Quandle Identities and Homology

TL;DR

This paper explores how quandle identities relate to homology, showing that identities produce 2-cycles and influence abelian extensions, with analysis on small connected quandles.

Contribution

It reveals the common structure linking quandle identities, 2-cycles, and abelian extensions, providing new insights into quandle homology theory.

Findings

Identities correspond to specific 2-cycles in quandle homology.

Abelian extensions with 2-cocycles vanishing on these cycles inherit the identities.

Analysis of small connected quandles illustrates the theoretical concepts.

Abstract

Quandle homology was defined from rack homology as the quotient by a subcomplex corresponding to the idempotency, for invariance under the type I Reidemeister move. Similar subcomplexes have been considered for various identities of racks and moves on diagrams. We observe common aspects of these identities and subcomplexes; a quandle identity gives rise to a $2$-cycle, the abelian extension with a $2$-cocycle that vanishes on the $2$-cycle inherits the identity, and a subcomplex is constructed from the identity. Specific identities are examined among small connected quandles.