Quandle Identities and Homology

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.