Bruno Cisneros, Marcelo Flores, Jesús Juyumaya, Christopher Roque-Márquez
We construct an Alexander type invariant for oriented doodles from a deformation of the Tits representation of the twin group and from the Chebyshev polynomials of second kind. Similar to the Alexander polynomial, our invariant vanishes on unlinked doodles with more than one component. We also include values of our invariant on several doodles.
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Study of certain isotopy classes of a finite collection of immersed circles without triple or higher intersections on closed oriented surfaces can be thought of as a planar analogue of virtual knot theory where the genus zero case corresponds to classical knot theory. Alexander and Markov theorems for the genus zero case are known where the role of groups is played by twin groups, a class of right angled Coxeter groups with only far commutativity relations. The purpose of this paper is to prove Alexander and Markov theorems for higher genus case where the role of groups is played by a new clas…
Alexander-Markov correspondence for doodles on closed surfaces
本文引入扭曲虚拟涂鸦并定义扭曲虚拟孪生群作为虚拟孪生群的扩展，建立相关亚历山大-马尔可夫型定理以统一扩展涂鸦理论，还研究了纯扭曲虚拟孪生群的结构与分解性质。
A complete invariant for doodles on a 2-sphere
本文通过构造取值于特定弦图级数的完全不变量，研究二维球面上涂鸦的分类问题，并证明其系数为阶至多 $2n$ 的有限型不变量。
A representation of Milnor's triple linking number by chord diagrams and doodle invariants
本文借鉴二阶纽结不变量的分解思路，证明二阶的Milnor三重环绕数可表示为特定弦图计数与doodle不变量的组合，给出了该环绕数的新表示形式。
We give a volume formula of hyperbolic knot complements using twisted Alexander invariants.
$L^2$-Alexander invariant for knots
本文研究一类新的纽结不变量：$L^2$ -Alexander不变量。本文的主要结果是给出了利用纽结群的缺省1的任意表示来计算纽结补的$L^2$ -Alexander不变量的方法。
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 …
A diagrammatic Alexander invariant of tangles
我们给出了一个新的结构的一个变量的亚历山大多项式的一个有向的纽结或链接，并表明它推广到一个向量值不变量的有向缠结。
Fibred knots and twisted Alexander invariants
我们引入了一种新的代数拓扑技术来检测非线性结在三个球体使用扭曲的亚历山大不变量。作为应用，我们证明了对于任何具有非平凡Alexander多项式的结的Seifert矩阵，存在无穷多个具有给定Seifert矩阵的非纤维结。我们说明的例子，有平凡的亚历山大多项式，但没有扭曲的亚历山大不变量的Anglered结的结。
A homological representation formula of colored Alexander invariants
利用辫群的同调表示给出了色Alexander不变量的一个公式，我们称之为截断Lawrence表示。这个公式推广了著名的亚历山大多项式的Burau表示公式。
Proof of a Conjectured Formula for the Alexander Invariant
本文证明了Bar-Natan和Dancso在[BND]中提出的一个计算Alexander不变量的公式的有效性。
Twisted Alexander invariants detect trivial links
它遵循西尔弗-威廉姆斯和作者的早期工作，扭曲亚历山大多项式检测unknot和霍普夫链接。我们现在表明，扭曲的亚历山大多项式也检测三叶草和图8结，扭曲的亚历山大多项式检测是否一个链接是分裂和扭曲的亚历山大模块检测平凡的链接。
Concordance of links with identical Alexander invariants
J. Davis证明了3-球面中链路的拓扑协调类是由其Alexander多项式唯一确定的，对于Alexander多项式为1的2-分量链路。对于亚历山大多项式为1的纽结，M.弗里德曼我们证明，这两种情况下是唯一的例外情况下，通过显示的链接和谐类是不确定的亚历山大不变量在任何其他情况下。
Twisting Alexander Invariants with Periodic Representations
扭曲亚历山大不变量已被定义为任何纽结及其群的线性表示。对于纽结群的换位子群的任何周期表示，推广了这些不变量。给出了新的扭曲不变量的性质。在适当的假设下，得到了零模的互易性和界。给出了不变量的Mahler测度的拓扑解释。关键词：纽结，扭曲亚历山大多项式，表示移位，马勒测度。
Given a homomorphism from a knot group to a fixed group, we introduce an element of a $K_1$-group, which is a generalization of (twisted) Alexander polynomials. We compare this $K_1$-class with other Alexander polynomials. In terms of semi-local rings, we compute the $K_1$-classes of some knots and show their non-triviality. We also introduce metabelian Alexander polynomials.
We study how the genus, the simplicial volume and the $L^2$-Alexander invariant of W. Li and W. Zhang can detect individual knots among all others. In particular, we use various techniques coming from hyperbolic geometry and topology to prove that the $L^2$-Alexander invariant contains strictly more information than the pair (genus, simplicial volume). Along the way we prove that the $L^2$-Alexander invariant detects the figure-eight knot $4_1$, the twist knot $5_2$ and an infinite family of cables on the figure-eight knot.
Three flavors of twisted invariants of knots
纽结的亚历山大多项式已经以三种不同的方式推广，以给出扭曲不变量。由此产生的不变量通常被称为扭曲的亚历山大多项式，高阶亚历山大多项式和$L^2$ -亚历山大不变量的结。我们快速回忆一下定义，并总结和比较它们的一些性质。我们还报告了作者的工作$L^2$ -亚历山大扭转，我们总结了几个apturtures上的$L^2$ -亚历山大扭转的文件。
Fox conjectured the Alexander polynomial of an alternating knot is trapezoidal, i.e. the coefficients first increase, then stabilize and finally decrease in a symmetric way. Recently, Hirasawa and Murasugi further conjectured a relation between the number of the stable coefficients in the Alexander polynomial and the signature invariant. In this paper we prove the Hirasawa-Murasugi conjecture for two-bridge knots.
Alexander invariants for virtual knots
给定一个虚纽结$K$，我们构造一个称为虚纽结群的群$VG_K$，并且我们使用$VG_K$的基本理想来定义$K$的不变量，称为虚亚历山大不变量。例如，与$k=0$理想是一个多项式$H_K(s,t,q)$在三个变量，我们称之为虚拟亚历山大多项式，我们表明，它是密切相关的广义亚历山大多项式$G_K(s,t)$介绍Sawollek，考夫曼，雷德福，和西尔弗威廉姆斯。我们定义了一个自然规范化的虚拟亚历山大多项式，并证明它满足绞公式。我们还介绍了扭曲的虚拟亚历山大多项式相关联的虚拟结$K$和表示$\varrho \colon VG_K \to GL_n(R)$，我们定义了一个规范化的扭曲虚拟亚历山大多项式。作为应用，我们从虚拟Alexander多项式和扭曲的虚拟Alexander多项式导出了虚拟结点的虚拟交叉数的界。
On the Links-Gould invariant and the square of the Alexander polynomial
本文给出了有向链环的Links-Gould不变量的精心选择的约化与Alexander-Conway多项式的幂之间的联系。我们证明这些公式的表示的辫群，我们得出专门的林克斯-古尔德多项式可以被看作是外部权力的副本Burau表示。
We revisit the issue of the existence of infinitely many distinct prime knots with the same Alexander invariant. We present infinitely many distinct families, each family made up of infinitely many distinct knots. Within each family, the Alexander invariant is the same. Unlike other examples in the literature, ours are elementary and based on a sub-collection of pretzel knots with three tassels.
Twisted Alexander invariant and non-abelian Reidemeister torsion for hyperbolic three-dimensional manifolds with cusps
本文研究了J. Porti关于具有尖点的完备双曲三维流形的双曲Reidemeister挠率（在文献中也称为非交换Reidemeister挠率）的一种计算方法.双曲纽结外部的扭曲亚历山大不变量的导数给出双曲挠率。我们证明了这样一个衍生公式的扭曲的亚历山大不变量的双曲链接外部像怀特黑德链接外部。我们提供的衍生公式的工作框架，其中包括假设的拓扑结构的流形和扭曲的亚历山大不变量的定义中所涉及的表示，并证明衍生公式在这种情况下。我们还探讨了扭曲的亚历山大不变量的对称性（符号），并证明它实际上是一个多项式不变量，像通常的亚历山大多项式。
Normalization of twisted Alexander invariants
纽结的扭曲亚历山大不变量定义良好，直到单位相乘。我们摆脱这种乘法的歧义，通过组合的方法，并定义规范化扭曲亚历山大不变量。然后，我们表明，不变量与符号确定的Reidemeister扭转在一个规范化的设置，并完善的对偶定理。进一步得到了纽结纤维化的不变量的必要条件，并研究了最高阶不变量的性质。
Lin and Wang defined a model of random walks on knot diagrams and interprete the Alexnader polynomials and the colored Jones polynomials as Ihara zeta functions, i.e. zeta functions defined by counting cycles on the knot diagram. Using this explanation, they gave a more conceptual proof for the Melvin-Morton conjecture. In this paper, we give an analogous zeta function expression for the twisted Alexander invariants.
The computation of higher order Alexander invariants
本文针对高阶Alexander不变量的计算难题，提出了新方法，使交叉点数多达100的纽结的Alexander多项式能够被有效计算，克服了传统Smith标准形算法在16交叉点纽结上就已过慢的瓶颈。
There are many studies about twisted Alexander invariants for knots and links, but calculations of twisted Alexander invariants for spatial graphs, handlebody-knots, and surface-links have not been demonstrated well. In this paper, we give some remarks to calculate the twisted Alexander ideals for spatial graphs, handlebody-knots and surface-links, and observe their behaviors. For spatial graphs, we calculate the invariants of Suzuki's theta-curves and show that the invariants are nontrivial for Suzuki's theta-curves whose Alexander ideals are trivial. For handlebody-knots, we give a remark on…
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.
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…
The Magnus representation and higher-order Alexander invariants for homology cobordisms of surfaces
从曲面到自身的带有边界标记的同调配边集具有自然幺半群结构。为了研究这个幺半群的结构，我们定义并研究了它的Magnus表示和Reidemeister扭转不变量，通过推广Kirk-Livingston-Wang关于弦链的Gassner表示的论点。此外，通过应用Cochran和Harvey的高阶（非交换）亚历山大不变量的框架，我们提取了几条信息的幺半群和相关对象。
Given a homomorphism from a link group to a group, we introduce a $K_1$-class in another way, which is a generalization of the 1-variable Alexander polynomial. We compare the $K_1$-class with $K_1$-classes in \cite{Nos} and with Reidemeister torsions. As a corollary, we show a relation to Reidemeister torsions of finite cyclic covering spaces, and show reciprocity in some senses.
On the n-loop Kontsevich invariant of knots having the same Alexander polynomial
本文研究具有相同 Alexander 多项式的纽结的高阶圈 Kontsevich 不变量，证明亏格有界时其 $n\geq2$ 圈像生成有限维子空间，并给出简单情形的具体计算。
As an extension of Reshetikhin and Turaev's invariant, Costantino, Geer and Patureau-Mirand constructed $3$-manifold invariants in the setting of relative $G$-modular categories, which include both semisimple and non-semisimple ribbon tensor categories as examples. In this paper, we follow their method to construct a $3$-manifold invariant from Viro's $\mathfrak{gl}(1\vert 1)$-Alexander polynomial. We take lens spaces $L(7, 1)$ and $L(7, 2)$ as examples to show that this invariant can distinguish homotopy equivalent manifolds.
Let $Δ_{L,ρ_n}(t)$ be the twisted Alexander polynomial with respect to the representation given by the composition of the lift of the holonomy representation of a certain hyperbolic link $L$ and the $n$-dimensional irreducible complex representation of $\text{SL}(2,\mathbb C)$. We consider a sequence of $Δ_{L,ρ_n}(t)$ and extract the volume of the complement of $L$ from the asymptotic behaviour of the sequence obtained by evaluating $t=1$ or $t=-1$.
Nontrivial Alexander polynomials of knots and links
本文提出了一系列的链接不变量，定义扭曲亚历山大多项式，并讨论其有效性，在区分结。特别是，我们重铸和延长几何平均最近的结果银和威廉姆斯的非平凡扭曲的亚历山大多项式的非平凡结。此外，在[FV06b]的结果的基础上，我们证明了这些不变量决定了一个属的一个结是纤维化的，我们还表明，这些不变量区分所有突变体多达12个交叉。
Alexander polynomial, finite type invariants and volume of hyperbolic knots
证明了当n>0时，存在一个双曲纽结K，它具有平凡的Alexander多项式，平凡的有限型不变量的阶为<=n，且K的补的体积大于n.这与已知的陈述相反，双曲交替结的补的体积由结的亚历山大多项式的系数的线性函数上界。作为我们主要结果的推论，我们得到，对于任何m>0，存在一个序列的双曲纽结平凡有限型不变量的顺序<=m，但任意大的体积。我们将讨论我们的结果如何适合的框架内的关系的有限型不变量和双曲结的体积，预测由Kashaev的双曲体积猜想。
Ribbon tangles are proper embeddings of tori and cylinders in the $4$-ball~$B^4$, "bounding" $3$-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathsf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. This invariant induces a functor in a certain category $\mathsf{R}ib_G$ of tangles, which restricts to the exterior powers of Burau-Gassner representation for ribbon braids, that are analogous to usual braids in this context. We define a circuit algebra $\mathsf{C}ob_G$ over the op…
We introduce an Alexander polynomial for MOY graphs. For a framed trivalent MOY graph $\mathbb{G}$, we refine the construction and obtain a framed ambient isotopy invariant $Δ_{(\mathbb{G},c)}(t)$. The invariant $Δ_{(\mathbb{G}, c)}(t)$ satisfies a series of relations, which we call MOY-type relations, and conversely these relations determine $Δ_{(\mathbb{G}, c)}(t)$. Using them we provide a graphical definition of the Alexander polynomial of a link. Finally, we discuss some properties and applications of our invariants.
We show that every periodic virtual knot can be realized as the closure of a periodic virtual braid and use this to study the Alexander invariants of periodic virtual knots. If $K$ is a $q$-periodic and almost classical knot, we show that its quotient knot $K_*$ is also almost classical, and in the case $q=p^r$ is a prime power, we establish an analogue of Murasugi's congruence relating the Alexander polynomials of $K$ and $K_*$ over the integers modulo $p$. This result is applied to the problem of determining the possible periods of a virtual knot $K$. One consequence is that if $K$ is an alm…
On reciprocality of twisted Alexander invariants
给定一个纽结和它的群的SL（n，C）表示，该表示与它的对偶共轭，即用它的逆转置替换每个矩阵的表示，相关联的扭曲里德迈斯特挠率是倒数。给出了一个纽结群和SL（3，Z）表示的例子，SL（3，Z）表示不与其对偶共轭，且扭曲的Reidemeister挠率不互逆.
Alexander polynomial, Dijkgraaf-Witten invariant, and Seifert fibred surgery
本文利用 Dijkgraaf-Witten 不变量研究沿纽结的 k/ℓ 手术产生的小 Seifert 3-流形，证明其重数与手术系数受 Alexander 多项式约束。
In this paper, a regional knot invariant is constructed. Like the Wirtinger presentation of a knot group, each planar region contributes a generator, and each crossing contributes a relation. The invariant is call a tridle of the link. As in the quandle theory, one can define Alexander quandle and get Alexander polynomial from it. For link diagram, one can also define a linear tridle and its presentation matrix. A polynomial invariant can be derive from the matrix just like the Alexander polynomial case.
Higher-order Alexander Invariants of Hypersurface Complements
我们定义了高阶Alexander模$A_{n,i}(\mathcal{U})$和高阶度$δ_{n,i}(\mathcal{U})$，它们是复超曲面补$\mathcal{U}$的不变量.这些不变量来自超曲面补的某些可解覆盖的同调的模结构。这种不变量最初是由T. Cochran在[1]和S. Harvey在[8]中提出的，并用于研究纽结和三维流形.本文推广了C. Leidy和L. Maxim [22]从平面曲线补到高维超曲面补。Zebriki观察到奇异复平面曲线上奇点的位置会影响曲线的拓扑结构。我关于高阶超曲面补的结果也表明，全局拓扑是由局部拓扑控制的。特别地，超曲面补的高阶度由局部链路对的高阶度的线性组合界定。
On elementary invariants of genus one knots and Seifert surfaces
本文提出了一种用于研究一阶结和塞弗特面的亚历山大形式不变量，通过分析三维同调柄体的性质来证明其不变性。
For a knot in the 3-sphere, the Upsilon invariant is a piecewise linear function defined on the interval [0,2]. For an L-space knot, the Upsilon invariant is determined only by the Alexander polynomial of the knot. We exhibit infinitely many pairs of hyperbolic L-space knots such that two knots of each pair have distinct Alexander polynomials, so they are not concordant, but share the same Upsilon invariant. Conversely, we examine the restorability of the Alexander polynomial of an L-space knot from the Upsilon invariant through the Legendre-Fenchel transformation.
The Bott-Cattaneo-Rossi invariant $(Z_k)_{k\in \mathbb N\setminus\{0,1\}}$ is an invariant of long knots $\mathbb R^n\hookrightarrow\mathbb R^{n+2}$ for odd $n$, which reads as a combination of integrals over configuration spaces. In this article, we compute such integrals and prove explicit formulas for (generalized) $Z_k$ in terms of Alexander polynomials, or in terms of linking numbers of some cycles of a hypersurface bounded by the knot. Our formulas, which hold for all null-homologous long knots in homology $\mathbb R^{n+2}$ at least when $n\equiv 1\mod 4$, conversely express the Reidemei…
A restriction on the Alexander polynomials of $L$-space knots
利用Rasmussen定义的一个不变量，我们扩展了Hedden和Watson给出的一个参数，该参数进一步限制了哪些Alexander多项式可以由$L$ -空间节点实现。
Higher-order Alexander invariants and filtrations of the knot concordance group
我们建立了光滑和拓扑纽结协调群的几个滤子的某些“非平凡性”结果。首先，关于K. Orr，P. Teichner和第一作者，我们通过包含Alexander模的信息来改进Cochran和Teichner最近的非平凡性结果。这些结果也推广了C.利文斯顿和第二作者。我们表现出类似的结构，在密切相关的对称Grope过滤的结和谐组考虑的第一作者和Teichner。我们还表明，Grope过滤的光滑和谐组是不平凡的使用的例子，不能区分的Ozsvath-Szabo tau不变量，也不是由J.拉斯穆森的s-不变量。我们更广泛的贡献是建立，在“相对的情况下”，关键的同源结果，其类似物Cochran-Orr-Teichner建立在“绝对的情况下”。如果K#（-J）是n-可解的，则称两个纽结K和J是模n-协调可解的.我们的主要结果是：对于任何经典Alexander多项式的次数大于2的纽结K，对于任何正整数n，存在无穷多个纽结K_i与K模n-可解性一致，但都是不同的模n.5-可解性.此外，K_i和K共享相同的经典Seifert矩阵和Alexander模，以及共享相同的高阶Alexander模和直到n-1阶的Seifert表示.
Polynomial cocycles of Alexander quandles and applications
余圈是由亚历山大量子点的多项式表达式构造的。作为应用，证明了某些quandle同调群的非平凡性，并研究了纽结的quandle上循环不变量。特别地，对于无限Quandle族，证明了Quandle同调群的非平凡性。
Configuration space integral for long n-knots, the Alexander polynomial and knot space cohomology
有一个更高维度的微扰陈-西蒙斯理论的类似物，在这个意义上，一个类似的微扰系列在3维，这是计算通过配置空间积分，产生一个不变量的高维结（Bott-Cattaneo-Rossi不变量），这是由博特构造的程度2和Cattaneo-Rossi为更高的程度。然而，它的特点还不清楚。在本文中，我们限制的研究长带状n-knots和特点的Bott-Cattaneo-Rossi不变量作为一个有限型不变量的长带状n-knots在[HKS]。因此，我们得到了一个非平凡的描述的Bott-Cattaneo-Rossi不变的亚历山大多项式。对于高余维纽结也给出了结果。在这些情况下，定义Bott-Cattaneo-Rossi不变量的类似微分形式产生无穷多个Emb（R^n，R^m）的上同调类，如果m，n>= 3奇数且m>n+2。我们观察到这些类中有一半是非平凡的，沿着类似于Cattaneo-CottaRamusino-Longoni [CCL]的线。
A generalization of several classical invariants of links
我们扩展了几个经典的不变量的链接在3-球面链接在所谓的准圆柱体。这些不变量包括环绕数、Seifert形式、Alexander模、Alexander-Conway多项式和Murasugi-Tristram-Levine签名。
Perturbed Alexander Invariants via Quantum Cluster Algebras
本文通过量子簇代数推导了结不变量的扰动展开，利用量子簇代数的簇变换和辅助参数ε，得到一个以海森堡代数生成元表示的扰动R矩阵，其零阶项为亚历山大多项式的倒数，高阶项产生符合Bar-Natan和Van der Veen构造的扰动-亚历山大不变量。
Gröbner Bases for Alexander Fitting Ideals of Links
针对多变量Alexander理论生成的无优选生成元的Fitting理想，固定系数域与单项序后，通过多项式收缩的简约Gröbner基表示这些理想得到纽结的Alexander-Gröbner不变量，还推导了行列式-除子互反性并计算了多种纽结的相关不变量。
Tensors, Gaussians and the Alexander Polynomial
本文基于Bar-Natan和Van der Veen的方法，利用扰动的高斯函数开发出一个高斯模型，用于计算定向结K的亚历山大多项式Δ_K(T)。通过海森堡代数和张量收缩形式化，将结与一个高斯函数关联起来，其配分函数恢复Δ_K(T)。这里，亚历山大模块的呈现矩阵扮演高斯函数的精度矩阵的角色。
Alexander and Thurston norms of fibered 3-manifolds
对于一个3-流形M，麦克马伦从M的亚历山大多项式导出了一个关于H ^1（M，R）的范数，称为亚历山大范数。他证明了H^1（M，R）上的瑟斯顿范数是亚历山大范数的上界，它度量了对偶曲面的复杂性。他问，当M根纤维在圆上时，这两个范数是否在所有的H^1（M，R）上都相等。在这里，我给的例子表明，这个问题的答案是断然没有。这个问题是有关的忠实性的加斯纳表示的辫群。用来理解这个问题的关键工具是来自组合群论的Bieri-Neumann-Strebel不变量。定理1.7是一个独立的定理，它把Alexander多项式与一个Bieri-Neumann-Strebel不变量联系起来。
The $L^2$-Alexander invariant detects the unknot
本文给出了Li和Zhang定义的纽结的$L^2$ -Alexander不变量的一些性质，其中一些性质类似于经典的Alexander多项式.值得注意的是，我们证明了$L^2$ -亚历山大不变量检测平凡结。
On a Generalization of Alexander Polynomial for Long Virtual Knots
我们构造了新的不变多项式的长虚拟纽结。它是Alexander多项式的推广。我们用$ζ$来表示它，这意味着与用于虚拟链路的$ζ$ -多项式的类比。$ζ$ -多项式的次数估计虚拟交叉数。本文描述了$ζ$ -多项式在研究关于虚交叉数的极小长虚图中的应用。
Ozsváth-Szabó proved the property that any coefficient of Alexander polynomial of lens space knot is either $\pm1$ or $0$ and the non-zero coefficients are alternating. Combining the formulas of the Alexander polynomial of lens space knots due to Kadokami-Yamada and Ichihara-Saito-Teragaito, we refine Ozsváth-Szabó's property as the existence of simple curves included in a region in ${\Bbb R}^2$. The existence of curves, that has no end-points connected, is just 1-component in a region, can search distribution of non-zero coefficients of the Alexander polynomial of the lens space knot. This cu…
Links with trivial Alexander module and nontrivial Milnor invariants
修复了小的印刷错误。Cochran用非环的Alexander模和一些非零Milnor不变量构造了许多环，用自由群中的不变量作为输入，用环的模作为不变量.我们提出了一个不同的和构造上完整的结构，它使用了clasper手术的基本属性，和一个不同的不变量，LMO不变量的树部分。我们的方法还构造平凡的高亚历山大模块和非平凡的Milnor不变量的链接。
We address the primary decomposition of the knot concordance group in terms of the solvable filtration and higher-order von Neumann $ρ$-invariants by Cochran, Orr, and Teichner. We show that for a nonnegative integer n, if the connected sum of two n-solvable knots with coprime Alexander polynomials is slice, then each of the knots has vanishing von Neumann $ρ$-invariants of order n. This gives positive evidence for the conjecture that nonslice knots with coprime Alexander polynomials are not concordant. As an application, we show that if K is one of Cochran-Orr-Teichner's knots which are the f…
Quotient Quandles and the Fundamental Latin Alexander Quandle
由Joyce和Matveev定义的基本困境是有向经典纽结的完全不变量。我们考虑从基本quandle的不变量定义的结的不变量。特别是，我们介绍了一个结的基本拉丁亚历山大quandle，并考虑其Gröbner基值不变量，推广的亚历山大多项式。我们通过例子表明，不变量不是由广义亚历山大多项式的虚拟节点。
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