Commutator theory for racks and quandles

Abstract

We adapt the commutator theory of universal algebra to the particular setting of racks and quandles, exploiting a Galois connection between congruences and certain normal subgroups of the displacement group. Congruence properties such as abelianness and centrality are reflected by the corresponding relative displacement groups, and so do the global properties, solvability and nilpotence. To show the new tool in action, we present three applications: non-existence theorems for quandles (no connected involutory quandles of order 2 k , no latin quandles of order ≡ 2 (mod 4)), a non-colorability theorem (knots with trivial Alexander polynomial are not colorable by latin quandles), and a strengthening of Glauberman's results on Bruck loops of odd order.

Principal and doubly homogeneous quandles

“…Most of the construction and tools used in quandle theory are based on groups [13,19] and modules [18]. In a recent paper [7] we developed a commutator theory for racks and quandles in the sense of [14], where the notion of abelianness and nilpotence are developed for general algebras. In this paper we are going to use this universal algebraic viewpoint to investigate some classes of quandles: principal, doubly homogeneous and cyclic quandles (see Section 2, 3 and 4.2 respectively).…”

“…The contents of the paper are organized as follows: in Section 1 we collect some basic results on quandles and we introduce two new relations for quandles: the first one is related to projection subquandles and the second one was already defined in [7] in connection with central congruences. We define the class of semiregular quandles as the class of quandles with semiregular displacement group (of which the class of principal quandles is an instance).…”

“…If α ∈ Con(A) the algebra A α is called factor algebra. We denote by {γ n (A) ∶ n ∈ N} and by {γ n (A) ∶ n ∈ N} the congruences in the lower central series and of the derived series of A and by ζ A its center (defined in analogy with group theory to capture the notion of solvability and centrality, see [7,14]). By H(A), S(A) and P(A) we denote respectively the set of isomorphism classes of factors, of subalgebras and of powers of A.…”

“…For every congruence α we can define the displacement group relative to the congruence α as Dis α = ⟨{L a L −1 b ∶ a α b}⟩ which is a normal subgroup of LMlt(Q) contained in the kernel Dis α . Properties as abeliannes and centrality of congruences are completely determined by the properties of the relative displacement groups (see [7,Theorem 1.1]).…”

“…The pair of mappings α ↦ Dis α , N ↦ con N provides a Galois correspondence between Con(Q) and N orm(Q) and its properties are described in Section 3.3 of [7]. This correspondence can be used to obtain information on the displacement group from the congruence lattice and vice versa.…”