The Cocycle structure of the Alexander $f$-quandles on finite fields

TL;DR

This paper computes the second, third, and fourth cohomology groups of Alexander $f$-quandles over finite fields, revealing their cocycle structures and algebraic properties.

Contribution

It provides explicit calculations of higher cohomology groups for a class of Alexander $f$-quandles over finite fields, advancing understanding of their algebraic topology.

Findings

Cohomology groups of Alexander $f$-quandles are explicitly determined

Identifies cocycle structures for specific finite field quandles

Enhances algebraic understanding of quandle cohomology in finite settings

Abstract

We determine the second, third, and fourth cohomology groups of Alexander $f$-quandles of the form $\mathbb{F}_q[T,S]/ (T-\omega, S-\beta)$, where $\mathbb{F}_q$ denotes the finite field of order $q$, $\omega \in \mathbb{F}_q\setminus \{0,1\}$, and $\beta \in \mathbb{F}_q$