p}^{*}(K) \in \mathbb{Z}[t^{\pm1}]/(t^p-1)$, as invariants of K. We prove that the equality $\Phi_p(K) = p\Phi_{p}^{*}(K)$ holds for any knot K. Also we prove that, if the p-coloring number of K is equal to p2, then $\Phi_{p}^{*}(K)$ has the form $p \sum_{i=0}^{p-1} t^{Ni^{2}}$ for some N ∈ ℤp. [ABSTRACT FROM AUTHOR]

Quandle cocycles are constructed from extensions of quandles. The theory is parallel to that of group cohomology and group extensions. An interpretation of quandle cocycle invariants as obstructions to extending knot colorings is given, and is extended to links component-wise. [ABSTRACT FROM AUTHOR]