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

Abstract

Biquandles and multiple conjugation biquandles are algebras which are related to links and handlebody-links in 3-space. Cocycles of them can be used to construct state-sum type invariants of links and handlebody-links. In this paper we discuss cocycles of a certain class of biquandles and multiple conjugation biquandles, which we call G-Alexander biquandles and G-Alexander multiple conjugation biquandles, with a relationship with group cocycles. We give a method to obtain a (biquandle or multiple conjugation biquandle) cocycle of them from a group cocycle.