图链理论导论

图-环的Khovanov同调

Khovanov homology of graph-links

我们定义Khovanov同调模2图链接。

关于连接图的Seifert图及其平行图

On the Seifert graphs of a link diagram and its parallels

最近，Dasbach，Futer，Kalfagianni，Lin和Stoltzfus通过将一组带状图（或等价地，嵌入图）关联到链接图来扩展Tait图的概念。在这里，我们专注于塞弗特图，这是一个结或链接图，从塞弗特状态产生的带状图。我们提供了塞弗特图的欧拉子图的一个特征。这个特征可以看作是塞弗特图是二部图的一个改进。我们继续研究家庭的带状图，所产生的形成平行的链接图，并确定如何属的带状图的$r$倍平行的链接图是相关的原始链接图。

完全图中的链计数

Counting Links in Complete Graphs

我们找到了任意7个顶点的完全$k$ -部图（包括至少有21个链环的$K_7$）的嵌入中的最小链环数.我们给出了8点完全$k$ -部图的最小链数的精确值或上、下界。我们也看看更大的完全二部图，并说明一个猜想有关的最小链接嵌入与最小的书嵌入。

Large-scale geometry of the saddle connection graph

We prove that the saddle connection graph associated to any half-translation surface is 4-hyperbolic and uniformly quasi-isometric to the regular countably infinite-valent tree. Consequently, the saddle connection graph is not quasi-isometrically rigid. We also characterise its Gromov boundary as the set of straight foliations with no saddle connections. In our arguments, we give a generalisation of the unicorn paths in the arc graph which may be of independent interest.

On ribbon graphs and virtual links

We introduce a new equivalence relation on decorated ribbon graphs, and show that its equivalence classes directly correspond to virtual links. We demonstrate how this correspondence can be used to convert any invariant of virtual links into an invariant of ribbon graphs, and vice versa.

替代链接、同构链接和图形

Alternative links, homogeneous links, and graphs

在本文中，我们回顾了同质和替代链接的定义。我们还给出了两个新的特征的替代链接图，一个在增强的棋盘图的上下文中，另一个从标记塞弗特图。

内连通图与偶环绕数

Intrinsically linked graphs and even linking number

我们研究内在链接图，我们要求每个嵌入的图不仅包含一个非分裂的链接，但一个链接，满足一些额外的属性。我们在本文中讨论的性质的例子是：一个两个组件链接lk（A，L）= k2 ^r，k不为0，一个非分裂的n-组件链接，其中所有的环绕数是偶数，或一个n-组件链接与组件L，A_i，其中lk（L，A_i）= 3 k，k不为0。与其他属性的链接也被考虑在内。对于给定的性质，我们证明了某个完全图的每个嵌入都包含一个具有该性质的链。完全图的大小由所讨论的性质决定。

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.

图链的Conway势函数

The Conway potential function of a graph link

本文给出了同调球面中任意图链的多元Conway势函数的闭公式。作为推论，我们回答了Walter Neumann关于图链的三个问题。

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…

Graphical virtual links and a polynomial of signed cyclic graphs

For a signed cyclic graph G, we can construct a unique virtual link L by taking the medial construction and convert 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In the article we shall prove that a virtual link L is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial F[G] for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between F[G] of a signed cyclic graph G and the …

有色链环与空间图的同伦理论

Homotopy theories of colored links and spatial graphs

本文基于Habegger-Lin理论，完成了有色弦链的有色链环同伦分类，并进一步分别实现了有色链环的有色链环同伦分类与空间图的分支同伦分类。

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。

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。

具有纽结分支的内链图

Intrinsically Linked Graphs with Knotted Components

我们构造一个图G，使得G到R^{3}的任何嵌入都包含两个分量的非分裂链，其中至少一个分量是非平凡结。此外，对于任意m < n，我们生成一个图H，使得H的每个嵌入都包含一个非分裂的n个分量链，其中至少m个分量是非平凡的结点。然后，我们把我们的注意力转向完全图，并表明，对于任何给定的n，每一个嵌入的足够大的完全图包含两个组件链接的环绕数是一个非零倍的n。

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。

Generalization of the Conway-Gordon theorem and intrinsic linking on complete graphs

Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and Kazakov and Korablev proved that for every spatial complete graph with arbitrary number of vertices greater than six, the sum of the linking numbers over all of the constituent two-component Hamiltonian links is even. In this paper, we show that for every spatial complete graph whose number of vertices is greater than six, the sum of the square of the linking numbers over all of the two-component Hamiltonian links is determin…

与图形的一般浸入相关联的链接

Links associated with generic immersions of graphs

作为代数链类的扩展，A'Campo、Gibson和Ishikawa构造了与二维圆盘中的浸入弧和树相关的链。通过扩展他们的论点，我们构建了相关的沉浸图在一个磁盘上的链接，并表明这样的链接是拟正的。