Related papers: Constructing an infinite family of quandles from a…
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…
We review the definition of a quandle, and in particular of the core quandle $\mathrm{Core}(G)$ of a group $G$, which consists of the underlying set of $G$, with the binary operation $x\lhd y = x y^{-1} x$. This is an involutory quandle,…
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…
The paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an…
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,…
We study the quandle counting invariant for a certain family of finite quandles with trivial orbit subquandles. We show how these invariants determine the linking number of classical two-component links up to sign.
Finite quandles with n elements can be represented as n-by-n matrices. We show how to use these matrices to distinguish all isomorphism classes of finite quandles for a given cardinality n, as well as how to compute the automorphism group…
This paper is devoted to qualgebras and squandles, which are quandles enriched with a compatible binary/unary operation. Algebraically, they are modeled after groups with conjugation and multiplication/squaring operations. Topologically,…
This paper gives a new way of characterizing L-space $3$-manifolds by using orderability of quandles. Hence, this answers a question of Adam Clay et al. [Question 1.1 of Canad. Math. Bull. 59 (2016), no. 3, 472-482]. We also investigate…
We define a two-variable polynomial invariant of finite quandles. In many cases this invariant completely determines the algebraic structure of the quandle up to isomorphism. We use this polynomial to define a family of link invariants…
The goal of this paper is to characterization generalized Alexander quandles of finite groups in the language of the underlying groups. Firstly, we prove that if finite groups $G$ are simple, then the quandle isomorphic classes of…
We find all words $W(x,y,z)$ in the free group $F(x,y,z)$, such that for every group $G$ and an element $c\in G$ the algebraic system $(G,*_{W,c})$ with the binary operation $*_{W,c}$ given by $a*_{W,c}b=W(a,b,c)$ for $a,b\in G$ is a…
A latin bitrade is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. Dr\'apal (\cite{Dr9}) showed that a latin bitrade is…
In this paper, we resolve a conjecture of Green and Liebeck [Disc. Math., 343 (8):117119, 2019] on codes in $PGL(2,q)$. To be specific, we show that: if $D$ is a dihedral subgroup of order $2(q+1)$ in $G=PGL(2,q)$, and $A=\{g\in G: g^{q+1}=…
We give a complete description of the associated group of any quandle as a central extension of the inner-automorphism group. As an application, we compute the second quandle homology groups of quandles of some families, including those of…
We define a class of quandle-like structures called pseudoquandles and analyze some of their algebraic properties.
The fundamental quandle is a powerful invariant of knots and links, but it is difficult to describe in detail. It is often useful to look at quotients of the quandle, especially finite quotients. One natural quotient introduced by Joyce is…
In this note, we show that the category of Latin (resp. commutative) medial quandles is equivalent to the category of affine modules over a certain Laurent polynomial ring (resp. the dyadic rationals). As applications, we describe free…
The aim of this paper is to propose a theory of derivations for quandles. Given a quandle $A$ admitting an action by a quandle $Q$, derivations from $Q$ to $A$ are introduced as twisted analogues of quandle homomorphisms. It is shown that…
A rack of order $n$ is a binary operation $\rack$ on a set $X$ of cardinality $n$, such that right multiplication is an automorphism. More precisely, $(X,\rack)$ is a rack provided that the map $x\mapsto x\rack y$ is a bijection for all…