Cocycles of \(G\)-Alexander biquandles and \(G\)-Alexander multiple conjugation biquandles
type: Item
instance of: scholarly article
title: Cocycles of \(G\)-Alexander biquandles and \(G\)-Alexander multiple conjugation biquandles
zbMATH Open document ID: 1476.57016
author: Atsushi Ishii
author: Masahide Iwakiri
author: Seiichi Kamada
author: Jieon Kim
author: Shosaku Matsuzaki
author: Kanako Oshiro
published in: Topology and its Applications
publication date: 2021-09-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/2002.11292
review text: 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.
reviewed by: Mahender Singh
Mathematics Subject Classification ID: 57K12
zbMATH DE Number: 7401498
zbMATH Keywords: quandles
zbMATH Keywords: biquandles
zbMATH Keywords: \(G\)-Alexander biquandles
zbMATH Keywords: \(G\)-Alexander multiple conjugation biquandles
zbMATH Keywords: cocycle invariants
MaRDI profile type: Publication
OpenAlex ID: W3111511067
arXiv ID: 2002.11292
cites work: Homology theory for the set-theoretic Yang–Baxter equation and knot invariants from generalizations of quandles
cites work: Homology for quandles with partial group operations
cites work: Quandle cohomology and state-sum invariants of knotted curves and surfaces
cites work: Augmented biracks and their homology
cites work: Biquandles and virtual links
cites work: RACKS AND LINKS IN CODIMENSION TWO
cites work: Trunks and classifying spaces
cites work: The rack space
cites work: Cohomology theory in abstract groups. I
cites work: Quandle homology and complex volume
cites work: A multiple conjugation quandle and handlebody-knots
cites work: A \(G\)-family of quandles and handlebody-knots
cites work: A multiple conjugation biquandle and handlebody-links
cites work: Biquandle (co)homology and handlebody-links
cites work: Partially multiplicative biquandles and handlebody-knots
cites work: A classifying invariant of knots, the knot quandle
cites work: Q4419466
cites work: Qualgebras and knotted 3-valent graphs
cites work: DISTRIBUTIVE GROUPOIDS IN KNOT THEORY
cites work: Some calculations of cohomology groups of finite Alexander quandles
cites work: Quandle cocycles from invariant theory
DOI: 10.1016/J.TOPOL.2020.107512
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description: scientific article