Some calculations of cohomology groups of finite Alexander quandles
type: Item
instance of: scholarly article
title: Some calculations of cohomology groups of finite Alexander quandles
zbMATH Open document ID: 1036.18009
DOI: 10.1016/S0022-4049(02)00323-7
author: Takuro Mochizuki
published in: Journal of Pure and Applied Algebra
publication date: 2003-04-28T00:00:00Z
review text: The subject of quandle cohomology is of interest to knot theorists, since every quandle 2-cocyle defines a state-sum invariant of links [see \textit{J. S. Carter}, \textit{D. Jelsovsky}, \textit{S. Kamada}, \textit{L. Langford} and \textit{M. Saito}, Trans. Am. Math. Soc. 355, 3947--3989 (2003; Zbl 1028.57003)]. Similarly, every quandle 3-cocyle defines an invariant of 2-knots (surfaces embedded in \(\mathbb{R}^4\)). In this paper, the author proves several results on the structure of the 2nd and 3rd cohomology groups of finite Alexander quandles. Specifically, the second cohomology groups of finite Alexander quandles with coefficients in \(\mathbb{Q}\) and in finite fields are determined, as are the third cohomology groups of Alexander quandles of the form \(\mathbb{Z}_p/(T-\omega)\) for \(p\) prime with coefficients in the finite field \(\mathbb{Z}_p\).
reviewed by: Sam Nelson
Mathematics Subject Classification ID: 18G60
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 55N35
zbMATH DE Number: 1901787
zbMATH Keywords: quandle cohomology
zbMATH Keywords: Alexander quandles
MaRDI profile type: Publication
cites work: Quandle cohomology and state-sum invariants of knotted curves and surfaces
cites work: Computations of quandle cocycle invariants of knotted curves and surfaces
cites work: Quandle homology groups, their Betti numbers, and virtual knots
cites work: The Betti numbers of some finite racks
cites work: The 2-twist-spun trefoil has the triple point number four
full work available at URL: s0022-4049(02)00323-7
OpenAlex ID: W1981781970
description: scientific article; zbMATH DE number 1901787