Related papers: Connected quandles and transitive groups

We define the notion of the orbit group of a quandle via its connectivity and compute the orbit groups for some basic quandles. We also show that the orbit group counts the number of orbits of certain quandles.

In the present paper, we introduce the new construction of quandles. For a group $G$ and its subset $A$ we construct a quandle $Q(G,A)$ which is called the $(G,A)$-quandle and study properties of this quandle. In particular, we prove that…

We undertake the study of profinite quandles. We provide several constructions of profinite quandles from profinite groups, and from other profinite quandle. We characterize which subquandles of profinite quandles are again profinite.…

Isomorphism classes of Alexander quandles of order 16 are determined, and classes of connected quandles are identified. This paper extends the list of known distinct connected finite Alexander quandles.

We prove that the automorphism group of the dihedral quandle with n elements is isomorphic to the affine group of the integers mod n, and also obtain the inner automorphism group of this quandle. In [9], automorphism groups of quandles (up…

To any finite ordered subset and any finite partition of a group a set of tuples of positive integers, named as configurations, is associated that describes the group's behavior. The present paper provides an exposition of this notion and…

We construct the Langlands correspondence for connected reductive groups over finite fields, which we call the finite Langlands correspondence. We discuss also its relation with the categorical local Langlands correspondence.

A continuous cohomology theory for topological quandles is introduced, and compared to the algebraic theories. Extensions of topological quandles are studied with respect to continuous 2-cocycles, and used to show the differences in second…

We are interested in abstract conditions that characterize homomorphic images of affine quandles. Our main result is a two-fold characterization of this class: one by a property of the displacement group, the other one by a property of the…

Special kinds of rank 2 vector bundles with (possibly irregular) connections on P^1 are considered. We construct an equivalence between the derived category of quasi-coherent sheaves on the moduli stack of such bundles and the derived…

We completely describe good involutions of free quandles and subquandles of twisted conjugation quandles of groups, including all Alexander quandles. As an application, we enumerate good involutions of linear quandles, and we provide…

We introduce a notion of natural orderings of elements of finite connected quandles of order $n$. When the elements of such a quandle $Q$ are already ordered naturally, any automophism on $Q$ is a natural ordering. Although there are many…

We prove a Dold-Kan type correspondence between the category of dendroidal abelian groups and a suitably constructed category of dendroidal complexes. Our result naturally extends the classical Dold-Kan correspondence between the category…

We describe all subdirectly irreducible medial quandles. We show that they fall within one of four disjoint classes. In particular, in the finite case they are either connected (and therefore Alexander quandles) or reductive. Moreover, we…

In this paper, we give a characterization of homogeneous quandles with abelian inner automorphism groups. In particular, we show that such a quandle is expressed as an abelian extension of a trivial quandle. Our construction is a…

We show that there is an unique connected quandle of order twice an odd prime number greater than 3. It has order 10 and is isomorphic to the conjugacy class of transpositions in the symmetric group of degree 5. This establishes a…

In this paper, we introduce the notion of circular orderability for quandles. We show that the set all right (respectively left) circular orderings of a quandle is a compact topological space. We also show that the space of right…

The axioms of a quandle imply that the columns of its Cayley table are permutations. This paper studies quandles with exactly one non-trivially permuted column. Their automorphism groups, quandle polynomials, (symmetric) cohomology groups,…

In this work, we establish a categorification of the classical Dold-Kan correspondence in the form of an equivalence between suitably defined $\infty$-categories of simplicial stable $\infty$-categories and connective chain complexes of…

Using the classification of transitive groups we classify indecomposable quandles of size <36. This classification is available in Rig, a GAP package for computations related to racks and quandles. As an application, the list of all…