Quandle Identities and Homology
W. Edwin Clark, Masahico Saito

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 -cycle, the abelian extension with a -cocycle that vanishes on the -cycle inherits the identity, and a subcomplex is constructed from the identity. Specific identities are examined among small connected quandles.
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Taxonomy
TopicsTopological and Geometric Data Analysis · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models
