Biquandle (co)homology and handlebody-links

Abstract

In this paper, we introduce the (co)homology group of a multiple conjugation biquandle. It is the (co)homology group of the prismatic chain complex, which is related to the homology of foams introduced by J. S. Carter, modulo a certain subchain complex. We construct invariants for S 1 -oriented handlebody-links using 2-cocycles. When a multiple conjugation biquandle X × Z type X Y is obtained from a biquandle X using n-parallel operations, we provide a 2-cocycle (or 3-cocycle) of the multiple conjugation biquandle X × Z type X Y from a 2-cocycle (or 3-cocycle) of the biquandle X equipped with an X-set Y .2010 Mathematics Subject Classification. 57M27, 57M25.

Cocycles of $G$-Alexander biquandles and $G$-Alexander multiple conjugation biquandles

“…We review chain complexes P * (X; Z) Y , D * (X; Z) Y and C * (X; Z) Y defined in [14] for multiple conjugation biquandles X with X-set Y . Refer to [14] for a geometric interpretation of the chain complexes and an application in knot theory. These chain complexes are an analogy of chain complexes defined in [2] for multiple conjugation quandles.…”

“…Remark 2.8. The notations of this paper are different from that of the paper [14], where we used more notations in order to prove propositions clearly.…”

“…[13,15]), although the terminologies and definitions of such generalizations vary in the literature. Homology theory on biquandles is found in [1,8] and homology theory on multiple conjugation biquandles is found in [14]. In [14] we discussed the (co)homology theory of multiple conjugation biquandles and we provided a method of constructing an invariant of handlebody-links from a 2 or 3-cocycle.…”

“…Homology theory on biquandles is found in [1,8] and homology theory on multiple conjugation biquandles is found in [14]. In [14] we discussed the (co)homology theory of multiple conjugation biquandles and we provided a method of constructing an invariant of handlebody-links from a 2 or 3-cocycle. Thus, in order to construct such an invariant of links or handlebody-links, it is important to find a cocycle of a biquandle or a multiple conjugation biquandle.…”

On set-theoretic Yang-Baxter cohomology groups of cyclic biquandles

“…Biquandles [16], a generalization of quandles, are special cases of set-theoretic Yang-Baxter operators. Since it is known that their cocycles can be used to define invariants of (virtual) knots and links as a state-sum formulation, called Yang-Baxter cocycle invariants [2], the homology theory has been modified in various ways to define invariants of knotted objects such as knotted surfaces [13,19] and handlebody-links [10]. Moreover, one can build homotopical link invariants by using geometric realizations of those homology theories.…”