Search

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 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]

The purpose of this paper is to determine the homology groups of trivial quandles with good involutions. We also show that the quandle cocycle invariants of surface-links obtained from trivial quandles with good involutions are equivalent to the triple linking numbers. [ABSTRACT FROM AUTHOR]

We define the fundamental quandle of a spatial graph and several invariants derived from it. In the category of graph tangles, we define an invariant based on the walks in the graph and cocycles from nonabelian quandle cohomology. [ABSTRACT FROM AUTHOR]

Cocycles are constructed by polynomial expressions for Alexander quandles. As applications, non-triviality of some quandle homology groups are proved, and quandle cocycle invariants of knots are studied. In particular, for an infinite family of quandles, the non-triviality of quandle homology groups is proved for all odd dimensions. [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]