Homology operations on homology of quandles

TL;DR

This paper introduces new homological operations on quandle homology, including quandle partial derivatives and extreme chains, expanding the algebraic tools available for studying quandles.

Contribution

It presents novel homological operations on quandle homology, notably quandle partial derivatives and the concept of extreme chains, along with degree one operations using the $k$-condition.

Findings

Defined quandle partial derivatives and extreme chains.

Constructed homological operations from extreme chains.

Developed degree one operations using the $k$-condition.

Abstract

We consider various homological operations on homology of quandles. We introduce the notion of quandle partial derivatives, and extreme chains on which appropriate partial derivatives vanish. Extreme chains yield homological operations. We also consider the degree one homology operations created using elements of the quandle satisfying the so-called $k$-condition.