Sriram Nagaraj
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.
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We study the structure of finite quandles in terms of subquandles. Every finite quandle $Q$ decomposes in a natural way as a union of disjoint $Q$-complemented subquandles; this decomposition coincides with the usual orbit decomposition of $Q$. Conversely, the structure of a finite quandle with a given orbit decomposition is determined by its structure maps. We describe a procedure for finding all non-connected quandle structures on a disjoint union of subquandles.
Reorienting quandle orbits
本文探讨了淬链轨道的重新定向概念，并指出该概念在媒介淬链中不适用。
Computing the associated groups of quandles with tools from group homology theory
本文利用群同调理论将拟群的关联群刻画为内自同构群的中心扩张，并据此计算了包括亚历山大拟群在内的若干族拟群的二阶同调群。
The second quandle homology group of the knot $n$-quandle
本文计算了纽结 $n$-quandle 的第二 quandle 同调群，发现其比纽结 quandle 的同调群信息更丰富，并证明 $3$-quandle 的情形可刻画平凡纽结、三叶结和五叶结。
We define a class of quandle-like structures called pseudoquandles and analyze some of their algebraic properties.
We prove that an Alexander quandle of prime order is generated by any pair of distinct elements. Furthermore, we prove for such a quandle that any ordered pair of distinct elements can be sent to any other such pair by an automorphism of the quandle.
$2$-knots with the same knot group but different knot quandles
本文通过分析扭旋的纽结quandle，构造出同纽结群但不同纽结quandle的2-纽结首例，并分类了具有有限纽结quandle的扭旋。
A. Ishii and K. Oshiro introduced the notion of an $f$-twisted Alexander matrix. This notion is a quandle version of the twisted Alexander matrix which was introduced by M. Wada. They showed that the twisted Alexander matrix of a pair of a link group and a group representation can be recovered from the $f$-twisted Alexander matrix of a pair of a link quandle and a quandle homomorphism. In this paper, we study a relationship between the $f$-twisted Alexander matrix and the quandle cocycle invariant. As an application, we show that an $f$-twisted Alexander invariant of knot quandles is a really …
In this note, residual finiteness of quandles is defined and investigated. It is proved that free quandles and knot quandles of tame knots are residually finite and Hopfian. Residual finiteness of quandles arising from residually finite groups (conjugation, core and Alexander quandles) is established. Further, residual finiteness of automorphism groups of some residually finite quandles is also discussed.
Cocycles are constructed by polynomial expressions for Alexander quandles. As applications, non-triviality of some quandle homology groups are proved, and quandle cocycle invariants of knots are studied. In particular, for an infinite family of quandles, the non-triviality of quandle homology groups is proved for all odd dimensions.
An embedding of spherical quandles into Lie groups
研究通过构造球拟环到李群共轭拟环的光滑嵌入，探讨在1和3维情况下与Bergman和Akita的结果对比。
It is a classical result in reduced homology of finite groups that the order of a group annihilates its homology. Similarly, we have proved that the torsion subgroup of rack and quandle homology of a finite quasigroup quandle is annihilated by its order. However, it does not hold for connected quandles in general. In this paper, we define an $m$-almost quasigroup ($m$-AQ) quandle which is a generalization of a quasigroup quandle and study annihilation of torsion in its rack and quandle homology groups.
We introduce the notion of a quandle with a good involution and its homology groups. Carter et al. defined quandle cocycle invariants for oriented links and oriented surface-links. By use of good involutions, quandle cocyle invariants can be defined for links and surface-links which are not necessarily oriented or orientable. The invariants can be used in order to estimate the minimal triple point numbers of non-orientable surface-links.
We introduce a new homology theory of quandles, called simplicial quandle homology, which is quite different from quandle homology developed by Carter et al. We construct a homomorphism from a quandle homology group to a simplicial quandle homology group. As an application, we obtain a method for computing the complex volume of a hyperbolic link only from its diagram.
Multivariate Alexander quandles, VI. Metabelian groups and 2-component links
本文研究了多变量亚历山大淬火与可解群的关系，证明了其与链接群商群的同构性，并揭示了二链环的 medial 淬火与简化亚历山大不变量的联系。
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.
We propose a simple method to produce quandle cocycles from group cocycles, as a modification of Inoue-Kabaya chain map. We further show that, in respect to "universal central extended quandles", the chain map induces an isomorphism between their third homologies. For example, all Mochizuki's quandle 3-cocycles are shown to be derived from group cocycles of some non-abelian group. As an application, we calculate some $\Z$-equivariant parts of the Dijkgraaf-Witten invariants of some cyclic branched covering spaces, via some cocycle invariant of links.
Nilpotent quandles
本文研究幂零拟群，证明其生成集易刻画且具霍普夫性质，构造自由幂零拟群，恢复阿贝尔拟群分类，并给出幂零性的辫子不变量刻画。
We give a construction of quandle cocycles from group cocycles, especially, for any integer p \geq 3, quandle cocycles of the dihedral quandle R_p from group cocycles of the cyclic group Z/p. We will show that a group 3-cocycle of Z/p gives rise to a non-trivial quandle 3-cocycle of R_p. When p is an odd prime, since dim_{F_p} H_Q^3(R_p; F_p) = 1, our 3-cocycle is a constant multiple of the Mochizuki 3-cocycle up to coboundary. Dually, we construct a group cycle represented by a cyclic branched covering branched along a knot K from the quandle cycle associated with a colored diagram of K.
A quandle is an algebraic structure whose axioms correspond to the Reidemeister moves of knot theory. S. Kamada introduced the notion of a quandle with a good involution, which is later called a symmetric quandle. We are interested in the algebraic structure of symmetric quandles. Given a group $G$, an element $z$ and a certain subgroup $H$, one can obtain the quandle. D. Joyce showed that every quandle is isomorphic to the disjoint union of such quandles. In this paper, given a group $G$, elements $z,r$ in $G$ and a certain subgroup $H$, we construct a symmetric quandle. Futhermore, we show t…
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 of each finite quandle. As an application, we classify finite quandles with up to 5 elements and compute the automorphism group for each quandle.
Characterizations of knot groups and knot symmetric quandles of surface-links
本文推广了已有可定向曲面链纽结群的刻画结论，利用plat表示刻画了含不可定向曲面链在内的曲面链纽结群，还给出了曲面链纽结对称广群的类似刻画，并证明任意带优对合的二面体广群都可实现为某曲面链的纽结对称广群。
On Profinite Quandles
本文研究投射有限广群，给出其多种构造方式、子广群为投射有限的判定条件及代数连通性的内自同构群刻画，补充了否定Stone拓扑广群均为投射有限的例子，结果有望应用于数域平展同伦理论。
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…
Dehn quandles of groups and orientable surfaces
本文引入群关于子集的Dehn quandle以统一多种quandle构造，证明其包络群性质，研究可序性，并计算可定向曲面Dehn quandle的生成集与自同构群，推广了若干已知结果。
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.
Associated groups of symmetric quandles
本文研究对称quandle关联群的结构，给出其群论刻画，探讨其与基础quandle关联群的关系，并证明可嵌入性及确定阿贝尔化。
We prove that if G is an abelian group of odd order then there is an isomorphism from the second quandle homology of the Takasaki quandle of G to the exterior square of G. In particular, for G=Z_k^n, k odd, we obtain Z_k^{n(n-1)/2}. Nontrivial second homology allows us to use 2-cocycles to construct new quandles from T(G), and to construct link invariants.
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 $G$.
This article establishes the algebraic covering theory of quandles. For every connected quandle we explicitly construct a universal covering, which in turn leads us to define the algebraic fundamental group as the automorphism group of the universal covering. We then establish the Galois correspondence between connected coverings and subgroups of the fundamental group. Quandle coverings are thus formally analogous to coverings of topological spaces, and resemble Kervaire's algebraic covering theory of perfect groups. A detailed investigation also reveals some crucial differences, which we illu…
Orderability of link quandles
本文发展了quandles可序性的一般理论，证明许多纤维素素数纽结的纽结quandles右可序，而多数非平凡环面链环的链环quandles不可右序，并探讨了其与包络群可序性的联系。
When a boudnary-parabolic representation of a link group to PSL(2,$\mathbb{C}$) is given, Inoue and Kabaya suggested a combinatorial method to obtain the developing map of the representation using the octahedral triangulation and the shadow-coloring of certain quandle. Quandle is an algebraic system closely related with the Reidemeister moves, so their method changes quite naturally under the Reidemeister moves. In this article, we apply their method to the potential function, which was used to define the optimsitic limit, and construct a saddle point of the function. This construction works f…
Hurwitz equivalence in the universal dihedral quandle
本文研究了m-辫群对m-折万有二面体淬火群的Hurwitz作用，引入三个可计算不变量并证明其可完全分类轨道，同时得到纯m-辫群、虚拟m-辫群及纯虚拟m-辫群的类似分类结果。
Algebraic homology and cohomology theories for quandles have been studied extensively in recent years. With a given quandle 2(3)-cocycle one can define a state-sum invariant for knotted curves(surfaces). In this paper we introduce another version of quandle (co)homology theory, say positive quandle (co)homology. Some properties of positive quandle (co)homology groups are given and some applications of positive quandle cohomology in knot theory are discussed.
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…
Endomorphism monoid and automorphism group of residually finite and profinite quandles
该研究探讨剩余有限与投射quandle，证明有限生成及一大类剩余有限quandle的自同态幺半群、自同构群为剩余有限，给出投射quandle拓扑刻画并建立其自同态幺半群与自同构群为投射的充要条件。
Connections between conjugation quandles and their underlying groups via residual finiteness and the Hopf property
本文研究了共轭quandle与其底层群的Hopf性质和剩余有限性联系，证明了Hopf性质的传递性并提供反例，同时利用Dehn quandle完全刻画了Baumslag-Solitar群共轭quandle的剩余有限性。
Relativization of symmetries on quandles
本文引入满射拟群同态的相对内自同构群和相对换位群，定义连通同态并建立最大连通覆盖分解，最后分类有限纤维的拟群结构。
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.
A quandle is a self-distributive algebraic structure that appears in quasi-group and knot theories. For each abelian group A and c \in A we define a quandle G(A, c) on \Z_3 \times A. These quandles are generalizations of a class of non-medial Latin quandles defined by V. M. Galkin so we call them Galkin quandles. Each G(A, c) is connected but not Latin unless A has odd order. G(A, c) is non-medial unless 3A = 0. We classify their isomorphism classes in terms of pointed abelian groups, and study their various properties. A family of symmetric connected quandles is constructed from Galkin quandl…
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…
Quandles from group actions and a Cayley-type embedding theorem
本文通过研究群作用相关的拟群构造，建立了有限拟群的Cayley型嵌入定理，并利用对称群的自然作用构造出能嵌入所有$n$元拟群的泛拟群。
Fundamental Quandles Do Not Determine the First Postnikov Invariant of 2-Knots
针对Plotnick--Suciu构造及Suciu的替代构造，研究发现基本quandle等对象无法决定定向2-纽结外部空间的同伦类型，还明确了Tanaka--Taniguchi例子的二阶同伦模性质。
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 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…
Homogeneous quandles with abelian inner automorphism groups and vertex-transitive graphs
该研究从简单图出发构造内自同构群为阿贝尔群的广群，证明其齐次性等价于对应图的顶点传递性，并给出有向实格拉斯曼流形子广群的实例，拓展了相关广群的家族。
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…
Lower bounds of betti numbers for homology groups of racks and quandles will be given using the quotient homomorphism to the orbit quandles. Exact sequences relating various types of homology groups are analyzed. Geometric methods of proving non-triviality of cohomology groups are also given, using virtual knots. The results can be applied to knot theory as the first step towards evaluationg the state-sum invariants defined from quandle cohomology.
On the quandles of isometries of the hyperbolic 3-space
本文通过构造与Kleinian群及其非平凡元素相关联的quandle $Q(Γ, γ)$，将纽结quandle推广到双曲纽结，并建立了到$\mathrm{PSL}(2, \mathbb{C})$共轭quandle的具有离散像的单射同态。
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…
We introduce a quandle invariant of classical and virtual links, denoted $Q_{tc} (L)$. This quandle has the property that $Q_{tc} (L) \cong Q_{tc} (L')$ if and only if the components of $L$ and $L'$ can be indexed in such a way that $L=K_1 \cup \dots \cup K_μ$, $L'=K'_1 \cup \dots \cup K'_μ$ and for each index $i$, there is a multiplier $ε_i \in \{-1,1\}$ that connects virtual linking numbers over $K_i$ in $L$ to virtual linking numbers over $K'_i$ in $L'$: $\ell_{j/i}(K_i,K_j)= ε_i \ell_{j/i}(K'_i,K'_j)$ for all $j \neq i$. We also extend to virtual links a classical theorem of Chen, which re…
Joyce has shown that the fundamental quandle of a classical knot can be derived from consideration of the fundamental group and the peripheral structure of the knot, and also that the group and much of the peripheral structure can be recovered from the quandle. We generalize these results to arbitrary dimensions, and also to virtual and welded knots and arcs.
Quandle homology and relative group homology
本文引入从quandle同调到相对群同调的链映射，构造若干quandle上循环，并将其与1维和2维链环的Seifert（超）曲面的三角剖分联系起来。
Symplectic quandle Method and $SL(2,\mathbb C)$-representations of 2-bridge Knots
本文提出了一种基于广义辛淬轮结构的方法，用于计算2-桥结群的SL(2,C)-表示的A多项式，从而获得许多新的结果。
More Automorphism Groups of Quandles
该研究证明了二面体广群位移群的同构结构与含平凡列广群的位移群和内自同构群的等价性，验证并计算了10阶及以内所有广群的自同构群、内自同构群与位移群。
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.
A homology and cohomology theory for topological quandles are introduced. The relation between these (co)homology groups and quandle (co)homology groups are studied. The 1 - topological quandle cocycles are used to compute state sum invariants corresponding to knot diagrams.
Homomorphisms on quandle cohomology groups that raise the dimensions by one are studied in relation to the cocycle state-sum invariants of knots and knotted surfaces. Skein relations are also studied.
Homogeneous quandles with abelian inner automorphism groups
本文刻画了具有阿贝尔内自同构群的齐次拟群，证明其可表示为平凡拟群的阿贝尔扩张，并推广了Furuki和Tamaru构造不连通平坦拟群的工作。
Quandle cocycles are constructed from extensions of quandles. The theory is parallel to that of group cohomology and group extensions. An interpretation of quandle cocycle invariants as obstructions to extending knot colorings is given, and is extended to links component-wise.
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