Cocycles of \(G\)-Alexander biquandles and \(G\)-Alexander multiple conjugation biquandles
It is well-known that quandle 2-cocycles and 3-cocycles can be used to construct invariants of knots and links. A multiple conjugation quandle is an algebra motivated by handlebody-links. The notion of a quandle and a multiple conjugation quandle can be generalised to biquandles and multiple conjugation biquandles, respectively. A (co)homology theory of multiple conjugation biquandles is also known and it can be used to construct invariants of handlebody-links through 2-cocycles and 3-cocycles. This makes the construction of such cocycles an interesting problem, which is considered in the paper under review. The authors introduce \(G\)-Alexander biquandles and \(G\)-Alexander multiple conjugation biquandles for a group \(G\). It is known due to \textit{T. Nosaka} that, for a given group \(G\), a \(G\)-module \(M\) and a \(G\)-invariant group cocycle of the \(G\)-family \(M\), a cocycle of the associated quandle \(M \times G\) can be constructed using invariant theory [Adv. Math. 245, 423--438 (2013; Zbl 1287.55002)]. The authors show that a cocycle of the \(G\)-Alexander multiple conjugation biquandle can be obtained from a group cocycle. Since any \(G\)-invariant multilinear map provides a \(G\)-invariant group cocycle, this yields many cocycles of the biquandle.
\(G\)-Alexander multiple conjugation biquandles
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