The torsion of a finite quasigroup quandle is annihilated by its order

Abstract

We prove that if Q is a finite quasigroup quandle, then |Q| annihilates the torsion of its rack homology.

On set-theoretic Yang-Baxter cohomology groups of cyclic biquandles

“…See [11,29] for details. The homology of special families of set-theoretic Yang-Baxter operators, known as racks and quandles, has been extensively studied, see [4,7,18,20,21,25,26,31], but other than for racks and quandles little is known. Therefore, it is important for the study of invariants of knotted objects to determine settheoretic Yang-Baxter (co)homology groups of biquandles and find explicit formulae of their cocycles.…”

Quandle cohomology is a Quillen cohomology

“…These invariants have found numerous applications, not only to knot theory. See [1,6,7] for instance, as well as [4,24,26,22,27,15] for various other approaches to rack and quandle (co)homology groups.…”

Foundations of topological racks and quandles

“…Quandles are non-associative algebraic structures (with the exception of the trivial quandles) that correspond to the axiomatization of the three Reidemeister moves in knot theory. Since 1982 when quandles were introduced by Joyce [12] and Matveev [13] independently, there have been investigations, (see for example [8,14,[16][17][18][19][20]), that have mostly focused on finite quandles. Joyce and Matveev proved that the fundamental quandle of a knot is a complete invariant up to orientation.…”