链接图中的法向面理论

用标记图计算曲面链环的Quandle内圆不变量

Computations of quandle cocyle invariants of surface-links using marked graph diagrams

利用Quandle上同调理论，通过破面图定义了有向链和面链的Quandle上循环不变量和影子Quandle上循环不变量。利用对称量子线，通过破面图，对无向链和面链定义了对称量子线上循环不变量。标记图是可能具有配备有标记的$4$价顶点的链接图。S.小洛莫纳科和K. Yoshikawa介绍了一种用标记图描述表面链接的方法。本文给出了这些Quandle上循环不变量的标记图的解释，并介绍了一种由标记图计算它们的方法。

经典链路和虚拟链路的最小图

Minimal diagrams of classical and virtual links

我们证明了满足两个条件的Khovanov同调的虚拟链接图是最小的，也就是说，没有虚拟图表示相同的链接具有较少的交叉数。这种方法适用于经典链路和虚拟链路

Presentation of immersed surface-links by marked graph diagrams

It is well known that surface-links in 4-space can be presented by diagrams on the plane of 4-valent spatial graphs with makers on the vertices, called marked graph diagrams. In this paper we extend the method of presenting surface-links by marked graph diagrams to presenting immersed surface-links. We also give some moves on marked graph diagrams that preserve the ambient isotopy classes of their presenting immersed surface-links.

关于面环标记图的Yoshikawa移动生成集

On generating sets of Yoshikawa moves for marked graph diagrams of surface-links

标记图是可能具有标记$4$ -价顶点的链接图。S.小洛莫纳科和K. Yoshikawa介绍了一种用标记图表示表面链接的方法。特别是K。吉川给出了标记图的局部移动，现在称为吉川移动。现在我们知道，代表等价表面链接的两个有标记的图形由这些Yoshikawa移动的有限序列联系起来。本文给出了表示无向面链和有向面链的标记图上的Yoshikawa移动的生成集。我们还讨论了某些吉川动作与其他动作的独立性。

Triple-crossing number and moves on triple-crossing link diagrams

Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of n-crossing diagrams for every n>1 allows the definition of the n-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-c…

射影空间中纽结与链环的图解解结

Diagrammatic unknotting of knots and links in the projective space

在经典的纽结理论中有一个著名的下降图概念。从一个任意的图，人们可以很容易地得到，通过一些交叉的变化，一个下降的图，这是一个图的解开或解开。本文将实空间中纽结和链环的降图概念推广到射影空间中无向纽结和链环的情况。它还表明，这个概念不能扩展到定向链接。

Integral region choice problems on link diagrams

Shimizu introduced a region crossing change unknotting operation for knot diagrams. As extensions, two integral region choice problems were proposed and the existences of solutions of the problems were shown for all non-trivial knot diagrams by Ahara and Suzuki, and Harada. We relate both integral region choice problems with an Alexander numbering for regions of a link diagram, and give alternative proofs of the existences of solutions for knot diagrams. We also discuss the problems on link diagrams. For each of the problems on the diagram of a two-component link, we give a necessary and suffi…

Simplicial volume of links from link diagrams

The hyperbolic volume of a link complement is known to be unchanged when a half-twist is added to a link diagram, and a suitable 3-punctured sphere is present in the complement. We generalize this to the simplicial volume of link complements by analyzing the corresponding toroidal decompositions. We then use it to prove a refined upper bound for the volume in terms of twists of various lengths for links.

The Jones-Krushkal polynomial and minimal diagrams of surface links

We prove a Kauffman-Murasugi-Thistlethwaite theorem for alternating links in thickened surfaces. It states that any reduced alternating diagram of a link in a thickened surface has minimal crossing number, and any two reduced alternating diagrams of the same link have the same writhe. This result is proved more generally for link diagrams that are adequate, and the proof involves a two-variable generalization of the Jones polynomial for surface links defined by Krushkal. The main result is used to establish the first and second Tait conjectures for links in thickened surfaces and for virtual l…

Lipson状态模型在表面链接标记图中的应用

Applying Lipson's state models to marked graph diagrams of surface-links

A. S.利普森构造了两个状态模型，产生相同的经典链路不变的考夫曼多项式$F(a,z)$。在本文中，我们应用Lipson的状态模型标记的图形图的表面链接，并观察当他们诱导表面链接不变量。

Minimal coloring numbers on minimal diagrams of torus links

We determine the minimal number of colors for non-trivial $\mathbb{Z}$-colorings on the standard minimal diagrams of $\mathbb{Z}$-colorable torus links. Also included are complete classifications of such $\mathbb{Z}$-colorings and of such $\mathbb{Z}$-colorings by only four colors, which are shown by using rack colorings on link diagrams.

刚性硬图与晃动硬图

Rigid and shaky hard link diagrams

本研究将硬图分为刚性硬图和晃动硬图，证明每个链环存在刚性硬图并给出交叉数上限，同时证明平凡结和平凡链环存在晃动硬图。

洛伦兹链的网格图

Grid diagrams of Lorenz links

在论文“A new twist on Lorenz links”（Journal of Topology 2（2009），227-248）中，Joan Birman和Ilya Kofman证明了Lorenz links类和twisted links类的重合。在这项工作中的证明是代数的。我们将确定这个类的网格图，并提供一个透明的几何参数的伯曼-科夫曼的结果。

虚拟链路的最小图：II

Minimal diagrams of virtual links: II

在本文中，我们汇集了极小条件，提出了在前两篇论文中，并提出了一些新的极小条件的经典和虚拟的节点和链接。

Multiflypes of rectangular diagrams of links

We introduce a new very large family of transformations of rectangular diagrams of links that preserve the isotopy class of the link. We provide an example when two diagrams of the same complexity are related by such a transformation and are not obtained from one another by any sequence of `simpler' moves not increasing the complexity of the diagram along the way.

OU数与III型Reidemeister移动对链接图的性质

The OU number and Reidemeister moves of type III for link diagrams

本文提出OU数和非自OU序列，用于确定链接图之间III型Reidemeister移动的下界。

New classes of minimal knot diagrams

We describe a new class of minimal link diagrams. This class includes certain alternating diagrams, the standard diagrams of all torus links, and numerous homogeneous diagrams whose minimality has not been proven before. Besides, we describe a new larger class of link diagrams with the least number of Seifert circles among all diagrams of a given link. Our approach refers to the Morton-Franks-Williams inequality.

Minimal coloring number on minimal diagrams for $\mathbb{Z}$-colorable links

It was shown that any $\mathbb{Z}$-colorable link has a diagram which admits a non-trivial $\mathbb{Z}$-coloring with at most four colors. In this paper, we consider minimal numbers of colors for non-trivial $\mathbb{Z}$-colorings on minimal diagrams of $\mathbb{Z}$-colorable links. We show, for any positive integer $N$, there exists a minimal diagram of a $\mathbb{Z}$-colorable link such that any $\mathbb{Z}$-coloring on the diagram has at least $N$ colors. On the other hand, it is shown that certain $\mathbb{Z}$-colorable torus links have minimal diagrams admitting $\mathbb{Z}$-colorings wit…

All Known Principal Congruence Links

This report lists the link diagrams in S^3 for all principal congruence link complements for which such a link diagram is known. Several unpublished link diagrams are included. Related to this, we also include one link diagram for an arithmetic regular tessellation link complement.

通过链环图着色研究链环的宽度

Widths of links via diagram colorings

本研究通过链环图着色定义链环不变量，证明其与标准Morse函数下的各类链环宽度概念一致，该方法算法化且易编程实现，还计算了万余个14交叉以内链环的最大宽度。

透镜空间中链环的两种图表示的等价性及本质不变量

Equivalence of two diagram representations of links in lens spaces and essential invariants

本文研究了透镜空间中链环的两种图解表示法：圆盘图和网格图之间的关系，并找到了如何从一种表示法过渡到另一种表示法。我们还研究了HOMFLY-PT不变量和Link弗洛尔同调是否是本质不变量，即这些不变量是否能够区分$\mathbf{S}^3$中相同链接所覆盖的$L(p,q)$中的链接。为了做到这一点，我们推广的组合定义的纽结弗洛尔同调在透镜空间的链接的情况下，我们分析如何改变这两个不变量时，我们切换的链接方向。

关于关于Reidemeister move I和II极小的链环图

On link diagrams that are minimal with respect to Reidemeister moves I and II

本文定义了关于Reidemeister move I和II极小的链环图，证明了无平凡分离分支链环的极小图及其交叉点数唯一，并探讨了Reidemeister move III对极小图的作用。

Checkerboard graph links and simply laced Dynkin diagrams

We define an equivalence relation on graphs with signed edges, such that the associated adjacency matrices of two equivalent graphs are congruent over $\mathbb{Z}$. We show that signed graphs whose eigenvalues are larger than $-2$ are equivalent to one of the simply laced Dynkin diagrams: $A_{n}$, $D_{n}$, $E_{6}$, $E_{7}$ and $E_{8}$. Checkerboard graph links are a class of fibred strongly quasipositive links which include positive braid links. We use the previous result to prove that a checkerboard graph link with maximal signature is isotopic to one of the links realized by the simply laced…

链接和平面图代码

Links and Planar Diagram Codes

在本文中，我们形式化的组合对象描述链接图称为平面图代码。PD-代码使用的结理论Mathematica软件包开发的Bar-Natan等人。我们提出了一套PD-代码作为一个独立的对象，并讨论了它与链接图的关系。我们给出了一个明确的算法重建的纽结图的表面上的PD码。我们还讨论了PD码的内在对称性（即，可逆性和手性）。还探索了类似于Reidemeister移动的移动，并且我们证明了给定的PD码集模这些组合Reidemeister移动等价于经典链接理论。

确定奇异链环和泛阿贝尔覆盖的拼接图

Splice diagram determining singularity links and universal abelian covers

对于有理同调球图流形，可以关联一个加权树不变量，称为拼接图。在本文中，我们证明了拼接图上的一个充分的数值条件的图流形是一个奇异链。我们还证明了，如果两个流形有相同的拼接图，那么它们的泛阿贝尔覆盖是同胚的。为了证明最后一个定理，我们必须把我们的概念推广到轨道。

平坦虚拟图的等价性

Equivalence of flat-virtual diagrams

本研究聚焦于作为加厚曲面中链环映射像的平坦虚拟图，探究其等价性与不变量，为利用平坦虚拟链环不变量构建链环不变量提供支撑。

交错虚链路的Jones多项式的跨度

Span of the Jones polynomial of an alternating virtual link

对于定向虚拟链路，L.H.考夫曼定义了f多项式（琼斯多项式）。虚拟链接图的支撑亏格是该图可嵌入的曲面的最小曲面的亏格。本文证明了交错虚环L的f-多项式的跨距由L的任意交错图的交叉数和图的支撑亏格决定。它是Kauffman-Murasugi-Thistlethwaite定理的推广。我们还证明了一个类似的结果，从一个交替的虚拟链路图虚拟一个真正的交叉得到的虚拟链路图。因此，这样的图不等同于经典的链接图。

利用切割图进行组合链接同痕

Combinatorial link concordance using cut-diagrams

引入切割图概念，定义切割同痕关系，证明一维切割图的幂零周边系统是切割同痕不变量，并给出Stallings定理的组合版本。

Grid diagram for singular links

In this paper, we define the set of singular grid diagrams $\mathcal{SG}$ which provides a unified description for singular links, singular Legendrian links, singular transverse links, and singular braids. We also classify the complete set of all equivalence relations on $\mathcal{SG}$ which induce the bijection onto each singular object. This is an extension of the known result of Ng-Thurston for non-singular links and braids.

弦图的定向与Khovanov同调

Orientations of Chord Diagrams and Khovanov Homology

通过在高斯图中添加或删除适当的结构，可以创建与虚拟链接相关的有用对象。本文研究了几个这类对象：扭曲的虚拟链接，推广虚拟链接;签署弦图之间的扭曲虚拟链接和考夫曼括号/ Khovanov同源中途停留;虚拟和经典链接之间的可替代的虚拟链接。这里最深刻的作用属于一种我们称之为和弦图取向的结构。将Khovanov同调推广到有向符号弦图和有向加厚曲面中的链环，使得链环投影实现曲面的第一Stiefel-Whitney类.在这篇论文发表后，V.O.Manturov成功地将具有任意系数的Khovanov同调扩展到任意虚链接，参见arXiv：math.GT/0601152。

由虚链路不变量导出的扭链路不变量

A twisted link invariant derived from a virtual link invariant

虚纽结理论是基于闭定向曲面上的高斯弦图和链图的纽结理论的推广。扭结是虚拟结的推广，其对应于可能非可定向曲面上的链接图。本文讨论了利用双覆盖由虚链路的JKSS不变量得到的扭链路的一个不变量。本文还讨论了双覆盖图的一些性质。

两组分链接图非自OU序列的特征化

Characterization of non-self OU sequences of two-component link diagrams

本文研究了非自OU序列能提取的信息，并完全特征化了两组分链接图中非自OU序列的配对，特别针对交叉数不超过五的素链接进行了具体特征化。

Lattice Diagrams of Knots and Diagrams of Lattice Stick Knots

We give a simple example showing that a knot or link diagram that lies in the ${\mathbb{Z}}^2$ lattice is not necessarily the projection of a lattice stick knot or link in the ${\mathbb{Z}}^3$ lattice, and we give a necessary and sufficient condition for when a knot or link diagram that lies in the ${\mathbb{Z}}^2$ lattice is in fact the projection of a lattice stick knot or link.

Volumes of Random Alternating Link Diagrams

We describe a model of random links based on random 4-valent maps, which can be sampled due to the work of Schaeffer. We will look at the relationship between the combinatorial information in the diagram and the hyperbolic volume. Specifically, we show that for random alternating diagrams, the expected hyperbolic volume is asymptotically linear in the number of crossings. For nonalternating diagrams, we compute the probability of finding a given, arbitrary tangle around a given crossing, and show that a random link diagram will be highly composite. Additionally, we present some results of comp…

焊接链接的并行化

Parallelization of Welded Links

本文提出了一种焊接链接图的并行化构造，展示了其等价性，并探讨了其在棋盘着色和非分裂条件中的应用。

Up-down colorings of virtual-link diagrams and the necessity of Reidemeister moves of type II

We introduce an up-down coloring of a virtual-link diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two 2-component virtual-link diagrams. By using the notion of a quandle cocycle invariant, we determine the necessity of Reidemeister moves of type II for a pair of diagrams of the trivial virtual-knot. This implies that for any virtual-knot diagram $D$, there exists a diagram $D'$ representing the same virtual-knot such that any sequence of generalized Reidemeister moves between them includes at least one Reidemeister…

虚拟纽结与链环的虚拟多重交叉与花瓣图

Virtual Multicrossings and Petal Diagrams for Virtual Knots and Links

本文将经典纽结的多重交叉概念推广至虚拟纽结与链环，并证明所有虚拟纽结均存在花瓣图。

An Unknotting Index for Virtual Links

Given a virtual link diagram $D$, we define its unknotting index $U(D)$ to be minimum among $(m, n)$ tuples, where $m$ stands for the number of crossings virtualized and $n$ stands for the number of classical crossing changes, to obtain a trivial link diagram. By using span of a diagram and linking number of a diagram we provide a lower bound for unknotting index of a virtual link. Then using warping degree of a diagram, we obtain an upper bound. Both these bounds are applied to find unknotting index for virtual links obtained from pretzel links by virtualizing some crossings

链接图的翘曲度

The warping degree of a link diagram

对于有向链图D，翘曲度d（D）是从D得到单调图所需的最小交叉变换次数。证明了d（D）+d（-D）+sr（D）小于或等于D的交叉数，其中-D表示D的逆，sr（D）表示至少有一个自交叉的分支数.并给出了等式成立的一个充要条件。我们还考虑了所有图D的最小d（D）+d（-D）+sr（D）。对于翘曲度和连接翘曲度，我们给出了它们与环绕数、解结数和分裂数的关系。

Milnor's triple linking number and Gauss diagram formulas of 3-bouquet graphs

In this paper, we introduce two functions such that the subtraction corresponds to the Milnor's triple linking number; the addition obtains a new integer-valued link homotopy invariant of $3$-component links. We also have found a series of integer-valued invariants derived from four terms whose sum equals the Milnor's triple linking number. We apply this structure to give invariants of $3$-bouquet graphs.

Delta Diagrams

We call a Delta Diagram any diagram of a knot or link whose regions (including the unbounded one) have 3, 4, or 5 sides. We prove that any knot or link admits a delta diagram. We define and estimate combinatorial link invariants stemming from this definition.

曲面链与切割图的Milnor型不变量

Milnor-type invariants for surface-links and cut-diagrams

将Milnor链环不变量推广到4-空间中的曲面链（可能带边界），通过引入切割图（Gauss图的二维类比）构造群并提取Milnor型不变量，证明其为同痕不变量和链同伦不变量，并给出实现与分类结果。

解统一过程与极小图

The Delunification Process and Minimal Diagrams

一个链图被称为无弯图，如果当被看作一个4-正则平面图时，它在任何一对节点之间都没有重边。我们证明了任何着色链接图都等价于一个具有相同颜色数的着色无月图。因此，任何具有最小颜色数的彩色链接图（称为最小图）都等价于具有相同颜色数的彩色无月图。我们把从一个链接图到一个等价的无弦月图的过程称为去统一过程。然后，我们引入了灰集的概念，以获得更高的最小颜色数的下界。我们计算这些高下界的一些素数模的帮助下，计算机程序。对于每一个穿过16的数，我们列出无月图，并给它们上色。如果颜色的数量等于相应的上下界，我们知道我们有一个最小的颜色数量。我们还介绍并列出了一个环节的无月交叉数，即，该链路的无弯月图所需的最小交叉数，以及其他相关链路不变量。

Successively almost positive links

As an extension of positive or almost positive diagrams and links, we introduce a notion of successively almost positive diagrams and links, and good successively almost positive diagrams and links. We review various properties of positive links or almost positive links, and explain how they can be extended to (good) successively almost positive links. Our investigation also leads to an improvement of known results of positive or almost positive links.

Polynomial of an oriented surface-link diagram via quantum A_2 invariant

It is known that every surface-link can be presented by a marked graph diagram, and such a diagram presentation is unique up to moves called Yoshikawa moves. G. Kuperberg introduced a regular isotopy invariant, called the quantum A_2 invariant, for tangled trivalent graph diagrams. In this paper, a polynomial for a marked graph diagram is defined by use of the quantum A_2 invariant and it is studied how the polynomial changes under Yoshikawa moves. The notion of a ribbon marked graph is introduced to show that this polynomial is useful for an invariant of a ribbon 2-knot.

从纽结到2-链环、弦图的映射以及增强瓦西里耶夫不变量的一种方法

Maps from knots to $2$-links, chord diagrams, and a way to enhance Vassiliev invariants

研究如何将瓦西里耶夫纽结不变量等推广到特定带框弦图，核心方法是探讨相关映射，主要贡献是提供推广方式。

几乎交错链环不可裂的一个判别准则

A criterion for almost alternating links to be non-splittable

给出了几乎交错链环图表示不可裂链环的一个充分条件。主定理给了我们一种方法，看看如果一个给定的几乎交替链接图表示一个可分裂的链接，而不增加的过程中的交叉图的数量。此外，我们表明，几乎交替链接超过两个组件是非平凡的。

纽结图和链环图的互补区域

Complementary Regions of Knot and Link Diagrams

一个递增的整数序列被称为对纽结和链环是普适的，如果每个纽结和链环都有一个到球面的投影，使得投影的每个互补面的边数来自给定的序列。本文研究了有限序列和无限序列的泛域性。我们还考虑如何尽量减少奇数侧的面的数量与n个组件的结和链接的投影。

广义链接图的一种编码方法

A method of encoding generalized link diagrams

我们描述了一种编码各种类型的链接图的方法，包括经典的，平坦的，刚性的，焊接的和虚拟的交叉点。我们表明，这种方法可以用来编码链接图，等价，在一个符号，其长度是一个三次函数的“冒口标记”的数量。对于经典结，这种标记的最小数量是桥索引的两倍，并且具有桥索引$b$的最小桥形式的经典结图可以在空间$\mathcal{O}(b^2)$中编码。定义了符号上的一组移动。作为演示的效用的符号，我们给出了另一个证明，岸野虚结是非经典的。

Regular Isotopy Classes of Link Diagrams From Thompson's Groups

In 2014, Vaughan Jones developed a method to produce links from elements of Thompson's group $F$, and showed that all links arise this way. He also introduced a subgroup $\vec{F}$ of $F$ and a method to produce oriented links from elements of this subgroup. In 2018, Valeriano Aiello showed that all oriented links arise from this construction. We classify exactly those regular isotopy classes of links that arise from $F$, as well as exactly those regular isotopy classes of oriented links that arise from $\vec{F}$, answering a question asked by Jones in 2018.

几乎正链接有负签名

Almost positive links have negative signature

我们分析了具有少量负交叉的图的链接的属性。我们表明，如果一个非平凡的链接有一个图，所有的交叉点，除了可能是一个积极的，那么签名的链接是负的。如果一个链接图有两个负的交叉点，我们表明，该链接的签名是非正的左手Hopf链接（可能平凡的组件）除外。我们还刻画了具有签名零的链路和具有两个负交叉的图。特别是，我们表明，如果一个非平凡的结有一个图有两个负交叉，然后签名的结是负的，除非结是一个扭曲结负扣。我们完全确定所有平凡的链接图有两个或更少的负交叉。对于一个有三个负交叉的纽结图，除了左手三叶结之外，纽结的签名都是非正的。这些结果推广了L. Rudolph，T. Cochran，E. Gompf，P. Traczyk，and J. H. Przytycki，解决了[P-2]的猜想5，并部分回答了[Co-G]的问题2.8关于纽结支配三叶结或平凡纽结的问题。我们还描述了所有解开的一号正结。

着色链接图由亚历山大quandles

Coloring link diagrams by Alexander quandles

本文利用Alexander Quandles研究了连通图的可着色性。证明了若约化的Alexander多项式$Δ_{L}(t)$为零，则$L$允许由任何非平凡的Alexander quandle $Q$进行的非平凡染色;若$Δ_{L}(t)=1$，则$L$只允许由任何Alexander quandle $Q$进行的平凡染色;还证明了若$Δ_{L}(t)\not=0, 1$，则$L$允许由Alexander quandle $Λ/(Δ_{L}(t))$进行的非平凡染色.

Link invariants derived from multiplexing of crossings

We introduce the multiplexing of a crossing, replacing a classical crossing of a virtual link diagram with multiple crossings which is a mixture of classical and virtual. For integers $m_{i}$ $(i=1,\ldots,n)$ and an ordered $n$-component virtual link diagram $D$, a new virtual link diagram $D(m_{1},\ldots,m_{n})$ is obtained from $D$ by the multiplexing of all crossings. For welded isotopic virtual link diagrams $D$ and $D'$, $D(m_{1},\ldots,m_{n})$ and $D'(m_{1},\ldots,m_{n})$ are welded isotopic. From the point of view of classical link theory, it seems very interesting that $D(m_{1},\ldots,…

关于链环图关于Reidemeister move I和III的有序序列

On ordered sequences for link diagrams with respect to Reidemeister moves I and III

本文证明了存在无数对无法通过特定I和III序列变换的平凡纽结图，并提出I-广义有序序列可连通所有经有限次I和III变换的链环图。