The second quandle homology of the Takasaki quandle of an odd abelian group is an exterior square of the group

TL;DR

This paper establishes an isomorphism between the second quandle homology of Takasaki quandles of odd order abelian groups and the exterior square of those groups, enabling new quandle constructions and link invariants.

Contribution

It proves a new isomorphism linking second quandle homology to exterior squares for odd order abelian groups, expanding understanding of quandle homology.

Findings

Second homology is isomorphic to the exterior square of the group

For G=Z_k^n with odd k, the homology is Z_k^{n(n-1)/2}

Nontrivial homology enables new quandle constructions and link invariants

Abstract

We prove that if G is an abelian group of odd order then there is an isomorphism from the second quandle homology of the Takasaki quandle of G to the exterior square of G. In particular, for G=Z_k^n, k odd, we obtain Z_k^{n(n-1)/2}. Nontrivial second homology allows us to use 2-cocycles to construct new quandles from T(G), and to construct link invariants.