Torsion in one-term distributive homology

TL;DR

This paper investigates the torsion properties of one-term distributive homology, revealing that finite spindles can have torsion and providing methods to compute and analyze these homologies.

Contribution

It demonstrates that finite spindles can exhibit torsion in their homology, introduces techniques for precise computation, and links group structures to homology torsion.

Findings

Finite spindles can have torsion in their homology.

Any finite group can appear as torsion subgroup of some finite spindle.

Certain shelves lead to trivial one-term homology.

Abstract

The one-term distributive homology was introduced by J.H.Przytycki as an atomic replacement of rack and quandle homology, which was first introduced and developed by R.Fenn, C.Rourke and B.Sanderson, and J.S.Carter, S.Kamada and M.Saito. This homology was initially suspected to be torsion-free, but we show in this paper that the one-term homology of a finite spindle can have torsion. We carefully analyze spindles of block decomposition of type (n,1) and introduce various techniques to compute their homology precisely. In addition, we show that any finite group can appear as the torsion subgroup of the first homology of some finite spindle. Finally, we show that if a shelf satisfies a certain, rather general, condition then the one-term homology is trivial.