PL-虚纽结

虚勒让德结的Vassiliev不变量

Vassiliev Invariants of Virtual Legendrian Knots

我们介绍了一个理论的虚拟勒让德结。虚勒让德纽结是一个有向曲面上的共定向波前，直到它的单位余切丛的升力的勒让德合痕以及远离波前的曲面的稳定和不稳定。我们表明，Vassiliev不变量的虚拟勒让德结和虚拟框架结的群体是同构的。特别是，瓦西里耶夫不变量不能用来区分虚拟勒让德结，这些虚拟勒让德结与虚拟框架结是同位素的，并且具有相等的虚拟马斯洛夫数。

Virtual Mosaic Knot Theory

Mosaic diagrams for knots were first introduced in 2008 by Lomanoco and Kauffman for the purpose of building a quantum knot system. Since then, many others have explored the structure of these knot mosaic diagrams, as they are interesting objects of study in their own right. Knot mosaics have been generalized by Garduño to virtual knots, by including an additional tile type to represent virtual crossings. There is another interpretation of virtual knots, however, as knot diagrams on surfaces, which inspires this work. By viewing classical mosaic diagrams as $4n$-gons and gluing edges of these …

曲面上纽结的稳定等价与虚纽结余约束

Stable Equivalence of Knots on Surfaces and Virtual Knot Cobordisms

我们在纽结图和曲面上的闭曲线上引入了一种等价关系，称为稳定等价。我们给出了抽象结点集、虚结点集和曲面上结点图的稳定等价类集之间的双射。利用这些双射，我们定义了虚拟链接的一致性和链接同源性。作为一个应用，它表明，考夫曼的虚拟纽结图的例子是不等价于一个经典的纽结图。

Concordance group of virtual knots

We study concordance of virtual knots. Our main result is that a classical knot K is virtually slice if and only if it is classically slice. From this we deduce that the concordance group of classical knots embeds into the concordance group of long virtual knots.

关于虚纽结的虚交叉数

On Virtual Crossing Numbers for Virtual Knots

本文的目的是证明一些家庭的虚拟节点的虚拟交叉的最小数量的增长二次关于经典交叉的最小数量。所有以前已知的估计虚拟交叉数主要是不超过线性的经典交叉的数量（或，什么是相同的，在虚拟节点图的边缘数），并没有发现虚拟结与虚拟交叉数大于的经典交叉数。

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。

从虚纽结到经典纽结的函子映射及宇称的推广

A Functorial Map from Virtual Knots to Classical Knots and Generalisations of Parity

我们构造了从虚拟节点到经典节点的各种函子映射（投影）。这些映射被定义在虚结点的图上;根据高斯图，它们中的每一个都可以被表示为一些弦的删除。这种解释依赖于对等的概念。作为推论，我们证明了经典纽结的最小经典交叉数。

Secondary Upsilon invariants of knots

The knot invariant Upsilon, defined by Ozsvath, Stipsicz, and Szabo, induces a homomorphism from the smooth knot concordance group to the group of piecewise linear functions on the interval [0,2]. Here we define a set of related secondary invariants, each of which assigns to a knot a piecewise linear function on [0,2]. These secondary invariants provide bounds on the genus and concordance genus of knots. Examples of knots for which Upsilon vanishes but which are detected by these secondary invariants are presented.

Gordian complexes of knots and virtual knots given by region crossing changes and arc shift moves

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in $\mathbb{S}^3$. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using $v$-move and forbidden moves. In this paper we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high dimensional simplex in both the Go…

虚结、虚辫和自由结的sl（3）、so（3）和G2不变量的图形构造

Graphical Constructions for the sl(3), so(3) and G2 Invariants for Virtual Knots, Virtual Braids and Free Knots

我们构造了虚结点的Kuperberg sl（3）和G2不变量的图值类似物。经典纽结的sl（3）或G2不变量的限制与通常的Homflypt sl（3）不变量和G2不变量一致。对于虚拟节点和图形，这些不变量提供了新的图形信息，允许证明极小定理，并为自由节点（抽象Reidemeister移动的无方向和无标记高斯码）构建新的不变量。这种方法的一个新的特点是，一些结是足够的复杂性，他们评估自己的意义上说，不变的是结本身被视为一个组合结构。本文将这些结构推广到虚拟辫子，并讨论了与原来的彭罗斯括号图着色的关系。

虚拟纽结群

Virtual Knot Groups

考夫曼定义的虚结是经典结的自然推广。结的大多数不变量以自然的方式扩展以给出虚拟结的不变量。在本文中，我们研究了虚纽结的基本群，并观察到一些新的和意想不到的现象。在经典的设定中，如果一个纽结的经度在纽结群中是平凡的，那么这个群是无限循环的。我们证明了对于任何经典的纽结群，都存在一个具有该群和平凡经度的虚纽结。众所周知，经典纽结群的第二同调是平凡的。我们提供了反例，这虚拟结。对任意群G，给出了存在一个虚纽结群的充要条件，该群在G的周边子群上具有特定的行为.这些条件简化了经典环境中出现的条件。

On the lower central series of some virtual knot groups

We study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is a virtual knot with residually nilpotent group. This gives a possibility to construct invariants of virtual knots using quotients by terms of the lower central series of knot groups. Also, we study decomposition of virtual knot groups as semi direct product and free product with amalgamation. In particular, we prove that the groups of some virtual knots are extensions of finitely generated free groups by infinite …

平面虚纯缠结的分类

Classification of Flat Virtual Pure Tangles

虚纽结理论由考夫曼提出，是经典纽结理论的推广，因为它的有限型不变量理论可能是Etingof和Kazhdan的李双代数量子化理论的拓扑解释。经典纽结注入到虚纽结中，而平坦虚纽结是虚纽结的商，其等于实正交叉和实负交叉，并且在这个意义上是虚纽结理论中的经典纽结理论的补充。我们对没有封闭成分的平坦虚拟缠结（纯缠结）进行了完全分类。这种分类可以用作虚拟纯缠结和虚拟辫子的不变量。

纤维结和虚拟结

Fibered Knots and Virtual Knots

介绍了一种用虚纽结理论研究经典纽结的新方法。设$K$是一个纽结，$J$是$K$与$\text{lk}(J,K)=0$的补的纽结。设存在覆盖空间$π_J: Σ\times (0,1) \to \bar{S^3\backslash V(J)}$，其中$V(J)$是$J$的正则邻域，满足$V(J) \cap \text{im}(K)=\emptyset$，$Σ$是连通紧致可定向2-流形.令$K'$是$Σ\times (0,1)$中的结，使得$π_J(K')=K$。然后，$K'$稳定为虚拟结$\hat{K}$，称为$K$相对于$J$的虚拟覆盖。我们调查什么可以说一个经典的结从它的虚拟覆盖的情况下，$J$是一个纤维结。几个例子和应用程序的经典节点。建立了虚拟覆盖的基本理论。

虚拟纽结与链环的虚拟化 Δ 移动

Virtualized Delta moves for virtual knots and links

该研究提出虚拟纽结与链环的虚拟化 Δ 移动，证明其对虚拟纽结是解结操作，并刻画了虚拟链环在该移动下等价的充要条件。

平坦虚拟纽结的连通和与交叉数

Connected sum and crossing numbers of flat virtual knots

研究平坦虚拟纽结连通和对图与基点的依赖性，证明复合纽结的极小交叉图必为连通和图，且交叉数在连通和下具有超加性。

Virtual Knot Cobordism

This paper defines a theory of cobordism for virtual knots and studies this theory for standard and rotational virtual knots and links. Non-trivial examples of virtual slice knots are given. Determinations of the four-ball genus of positive virtual knots are given using the results of a companion paper by the author and Heather Dye and Aaron Kaestner. Problems related to band-passing are explained, and a theory of isotopy of virtual surfaces is formulated in terms of a generalization of the Yoshikawa moves.

Forbidden detour number on virtual knot

We show that the forbidden detour move, essentially introduced by Kanenobu and Nelson, is an unknotting operation for virtual knots. Then we define the forbidden detour number of a virtual knot to be the minimal number of forbidden detour moves necessary to transform a diagram of the virtual knot into the trivial knot diagram. Some upper and lower bounds on the forbidden detour number are given in terms of the minimal number of real crossings or the coefficients of the affine index polynomial of the virtual knot.

The Alexander polynomial for Virtual Twist Knots

We define a family of virtual knots generalizing the classical twist knots. We develop a recursive formula for the Alexander polynomial $Δ_0$ (as defined by Silver and Williams) of these virtual twist knots. These results are applied to provide evidence for a conjecture that the odd writhe of a virtual knot can be obtained from $Δ_0$.

虚拟纽结的矩形马赛克

Rectangular mosaics for virtual knots

本文引入虚拟纽结的矩形马赛克定义，修改Ganzell和Henrich的马赛克移动至矩形设定，提供若干不变量及常见虚拟纽结不变量的计算算法。

An unknotting index for virtual knots

In this paper we introduce the notion of an unknotting index for virtual knots. We give some examples of computation by using writhe invariants, and discuss a relationship between the unknotting index and the virtual knot module. In particular, we show that for any non-negative integer $n$ there exists a virtual knot whose unknotting index is $(1,n)$.

Intersection formulas for parities on virtual knots

We prove that parities on virtual knots come from invariant 1-cycles on the arcs of knot diagrams. In turn, the invariant cycles are determined by quasi-indices on the crossings of the diagrams. The found connection between the parities and the (quasi)-indices allows to define a new series of parities on virtual knots.

平坦虚拟纽结：引言与若干不变量

Flat-virtual knot: introduction and some invariants

该研究围绕从经典纽结到虚拟纽结的映射构建需求展开，定义了平坦虚拟纽结，得到了加厚环面中纽结到平坦虚拟纽结的系列映射，并给出了对应的类Alexander多项式与图像值Kauffman括号不变量。

多虚拟纽结的多元拧扭多项式

Multivariate writhe polynomial of multi-virtual knots

将拧扭多项式不变量从虚拟纽结推广到多虚拟纽结，并解答了[16]中的若干问题。

双环面的一个例子及通过其定义的长虚拟纽结的不变量

An example of biquandle and an invariant of long virtual knots defined through it

本文引入双环面的一个例子，定义长虚拟纽结的不变量，并给出两个可交换长虚拟纽结的必要条件。

关于虚拟纽结的经度的注记

A note on longitudes of virtual knots

该研究指出经典纽结经度的换位子群性质可推广至虚拟纽结，为虚拟纽结的相关研究提供了新的结论

虚拟节点和链接

Virtual Knots and Links

本文是对虚纽结理论的一个介绍，并结合虚纽结的一些具体的新定理进行了讨论。新的结果如下：我们证明，使用3维拓扑方法，如果两个虚拟节点$K_{1}$和$K_{2}$的连接和是平凡的，那么$K_{1}$和$K_{2}$。我们建立了一个算法，使用Haken-Matveev技术，识别虚拟节点。本文既可作为导言，也可作为研究论文阅读。

Maps from knots in the cylinder to flat-virtual knots

In the present paper, we address the problem how to get a map from knots in the cylinder and on the thickened torus to some (generalisation of) virtual knots called virtual-flat knots. The main construction takes a diagram on a cylinder (torus) and adds some ``invisible'' crossings which gives rise to a diagram which can be formally immersed but not embedded (drawn) on the cylinder (torus) and living comfortably in thickened surfaces of higher genera. This allows one to ``pull back'' invariants of virtual theory to the theory of knots in the thickened cylinder (torus) where the parity bracket …

虚纽结的一个多项式不变量

A polynomial invariant of virtual knots

本文的目的是介绍一个多项式不变的$f_K(t)$的虚拟节点。我们表明，$f_K(t)$可以用来区分一些虚拟结从它的逆和镜像。给出了$f_K(t)$在连通和下的性质.最后讨论了对于某些虚结点$K$，哪种多项式可以实现为$f_K(t)$.

A transcendental function invariant of virtual knots

In this work we describe a new invariant of virtual knots. We show that this transcendental function invariant generalizes several polynomial invariants of virtual knots, such as the writhe polynomial, the affine index polynomial and the zero polynomial.

Arc shift number and region arc shift number for virtual knots

In this paper, we formulate a new local move on virtual knot diagram, called arc shift move. Further, we extend it to another local move called region arc shift defined on a region of a virtual knot diagram. We establish that these arc shift and region arc shift moves are unknotting operations by showing that any virtual knot diagram can be turned into trivial knot using arc shift (region arc shift) moves. Based upon the arc shift move and region arc shift move, we define two virtual knot invariants, arc shift number and region arc shift number respectively.

Higher-order finite type invariants of classical and virtual knots and unknotting operations

Vassiliev introduced filtered invariants of knots using an unknotting operation, called crossing changes. Goussarov, Polyak, and Viro introduced other filtered invariants of virtual knots, which order is called GPV-order, using an unknotting operation, called virtualization. We defined other filtered invariants, which order is called $F$-order, of virtual knots using an unknotting operation, called forbidden moves. In this paper, we show that the set of virtual knot invariants of $F$-order $\le n+1$ is strictly stronger than that of $F$-order $\le n$ and that of GPV-order $\le 2n+1$. To obtain…

Finite-Type Invariants of Order One for Framed Virtual Knots

A. Henrich proved the existence of the universal finite-type invariant of order one for virtual knots. We extend the construction and the methods of her paper to framed virtual knots. To do so, we introduce the notions of virtual strings and based matrices for framed flat virtual knots.

Writhe polynomials and shell moves for virtual knots and links

The writhe polynomial is a fundamental invariant of an oriented virtual knot. We introduce a kind of local moves for oriented virtual knots called shell moves. The first aim of this paper is to prove that two oriented virtual knots have the same writhe polynomial if and only if they are related by a finite sequence of shell moves. The second aim of this paper is to classify oriented $2$-component virtual links up to shell moves by using several invariants of virtual links.

虚拟纽结的曲径图

Meander diagrams of virtual knots

本文将经典纽结的（半）曲径图概念推广到虚拟纽结，证明其曲径图类的通用性，并引入虚拟k-弧交叉数这一新不变量，通过Manturov投影证明经典纽结的虚拟与经典k-弧交叉数相等。

用于虚拟结的双量子线

Biquandles for Virtual Knots

本文研究了虚纽结的一个代数不变式--双量子。双量子推广了基本群和虚纽结量子。在本文中采取的方法biquandle强调理解其结构方面的组合物的态射，其中基本态射相关联的定向经典和虚拟交叉的图。

虚纽结的三圈同位素和框架同痕不变量

The Three Loop Isotopy and Framed Isotopy Invariants of Virtual Knots

本文介绍了两个虚拟的纽结理论“类比”的一个著名的家庭不变量的加厚曲面：Grishanov-Vassiliev有限型不变量的二阶。第一个称为三圈同痕不变量，是虚纽结的不变量，而第二个称为三圈框架同痕不变量，是框架虚纽结的正则同痕不变量。这些不变量的性质进行了研究的长度。此外，我们使“类比”的非正式概念精确化。利用这个形式定义，证明了广义三圈不变量是二阶Grishanov-Vassiliev不变量的推广的虚拟纽结理论模拟。

交错虚纽结的Kauffman-Harary猜想

Kauffman-Harary conjecture for alternating virtual knots

1999年，Kauffman-Harary证明了具有素行列式$p$的约化交错纽结图的每一个非平凡Fox $p$ -染色都是异质的. 10年后，这个猜想被W.马特曼和P.索利斯。马修·威廉姆森将这个猜想推广到交替虚纽结，并证明了它的某些家庭的虚纽结。在本文中，我们使用W. Mattman和P. Solis对交错虚纽结的Kauffman-Harary猜想给出了肯定的回答。

虚纽结的一阶Vassiliev不变量序列

A Sequence of Degree One Vassiliev Invariants for Virtual Knots

对于R3中的普通纽结，不存在一次Vassiliev不变量。然而，对于虚结点，一阶瓦西里耶夫不变量的空间是无穷维的。我们引入了一个序列的三度一Vassiliev不变量的虚拟节点的强度增加。我们证明，最强的不变量是一个普遍的Vassiliev不变量的程度为一个虚拟的结在这个意义上，任何其他程度的一个Vassiliev不变量可以从它恢复一定的自然建设。为了证明这些结果，我们扩展的基础矩阵不变量引入的Turaev虚拟字符串类的奇异虚拟结的一个双点。

Alexander多项式在长虚纽结上的推广

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。

虚拟桥一号结

Virtual Bridge Number One Knots

我们定义了虚拟桥数$vb(K)$和虚拟解结数$vu(K)$不变量。对于普通纽结$K$，它们与桥数$b(K)$和解纽结数$u(K)$密切相关，我们有$vu(K)\leq u(K), vb(K)\leq b(K).$没有普通纽结$K$与$b(K)=1.$我们证明了存在无穷多个虚拟纽结的同伦类，每个同伦类包含无穷多个$K$与$vb(K)=1.$的同伦类事实上，对于每个$i\in \N$，存在$K$虚拟同伦（但实际上不是同位素）与$vb(K)=1$和$vu(K)=i.$的解结

Virtual and welded periods of classical knots

We show that any virtual or welded period of a classical knot $K$ can be realized as a classical period. A direct consequence is that a classical knot admits only finitely many virtual or welded periods.

1-和2-串括号多项式不可检测的虚结点

Virtual knots undetected by 1 and 2-strand bracket polynomials

岸野结不能被基本群或括号多项式检测到;这些不变量不能区分岸野结和非结。然而，我们可以证明，岸野的结是不等同于unknot通过应用3-strand括号多项式或表面括号多项式。在本文中，我们构造了两个非平凡的虚拟纽结图，$ K_D $和$ K_m $，这是不是由括号多项式或2-链括号多项式检测。从这些图中，我们构造了两个无限族的非经典虚拟结图，没有检测到的括号多项式。此外，我们注意到这些虚结图是平凡的单位。

着色quandle图及其在缠结嵌入和虚纽结中的应用

On colored quandle longitudes and its applications to tangle embeddings and virtual knots

我们定义了新的不变量的纽结通过quandle着色和纵向信息。这些不变量可以应用于一个缠结嵌入问题和识别非经典的虚拟节点。

平坦虚纽结的一个不变量

An invariant for flat virtual knots

由于定义的模板出现严重错误，该论文已被作者撤回。

Virtual knot groups and almost classical knots

We define a group-valued invariant of virtual knots and relate it to various other group-valued invariants of virtual knots, including the extended group of Silver-Williams and the quandle group of Manturov and Bardakov-Bellingeri. A virtual knot is called almost classical if it admits a diagram with an Alexander numbering, and in that case we show that the group factors as a free product of the usual knot group and Z. We establish a similar formula for mod p almost classical knots, and we use these results to derive obstructions to a virtual knot K being mod p almost classical. Viewed as knot…

A note on coverings of virtual knots

For a virtual knot $K$ and an integer $r\geq 0$, the $r$-covering $K^{(r)}$ is defined by using the indices of chords on a Gauss diagram of $K$. In this paper, we prove that for any finite set of virtual knots $J_0,J_2,J_3,\dots,J_m$, there is a virtual knot $K$ such that $K^{(r)}=J_r$ $(r=0\mbox{ and }2\leq r\leq m)$, $K^{(1)}=K$, and otherwise $K^{(r)}=J_0$.

Khovanov Homology, Lee Homology and a Rasmussen Invariant for Virtual Knots

The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov-Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies to classical knot and classical knot cobordisms. To do so, we give an alternate formulation for the Manturov definition of Khovanov homology for virtual knots and links with arbitrary coefficients. This approach uses cut loci on the knot diagram to induce a conjugation operator in the Frobenius algebra. We then discuss the implications of the maps induced in the aforementioned theory to the…

Alexander invariants of periodic virtual knots

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…

Virtual knot cobordism and bounding the slice genus

In this paper, we compute the slice genus for many low-crossing virtual knots. For instance, we show that 1295 out of 92800 virtual knots with 6 or fewer crossings are slice, and that all but 248 of the rest are not slice. Key to these results are computations of Turaev's graded genus, which we show extends to give an invariant of virtual knot concordance. The graded genus is remarkably effective as a slice obstruction, and we develop an algorithm that applies virtual unknotting operations to determine the slice genus of many virtual knots with 6 or fewer crossings.

半拟群的一个例子及利用该半拟群着色得到的平坦虚纽结的u-多项式

An example of semiquandle and u-polynomials for flat virtual knots via this semiquandle coloring

本文给出一个半拟群例子，利用其着色定义平坦虚纽结的不变量，并阐明其与u-多项式的联系。

Recurrent Generalization of F-Polynomials for Virtual Knots and Links

F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in 2018 using flat virtual knot invariants. These polynomials naturally generalize Kauffman's affine index polynomial and use smoothing in classical crossing of a virtual knot diagram. In this paper we introduce weight functions for ordered orientable virtual and flat virtual link. A flat virtual link is an equivalence class of virtual links in respect to a local symmetry changing type of classical crossing in a diagram. By considering three types of smoothings in classical crossings of a virtual link diagram and suitab…

定向虚拟纽结与链环的虚拟化 Delta、sharp 和 pass 移动

Virtualized Delta, sharp, and pass moves for oriented virtual knots and links

本文研究定向虚拟链环上的虚拟化 Delta、sharp 与 pass 移动，给出链环经这些局部移动等价的充要条件，并确立可达到的解结数下界。

Alexander invariants for virtual knots

Given a virtual knot $K$, we construct a group $VG_K$ called the virtual knot group, and we use the elementary ideals of $VG_K$ to define invariants of $K$ called the virtual Alexander invariants. For instance, associated to the $k=0$ ideal is a polynomial $H_K(s,t,q)$ in three variables which we call the virtual Alexander polynomial, and we show that it is closely related to the generalized Alexander polynomial $G_K(s,t)$ introduced by Sawollek, Kauffman-Radford, and Silver-Williams. We define a natural normalization of the virtual Alexander polynomial and show it satisfies a skein formula. W…