Quandle cohomology is a Quillen cohomology

Abstract

Racks and quandles are fundamental algebraic structures related to the topology of knots, braids, and the Yang-Baxter equation. We show that the cohomology groups usually associated with racks and quandles agree with the Quillen cohomology groups for the algebraic theories of racks and quandles, respectively. This makes available the entire range of tools that comes with a Quillen homology theory, such as long exact sequences (transitivity) and excision isomorphisms (flat base change).MSC: 18G50 (18C10, 20N02, 55U35, 57M27)

The homotopy types of free racks and quandles

“…On the one hand, the Quillen homology groups of Q are given as the homotopy groups of the abelianization Q ab of Q. It is shown in [47] that this agrees with the usual quandle homology groups of Q when those are defined. On the other hand, there are the homology groups of the stabilization Q st of Q.…”

“…If the theory T is the theory of racks or quandles, Quillen homology and cohomology is discussed in detail in [46,47], where the Quillen cohomology groups were shown to be isomorphic (up to a shift in degrees) with cohomology groups defined earlier using less universally applicable methods [10,22,24,25]. This point of view has already led to new computations [32] of rack homology.…”

“…Let K be a knot, and let Q K be its knot quandle. We need to recall a few facts about the quandle homology H n (Q) of Q = Q K , and we will do this by referring to the usual indexing of quandle homology; Quillen's H n (Q) is the usual H n+1 (Q) [47]. The first quandle homology group H 1 (Q K ) is easily seen to be infinite cyclic.…”

“…One of the applications that our analogs of Milnor's theorem have is toward a deeper understanding of the homological invariants of racks and quandles. Indeed, these two structures have already been studied using homological methods, first using explicit chain complexes [10,24,25], and then in the more versatile context of Quillen's homotopical algebra [40,47], leading to new homological knot invariants [46] and new computations of rack homology [32].…”

Groups, conjugation and powers

“…This basic understanding of abelian power quandles is enough to develop the Quillen homology theory for power quandles by deriving the abelianization functor. We will not do this here and refer to [16], where one of us treated the case of plain quandles, without power structure.…”