Some calculations of cohomology groups of finite Alexander quandles
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 \(\mathbb{Z}_p\).
scientific article; zbMATH DE number 1901787