具有自分配操作的集合（以$(a \triangleleft b) \triangleleft c = (a \triangleleft c) \triangleleft (b \triangleleft c)$的意义），特别是淬轮，在结和辫理论、霍普夫代数分类、杨- Baxter方程研究及其他领域中出现。淬轮的重要不变量是其结构群。有限淬轮的结构群已知要么是“无聊的”（自由阿贝尔），要么是“有趣的”（非阿贝尔且有 torsion）。在本文中，我们明确描述了所有具有阿贝尔结构群的有限淬轮。为此，我们证明此类淬轮是阿贝尔的（即满足$(a \triangleleft b) \triangleleft c = (a \triangleleft c) \triangleleft b)$）；展示任何阿贝尔淬轮的结构群作为自由阿贝尔群的中央扩展，由一个显式的有限阿贝尔群扩展；并确定后者何时为平凡。在论文的第二部分，我们将任何淬轮的结构群与其第二同调群$H_2$相关联。我们利用这一点来证明有限淬轮具有阿贝尔结构群的$H_2$是无 torsion 的，但一般的阿贝尔淬轮可能表现出 torsion。$H_2$中的 torsion 对构造结不变量和指向霍普夫代数很重要。
英文摘要：
Sets with a self-distributive operation (in the sense of $(a \triangleleft b) \triangleleft c = (a \triangleleft c) \triangleleft (b \triangleleft c))$, in particular quandles, appear in knot and braid theories, Hopf algebra classification, the study of the Yang-Baxter equation, and other areas. An important invariant of quandles is their structure group. The structure group of a finite quandle is known to be either "boring" (free abelian), or "interesting" (non-abelian with torsion). In this paper we explicitly describe all finite quandles with abelian structure group. To achieve this, we show that such quandles are abelian (i.e., satisfy $(a \triangleleft b) \triangleleft c = (a \triangleleft c) \triangleleft b)$; present the structure group of any abelian quandle as a central extension of a free abelian group by an explicit finite abelian group; and determine when the latter is trivial. In the second part of the paper, we relate the structure group of any quandle to its 2nd homology group $H_2$. We use this to prove that the $H_2$ of a finite quandle with abelian structure group is torsion-free, but general abelian quandles may exhibit torsion. Torsion in $H_2$ is important for constructing knot invariants and pointed Hopf algebras.
A quandle will be called quasi-affine, if it embeds into an affine quandle. Our main result is a characterization of quasi-affine quandles, by group-theoretic properties of their displacement group, by a universal algebraic condition coming from the commutator theory, and by an explicit construction over abelian groups. As a consequence, we obtain efficient algorithms for recognizing affine and quasi-affine quandles, and we enumerate small quasi-affine quandles. We also prove that the "abelian implies quasi-affine" problem of universal algebra has affirmative answer for the class of quandles.
We show that the adjoint group of the Alexander quandle associated to an abelian group M and an automorphism T has a nice description in terms of M and T.
Let $G$ be a group and $φ\in \Aut(G)$. Then the set $G$ equipped with the binary operation $a*b=φ(ab^{-1})b$ gives a quandle structure on $G$, denoted by $\Alex(G, φ)$ and called the generalised Alexander quandle. When $G$ is additive abelian and $φ= -\id_G$, then $\Alex(G, φ)$ is the well-known Takasaki quandle. In this paper, we determine the group of automorphisms and inner automorphisms of Takasaki quandles of abelian groups with no 2-torsion, and Alexander quandles of finite abelian groups with respect to fixed-point free automorphisms. As an application, we prove that if $G\cong (\mathbb…
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 for each quandle $Q$ there exists a unique $Q$-quandle $\mathcal{A}_Q$ (the derived quandle of $Q$) such that derivations from $Q$ to any $Q$-quandle $A$ are in bijective correspondence with $Q$-quandle homomorphisms from $\mathcal{A}_Q$ to $A$. Further, it is proved that the set of all derivations to an abelian $Q$-quandle $A$ has the structure of an abelian…
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 abelian extension of quandles. A non-abelian counterpart of this sequence involving dynamical cohomology classes is also established, and some applications to lifting of quandle automorphisms are given. Viewing the construction of the conjugation, the core and the generalised Alexander quandle of a group as an adjoint functor of some appropriate functor from the c…
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 $\equiv2\pmod4$), a non-colorabil…
We generalise the construction of $Q$-family of quandles and $G$-family of quandles which were introduced in the paper of A. Ishii, M. Iwakiri, Y. Jang, K. Oshiro, and find connection with other constructions of quandles. We define a composition of quandl's structures, which are defined on the same set and find conditions under which this composition gives a quandle. Further we prove that under this multiplication we get a group and show that this group is abelian.
We study rack and quandle coverings from a universal algebraic viewpoint and we show how they can be understood using the notion of strongly abelian congruences. We provide an abstract characterization of several particular types of covering extensions, such as central and abelian ones. We give a new characterization of simply connected quandles and we show that the categorical notion of normal extension coincides with the notion of central covering. We answer several questions from the papers of Clark, Saito and Vendramin \cite{CS} and \cite{CSV} about identities preserved by quandle covering…
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 if $Q$ is a quandle such that the natural map $Q\to G_Q$ from $Q$ to its enveloping group $G_Q$ is injective, then $Q$ is the $(G,A)$-quandle for an appropriate group $G$ and its subset $A$. Also we introduce the free product of quandles and study this construction for $(G,A)$-quandles. In addition, we classify all finite quandles with enveloping group $\math…
We establish a canonical correspondence between connected quandles and certain configurations in transitive groups, called quandle envelopes. This correspondence allows us to efficiently enumerate connected quandles of small orders, and present new proofs concerning connected quandles of order p and 2p. We also present a new characterization of connected quandles that are affine.
For any twisted conjugate quandle $Q$, and in particular any Alexander quandle, there exists a group $G$ such that $Q$ is embedded into the conjugation quandle of $G$.
A (left) quandle is connected if its left multiplication group acts transitively. In 2014, Eisermann introduced the concept of quandle coverings, corresponding to so-called constant quandle cocycles that form a subset of quandle cocycles. A connected quandle is said to be \emph{simply connected} if it has no nontrivial coverings, or, equivalently, if its second constant cohomology groups are trivial. In this paper we develop a combinatorial approach to constant cohomology. Upon applying our theory, we prove that connected quandles that are affine over cyclic groups are simply connected (extend…
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 isomorphisms) of order less than or equal to 5 were given. With the help of the software Maple, we compute the inner and automorphism groups of all seventy three quandles of order six listed in the appendix of [4]. Since computations of automorphisms of quandles relates to the problem of classification of quandles, we also describe an algorithm implemente…
This paper develops an approach for describing centrally extended groups, as determining the adjoint groups associated with quandles. Furthermore, we explicitly describe such groups of some quandles. As a corollary, we determine some second quandle homologies.
Quandle is an algebraic system with one binary operation, but it is quite different from a group. Quandle has its origin in the knot theory and good relationships with the theory of symmetric spaces, so it is well-studied from points of view of both areas. In the present paper, we investigate a special kind of quandles, called generalized Alexander quandles $Q(G,ψ)$, which is defined by a group $G$ together with its group automorphism $ψ$. We develop the quandle invariants for generalized Alexander quandles. As a result, we prove that there is a one-to-one correspondence between generalized Al…
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 generalized Alexander quandles of $G$ one-to-one correspond to the conjugacy classes of the automorphism groups of $G$. This correspondence can be also claimed for the case of symmetric groups. Secondly, we give a characterization of generalized Alexander quandles of finite groups $G$ under some assumptions in terms of $G$. As corollaries of this characterization, we…
In this article, we characterize the (covariant) isotropy groups of free, finitely generated racks and quandles. As a consequence, we show that the usual inner automorphisms of such racks and quandles are precisely those automorphisms that are "coherently extendible". We then use this result to compute the global isotropy groups of the categories of racks and quandles, i.e. the automorphism groups of the identity functors of these categories.
In this paper we study different questions concerning automorphisms of quandles. For a conjugation quandle $Q={\rm Conj}(G)$ of a group $G$ we determine several subgroups of ${\rm Aut}(Q)$ and find necessary and sufficient conditions when these subgroups coincide with the whole group ${\rm Aut}(Q)$. In particular, we prove that ${\rm Aut}({\rm Conj}(G))={\rm Z}(G)\rtimes {\rm Aut}(G)$ if and only if either ${\rm Z}(G)=1$ or $G$ is one of the groups $\mathbb{Z}_2$, $\mathbb{Z}_2^2$ or $\mathbb{Z}_3$. For a big list of Takasaki quandles $T(G)$ of an abelian group $G$ with $2$-torsion we prove th…
Residual finiteness is known to be an important property of groups appearing in combinatorial group theory and low dimensional topology. In a recent work [2] residual finiteness of quandles was introduced, and it was proved that free quandles and knot quandles are residually finite. In this paper, we extend these results and prove that free products of residually finite quandles are residually finite provided their associated groups are residually finite. As associated groups of link quandles are link groups, which are known to be residually finite, it follows that link quandles are residually…
Quandles are self-distributive, right-invertible, idempotent algebras. A group with conjugation for binary operation is an example of a quandle. Given a quandle $(Q, \ast)$ and a positive integer $n$, define $a\ast_n b = (\cdots (a\ast \underbrace{b)\ast \cdots )\ast b}_{n}$, where $a, b \in Q$. Then, $(Q, \ast_n)$ is again a quandle. We set forth the following problem. ``Find $(Q, \ast)$ such that the sequence $\{(Q, \ast_n)\, :\, n\in \mathbb{Z}^+ \}$ is made up of pairwise non-isomorphic quandles.'' In this article we find such a quandle $(Q, \ast)$. We study the general linear group of $2$…
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 quandle. Such quandles are called verbal quandles with one parameter.
We prove that for n>2 there exists a quandle of cyclic type of size n if and only if n is a power of a prime number. This establishes a conjecture of S. Kamada, H. Tamaru and K. Wada. As a corollary, every finite quandle of cyclic type is an Alexander quandle. We also prove that finite doubly transitive quandles are of cyclic type. This establishes a conjecture of H. Tamaru.
In this paper, a theory of quandle rings is proposed for quandles analogous to the classical theory of group rings for groups, and interconnections between quandles and associated quandle rings are explored.
Hayashi property for conjugation quandles over finite Coxeter groups
Indian Institute of Technology Jodhpur（印度技术学院乔德普尔分校）；School of Advanced Engineering, UPES（UPES高级工程学院）；Indian Institute of Science Education and Research Mohali（印度科学教育与研究学院莫哈利分校）
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, i.e., satisfies the identity $x\lhd (x\lhd y) = y$ in addition to the other identities defining a quandle. Trajectories $(x_i)_{i\in\mathbb{Z}}$ in groups and in involutory quandles (in the former context, sequences of the form $x_i = x z^i$ where $x,z\in G,$ among other characterizations; in the latter, sequences satisfying $x_{i+1}= x_i\lhd\,x_{i-1})$ are ex…
This paper studies the left (right) middle translations on finite involutory latin quandles and their representations. It also shows that a left involutory latin quandle of odd order n can be constructed from a cyclic group of odd order by the application of left middle translations. Furthermore, the concept of spins of involutory latin quandle is investigated, and it is shown that if Q is a left involutory latin quandle of odd order n, then the set of all right spins (r-spins) is a cyclic group of odd order n under composition of mapping.
T. Mochizuki determined all 3-cocycles of the third quandle cohomologies of Alexander quandles on finite fields. We show that all the 3-cocycles, except those of 2-cocycle forms, are derived from group 3-cocycles of a meta-abelian group. Further, the quandle cocycle invariant of a link using Mochizuki's 3-cocycle is equivalent to a $\Z$-equivariant part of the Dijkgraaf-Witten invariant of a cyclic covering of $S^3$ branched over the link using the group. We compute some Massey triple products via the former invariant.
We give a generating set of the generalized Reidemeister moves for oriented singular links. We use it to introduce an algebraic structure arising from the study of oriented singular knots. We give some examples, including some non-isomorphic families of such structures over non-abelian groups. We show that the set of colorings of a singular knot by this new structure is an invariant of oriented singular knots and use it to distinguish some singular links.
In the paper we describe the class of principal quandles and show that connected quandles can be decomposed as a disjoint union of principal quandles. We also prove that simple affine quandles are finite and they can be characterized among finite simple quandles by several different equivalent properties, as for instance being doubly homogeneous (i.e. having a doubly transitive automorphism group). A complete description of finite doubly-homogeneous quandles is provided extending the result of \cite{V} and solving \cite[Problem 6.7]{2trans}. We also provide a classification of connected cyclic…
