Toshiyuki Akita
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 $G$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Quandles with good involutions, which are called symmetric quandles, can be used to define invariants of unoriented knots and links. In this paper, we determine the necessary and sufficient condition for good involutions of a generalized Alexander quandle to exist. Moreover, we classify all good involutions of a connected generalized Alexander quandle up to symmetric quandle isomorphisms.
Classification of generalized Alexander quandles
本文利用基础群及其自同构，为广义Alexander quandle的同构类提供了新的刻画以推广先前结论，并计算了阶数不超过127的群及其群自同构导出的广义Alexander quandle数量。
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…
Ishii and Oshiro introduced the notion of an $f$-twisted Alexander matrix, which is a quandle version of a twisted Alexander matrix and defined an invariant of finitely presented quandles. In this paper, we study $f$-twisted Alexander matrices of certain quandles with the Alexander pair obtained from a quandle 2-cocycle. We show that the 0-th elementary ideal of $f$-twisted Alexander matrix of the knot quandle of a surface knot with the Alexander pair obtained from a quandle 2-cocycle can be described with the Carter-Saito-Satoh's invariant. We also discuss a relationship between $f$-twisted A…
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 …
Good involutions of twisted conjugation subquandles and Alexander quandles
本文完整刻画了自由拟群、扭共轭子拟群及所有 Alexander 拟群的优良对合，并枚举线性拟群的对合，给出阶数不超过 23 的显式映射，同时刻画了连通对合型 Alexander 拟群。
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.
Joyce observed that the Alexander invariant and the medial quandle of a classical knot are equivalent to each other, as invariants. In the present paper, we discuss the rather complicated extension of Joyce's observation to several different medial quandles and reduced (one-variable) Alexander modules associated with classical links. The theme is that for links, medial quandles provide stronger invariants than reduced Alexander modules.
In this paper, we study the colorability of link diagrams by the Alexander quandles. We show that if the reduced Alexander polynomial $Δ_{L}(t)$ is vanishing, then $L$ admits a non-trivial coloring by any non-trivial Alexander quandle $Q$, and that if $Δ_{L}(t)=1$, then $L$ admits only the trivial coloring by any Alexander quandle $Q$, also show that if $Δ_{L}(t)\not=0, 1$, then $L$ admits a non-trivial coloring by the Alexander quandle $Λ/(Δ_{L}(t))$.
We describe an algorithm for determining whether a finite quandle is isomorphic to an Alexander quandle by finding all possible Alexander presentations of the quandle. We give an implementation of this algorithm in Maple.
A condition of admissibility for generalized Alexander quandles
本文针对 quandle 的可容性问题，聚焦广义亚历山大 quandles，基于几何性质给出充分条件，推广了相关前期研究结论。
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…
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.
A quandle is an algebra whose axioms are motivated from knot theory. A linear extension of a quandle can be described by using a pair of maps called an Alexander pair. In this paper, we show that a linear extension of a multiple conjugation quandle can be described by using a pair of maps called an MCQ Alexander pair, where a multiple conjugation quandle is an algebra whose axioms are motivated from handlebody-knot theory.
Defined by Joyce and Matveev, the fundamental quandle is a complete invariant of oriented classical knots. We consider invariants of knots defined from quotients of the fundamental quandle. In particular, we introduce the fundamental Latin Alexander quandle of a knot and consider its Gröbner basis-valued invariants, which generalize the Alexander polynomial. We show via example that the invariant is not determined by the generalized Alexander polynomial for virtual knots.
A multiple conjugation quandle is an algebra whose axioms are motivated from handlebody-knot theory. Any linear extension of a multiple conjugation quandle can be described by using a pair of maps called an MCQ Alexander pair. In this paper, we show that any affine extension of a multiple conjugation quandle can be described by using a quadruple of maps, called an augmented MCQ Alexander pair.
The multivariate Alexander module of a link L has several subsets that admit quandle operations defined using the module operations. One of them, the fundamental multivariate Alexander quandle, determines the link module sequence of L.
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.
Multivariate Alexander quandles, V. Constructing the medial quandle of a link
本文提出通过减少的亚历山大模的周边结构来构建链接的媒介淬火方法。
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。
Two finite Alexander quandles with the same number of elements are isomorphic iff their Z[t,t^-1]-submodules Im(1-t) are isomorphic as modules. This yields specific conditions on when Alexander quandles of the form Z_n[t,t^-1]/(t-a) where gcd(n,a)=1 (called linear quandles) are isomorphic, as well as specific conditions on when two linear quandles are dual and which linear quandles are connected. We apply this result to obtain a procedure for classifying Alexander quandles of any finite order and as an application we list the numbers of distinct and connected Alexander quandles with up to fift…
This paper is a brief overview of some of our recent results in collaboration with other authors. The cocycle invariants of classical knots and knotted surfaces are summarized, and some applications are presented.
We determine the third cohomology of Alexander quandles of the form F_q[T]/(T-omega), where F_q denotes the finite field of order q and omega is an element of F-q which is neither 0 nor 1. As a result, we obtain many concrete examples of non-trivial 3-cocycles.
In a recent paper, Allen and Swenberg investigated which link polynomials are capable of detecting causality in (2+1)-dimensional globally hyperbolic spacetimes. They ultimately suggested it is likely that the Jones Polynomial accomplishes this, while the Alexander-Conway polynomial, independently, is insufficient. As the Alexander-Conway polynomial on its own is not likely to detect causality an additional piece of information must be supplemented to the link polynomial. This paper aims to examine the ability of Alexander quandles, to distinguish the connected sum of two Hopf links and Allen-…
We determine the second, third, and fourth cohomology groups of Alexander $f$-quandles of the form $\mathbb{F}_q[T,S]/ (T-ω, S-β)$, where $\mathbb{F}_q$ denotes the finite field of order $q$, $ω\in \mathbb{F}_q\setminus \{0,1\}$, and $β\in \mathbb{F}_q$
If $L$ is a classical link then the multivariate Alexander quandle, $Q_A(L)$, is a substructure of the multivariate Alexander module, $M_A(L)$. In the first paper of this series we showed that if two links $L$ and $L'$ have $Q_A(L) \cong Q_A(L')$, then after an appropriate re-indexing of the components of $L$ and $L'$, there will be a module isomorphism $M_A(L) \cong M_A(L')$ of a particular type, which we call a"Crowell equivalence." In the present paper we show that $Q_A(L)$ (up to quandle isomorphism) is a strictly stronger link invariant than $M_A(L)$ (up to re-indexing and Crowell equival…
Frobenius Alexander quandles, knot colorings, and isogeny classes of abelian varieties over finite fields
该研究引入有限域上阿贝尔簇的Frobenius Alexander纽胚概念，确定纽被其非平凡着色的素数条件，证明其同构类可确定阿贝尔簇同构类，为纽理论与算术研究提供新工具。
The theory of quandle (co)homology and cocycle knot invariants is rapidly being developed. We begin with a summary of these recent advances. One such advance is the notion of a dynamical cocycle. We show how dynamical cocycles can be used to color knotted surfaces that are obtained from classical knots by twist-spinning. We also demonstrate relations between cocycle invariants and Alexander matrices.
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 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.
We improve the lower bound for the minimum number of colors for linear Alexander quandle colorings of a knot given in Theorem 1.2 of Colorings beyond Fox: The other linear Alexander quandles (Linear Algebra and its Applications, Vol. 548, 2018). We express this lower bound in terms of the degree k of the reduced Alexander polynomial of the considered knot. We show that it is exactly k + 1 for L-space knots. Then we apply these results to torus knots and Pretzel knots P(-2,3,2l + 1), l>=0. We note that this lower bound can be attained for some particular knots. Furthermore, we show that Theorem…
We give a formula of the connected component decomposition of the Alexander quandle: $\mathbb{Z}[t^{\pm1}]/(f_1(t),\ldots, f_k(t))=\bigsqcup^{a-1}_{i=0}\mathrm{Orb}(i)$, where $a=\gcd (f_1(1),\ldots, f_k(1))$. We show that the connected component $\mathrm{Orb}(i)$ is isomorphic to $\mathbb{Z}[t^{\pm1}]/J$ with an explicit ideal $J$. By using this, we see how a quandle is decomposed into connected components for some Alexander quandles. We introduce a decomposition of a quandle into the disjoint union of maximal connected subquandles. In some cases, this decomposition is obtained by iterating a…
The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to group extension theories for group cocycles. Explicit formulas for cocycles corresponding to extensions are given, and used to prove non-triviality of cohomology groups for some quandles. The corresponding generalization of the quandle cocycle knot invariants is given, by using the…
We define a functor $\mathcal{Q}$ from the category of multiple conjugation biquandles to that of multiple conjugation quandles. We show that for any multiple conjugation biquandle $X$, there is a one-to-one correspondence between the set of $X$-colorings and that of $\mathcal{Q}(X)$-colorings diagrammatically for any handlebody-link and spatial trivalent graph. In particular, we prove that the set of $G$-family of Alexander biquandles colorings is isomorphic to that of $G$-family of Alexander quandles colorings as modules.
Multivariate Quandles as Groupoid Invariants
本文提出基于群胚的多元Alexander quandle构造框架，实现Alexander模的oid化，用于枚举新quandle类并寻找链环新着色不变量。
Joyce showed that for a classical knot $K$, the involutory medial quandle $\text{IMQ}(K)$ is isomorphic to the core quandle of the homology group $H_1(X_2)$, where $X_2$ is the cyclic double cover of $\mathbb S ^3$, branched over $K$. It follows that $|\text{IMQ}(K)| = | \det K |$. In the present paper, the extension of Joyce's result to classical links is discussed. Among other things, we show that for a classical link $L$ of $μ\geq 2$ components, the order of the involutory medial quandle is bounded as follows: \[ \frac{μ| \det L |}{2} \geq |\text{IMQ}(L)| \geq \frac{ μ| \det L |} {2^{μ-1}}.…
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。
Detecting Causality with Conjugation Quandles over Dihedral Groups
针对(2+1)维时空链环因果检测难题，发现二面体群D₅的共轭广群可区分因果相关与无关的链环，仅靠计数不变量就能实现简单可计算的因果检测。
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.
The Cayley graph of a quandle
本文研究了 quandle 的 Cayley 图的结构特性，描述了若干重要 quandle 类型的图结构，证明了 Alexander quandle 的连通分量与子群 coset 的关系，并展示了广义 Alexander quandle 的 Cayley 图的正则性。
We define biquandle structures on a given quandle, and show that any biquandle is given by some biquandle structure on its underlying quandle. By determining when two biquandle structures yield isomorphic biquandles, we obtain a relationship between the automorphism group of a biquandle and the automorphism group of its underlying quandle. As an application, we determine the automorphism groups of Alexander and dihedral biquandles. We also discuss product biquandles and describe their automorphism groups.
The set consisting of all rotations of the Euclidean plane is equipped with a quandle structure. We show that a knot is colorable by this quandle if and only if its Alexander polynomial has a root on the unit circle in $\mathbb{C}$. Further we enumerate all non-trivial colorings of a torus knot diagram by the quandle using PL trochoids. As an application of these results, we have the complete factorization of the Alexander polynomial of the torus knot.
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 $\math…
We denote by Q_F the family of the Alexander quandle structures supported by finite fields. For every k-component oriented link L, every partition P of L into h:=|P| sublinks, and every labelling z of such a partition by the natural numbers z_1,...,z_n, the number of X-colorings of any diagram of (L,z) is a well-defined invariant of (L,P), of the form q^(a_X(L,P,z)+1) for some natural number a_X(L,P,z). Letting X and z vary in Q_F and among the labellings of P, we define a derived invariant A_Q(L,P)=sup a_X(L,P,z). If P_M is such that |P_M|=k, we show that A_Q(L,P_M) is a lower bound for t(L),…
Quandles from group actions and a Cayley-type embedding theorem
本文通过研究群作用相关的拟群构造，建立了有限拟群的Cayley型嵌入定理，并利用对称群的自然作用构造出能嵌入所有$n$元拟群的泛拟群。
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。
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…
Carter, Jelsovsky, Kamada, Langford and Saito have defined an invariant of classical links associated to each element of the second cohomology of a finite quandle. We study these invariants for Alexander quandles of the form Z[t,t^{-1}]/(p, t^2 + kappa t + 1), where p is a prime number and t^2 + kappa t + 1 is irreducible modulo p. For each such quandle, there is an invariant with values in the group ring Z[C_p] of a cyclic group of order p. We shall show that the values of this invariant all have the form Gamma_p^r p^{2s} for a fixed element Gamma_p of Z[C_p] and integers r >= 0 and s > 0. We…
If a knot has the Alexander polynomial not equal to 1, then it is linear $n$-colorable. By means of such a coloring, such a knot is given an upper bound for the minimal quandle order, i.e., the minimal order of a quandle with which the knot is quandle colorable. For twist knots, we study the minimal quandle orders in detail.
Detecting Causality with Symplectic Quandles
本文证明辛Quandle结合Alexander-Conway多项式可区分Allen-Swenberg 2-sky类链环与两个Hopf链环的连通和，从而为检测(2+1)维全局双曲时空中的因果性提供了新的拓扑不变量工具。
We show that the lower bounds for Betti numbers given in math.GT/9909161 are equalities for a class of racks that includes dihedral and Alexander racks. We confirm a conjecture from the same paper by defining a splitting for the short exact sequence of quandle chain complexes. We define isomorphisms between Alexander racks of certain forms, and we also list the second and third homology groups of some dihedral and Alexander quandles.
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.
On embeddings of homogeneous quandles
本文研究了同构淬表的嵌入问题，提出了一个必要且充分条件，并应用该结果重新诠释了核心淬表的嵌入，同时构建了多个几何示例的显式嵌入。
Representations of knot groups in $\textrm{AGL}_{1}(\mathbb{C})$ and Alexander invariants
本文通过纽结群在复仿射群AGL1(C)中的表示簇重新诠释Alexander不变量，证明表示簇坐标环与Alexander模对称代数同构，并构造拓扑量子场论以提供计算方法。
This article is about applications of linear algebra to knot theory. For example, for odd prime p, there is a rule (given in the article) for coloring the arcs of a knot or link diagram from the residues mod p. This is a knot invariant in the sense that if a diagram of the knot under study admits such a coloring, then so does any other diagram of the same knot. This is called p-colorability. It is also associated to systems of linear homogeneous equations over the residues mod p, by regarding the arcs of the diagram as variables and assigning the equation "twice the over-arc minus the sum of t…
We classify indecomposable racks of order p^2 (p a prime). There are 2p^2 - 2p - 2 isomorphism classes, among which 2p^2 - 3p - 1 correspond to quandles. In particular, we prove that an indecomposable quandle of order p^2 is affine (=Alexander). One of the results yielding this classification is the computation of the quandle nonabelian second cohomology group of an indecomposable quandle of prime order; which turns out to be trivial, as in the abelian case.
We prove that if Q is a finite quasigroup quandle, then |Q| annihilates the torsion of its homology. It is a classical result in reduced homology of finite groups that the order of a group annihilates its homology. From the very beginning of the rack homology (between 1990 and 1995) the analogous result was suspected. The first general results in this direction were obtained independently about 2001 by R.A.Litherland and S.Nelson, and P.Etingof and M.Grana. In Litherland-Nelson paper it is proven that if (Q;*) is a finite homogeneous rack (this includes quasigroup racks) then the torsion of ho…
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。
To study embeddings of tangles in knots, we use quandle cocycle invariants. Computations are carried out for the tables of knots and tangles, to investigate which tangles may or may not embed in knots in the tables.
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。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。