Minimal diagrams of classical and virtual links
Vassily Olegovich Manturov
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
We prove that a virtual link diagrams satisfying two conditions on the Khovanov homology is minimal, that is, there is no virtual diagram representing the same link with smaller number of crossings. This approach works for both classical and virtual links
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A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative.In this paper, we introduce a method of converting a virtual link diagram to a normal virtual link diagram by use of the double covering technique. We show that the normal virtual link diagrams obtained from two equivalent virtual link diagrams are related by generalized Reidemeister moves and Kauffman flypes.
A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative. Normal virtual links have some properties similar to classical links.In this paper, we introduce a method of converting a virtual link diagram to a normal virtual link diagram. We show that the normal virtual link diagrams obtained by this method from two equivalent virtual link diagrams are equivalent. We discuss the relationship between this method and some invariants of virtual links.
A virtual link diagram is called mod $m$ almost classical if it admits an Alexander numbering valued in integers modulo $m$, and a virtual link is called mod $m$ almost classical if it has a mod $m$ almost classical diagram as a representative. In this paper, we introduce a method of constructing a mod $m$ almost classical virtual link diagram from a given virtual link diagram, which we call an $m$-fold cyclic covering diagram. The main result is that $m$-fold cyclic covering diagrams obtained from two equivalent virtual link diagrams are equivalent. Thus we have a well-defined map from the se…
We define a second Steenrod square for virtual links, which is stronger than Khovanov homology for virtual links, toward constructing Khovanov-Lipshitz-Sarkar stable homotopy type for virtual links. This induces the first meaningful nontrivial example of the second Steenrod square operator on the Khovanov homology for links in a 3-manifold other than the 3-sphere.
Virtual index cocycle is the 1-cochain that counts virtual crossings in the arcs of a virtual link diagram. We show how this cocycle can be used to reformulate and unify some known invariants of virtual links.
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。
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。
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…
Oikawa defined an unknotting operation on virtual knots, called a CF-move, and gave a classification of 2-component virtual links up to CF-moves by the virtual linking number and his $n$-invariant. In particular, it was proved that a CF-move characterizes the information contained in the virtual linking number for 2-component odd virtual links. In this paper, we extend this result by classifying odd virtual links and almost odd virtual links with arbitrary number of components up to CF-moves, using the virtual linking number. Moreover, we extend Oikawa's $n$-invariant and introduce two invaria…
Strict Equivalence of Multi-Virtual Linkoids
利用多虚拟纽结理论研究严格虚拟链环的等价关系，通过闭包构造和广义括号多项式获得不变量。
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。
Virtual Knot Diagrams and the Witten-Reshetikhin-Turaev Invariant
将经典链图的Witten-Reshetikhin-Turaev不变量推广到虚链图。这个不变量不受框架Reidemeister移动和Kirby演算的影响。因此，它也是由经典链接图表示的3-流形的不变量。这个推广被用来证明存在具有非平凡Witten-Reshetikhin-Turaev不变量和平凡3-流形基本群的虚纽结图。
Polyak type equations for virtual arrow diagram formulas in the annulus
本文从M.波利亚克作为一个主要应用，我们对平面链不变量的Grishanov-Vassiliev定理作了一些改进。
Jin and Lee proved the following: Suppose that $D_1, \dots, D_n$ are link diagrams. Given a link $L$ which is partitioned into sublinks $L_1, \dots, L_n$ admitting diagrams $D_1, \dots, D_n$ respectively, there is a diagram $D$ of $L$ whose restrictions to $L_1, \dots, L_n$ are isotopic to $D_1, \dots, D_n$, respectively. In this paper we show that a similar result does hold for welded links and does not for virtual links.
A virtual link is a generalization of a classical link that is defined as an equivalence class of certain diagrams, called virtual link diagrams. It is further generalized to a twisted link. Twisted links are in one-to-one correspondence with stable equivalence classes of links in oriented thickenings of (possibly non-orientable) closed surfaces. By definition, equivalent virtual links are also equivalent as twisted links. In this paper, we discuss when two virtual links are equivalent as twisted links, and give a necessary and sufficient condition for this to be the case.
Unknotting virtual knots with Gauss diagram forbidden moves
禁止的移动可以与高斯图Reidemeister移动相结合，以获得移动序列，我们可以将任何高斯图（因此任何虚拟结）改变为任何其他，特别是包括未结图
In this paper, we define the notion of a virtually symmetric representation of representations of virtual braid groups and prove that many known representations are equivalent to virtually symmetric. Using one such representation, we define the notion of virtual link groups which is an extension of virtual link groups defined by Kauffman. Moreover, we introduce the concept of marked Gauss diagrams as a generalisation of Gauss diagrams and their interpretation in terms of knot-like diagrams. We extend the definition of virtual link groups to marked Gauss diagrams and define their peripheral str…
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。
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。
Virtual Genus of Satellite Links
虚拟卫星链路的虚拟亏格等于其同伴的虚拟亏格。
We define a generalization of virtual links to arbitrary dimensions by extending the geometric definition due to Carter et al. We show that many homotopy type invariants for classical links extend to invariants of virtual links. We also define generalizations of virtual link diagrams and Gauss codes to represent virtual links, and use such diagrams to construct a combinatorial biquandle invariant for virtual $2$-links. In the case of $2$-links, we also explore generalizations of Fox-Milnor movies to the virtual case. In addition, we discuss definitions extending the notion of welded links to h…
Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed oriented surfaces. A twisted knot is a generalization of a virtual knot, which corresponds to a link diagram on a possibly non-orientable surface. In this paper, we discuss an invariant of twisted links which is obtained from the JKSS invariant of virtual links by use of double coverings. We also discuss some properties of double covering diagrams.
We consider the group of unrestricted virtual braids, describe its structure and explore its relations with fused links. Also, we define the groups of flat virtual braids and virtual Gauss braids and study some of their properties, in particular their linearity.
Checkerboard framings are an extension of checkerboard colorings for virtual links. According to checkerboard framings, in 2017, Dye obtained an independent invariant of virtual links: the cut point number. Checkerboard framings and cut points can be used as a tool to extend other classical invariants to virtual links. We prove that one of the conjectures in Dye's paper is correct. Moreover, we analyze the connection and difference between checkerboard framing obtained from virtual link diagram by adapting cut points and twisted link diagram obtained from virtual link diagram by introducing ba…
Virtual links are algorithmically recognisable
我们证明了存在一个判定两条虚链路是否等价的算法
Cobordism of virtual string links on $n$ strands is a combinatorial generalization of link cobordism. There exists a bijection between virtual string links up to cobordisms and elements of the group $\mathbb{Z}^{n(n-1)}$. This paper also shows that virtual string links up to unwelded equivalence are classified by those groups. Finally, the related theory of welded string link cobordism is defined herein and shown to be trivial for string links with one component.
Gauss diagrams of real and virtual knots in the solid torus
我们定义了一种新的高斯图来描述固体环面中的节点与环中的投影。我们看到，它提供了一个有效的工具，显示一个结图可以完全恢复从其装饰高斯图，我们用它来建立一个封闭的辫子装饰高斯图的表征。
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。
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
We extend some classical results of Bankwitz, Crowell, and Murasugi to the setting of virtual links. For instance, we show that an alternating virtual link is split if and only if it is visibly split, and that the Alexander polynomial of any almost classical alternating virtual link is alternating. The first result is a consequence of an inequality relating the link determinant and crossing number for any non-split alternating virtual link. The second is a consequence of the matrix-tree theorem of Bott and Mayberry. We extend the first result to semi-alternating virtual links. We discuss the T…
The Wirtinger number of a virtual link is the minimum number of generators of the link group over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a virtual link equals its virtual bridge number. Since the Wirtinger number is algorithmically computable, it gives a more effective way to calculate an upper bound for the virtual bridge number from a virtual link diagram. As an application, we compute upper bounds for the virtual bridge numbers and the quandle counting invariants of virtual knots with…
We use virtual knot theory to detect the non-invertibility of some classical links in $\mathbb{S}^3$. These links appear in the study of virtual covers. Briefly, a virtual cover associates a virtual knot $\upsilon$ to a knot $K$ in a $3$-manifold $N$, under certain hypotheses on $K$ and $N$. Virtual covers of links in $\mathbb{S}^3$ come from taking $K$ to be in the complement $N$ of a fibered link $J$. If $J \sqcup K$ is invertible and $K$ is "close to" a fiber of $J$, then $\upsilon$ satisfies a symmetry condition to which some virtual knot polynomials are sensitive. We also discuss virtual …
In this paper we investigate the virtual string links via a probabilistic interpretation. This representation can be used to distinguish some virtual string links from classical string links. In order to study the algebraic structure behind this probabilistic interpretation we introduce the notion of virtual flat biquandle. The cocycle invariants associated with virtual flat biquandle is discussed.
On the Jones polynomials of checkerboard colorable virtual knots
引入了虚拟链接和抽象链接的棋盘可染性概念。研究了虚链路和抽象链路的Jones多项式。证明了经典链环的琼斯多项式的某些性质对允许棋盘着色的虚链环也是有效的。
Biquandles with structures related to virtual links and twisted links
我们在双量子点上引入了两种结构，称为v-结构和t-结构。这些结构用于虚拟链路和扭曲链路的图的着色，使得着色的数量是不变的。给定一个双量子线或一个量子线，我们给出了用这些结构构造一个双量子线的方法。使用着色的数量，我们表明，Bourgoin的twofoil和不可定向的虚拟$m$ -箔不代表虚拟链接。
In this paper we give two new criteria of detecting the checkerboard colorability of virtual links by using odd writhe and arrow polynomial of virtual links, respectively. By applying new criteria, we prove that 6 virtual knots are not checkerboard colorable, leaving only one virtual knot whose checkerboard colorability is unknown among all virtual knots up to four classical crossings.
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。
Two classes of virtually fibered Montesinos links of type $\widetilde{SL_2}$
我们发现两个新的类几乎纤维的经典Montesinos链接类型$\widetilde{SL_2}$。
A weak chord index $Ind'$ is constructed for self crossing points of virtual links. Then a new writhe polynomial $W$ of virtual links is defined by using $Ind'$. $W$ is a generalization of writhe polynomial defined in [6]. Based on $W$, three invariants of virtual links are constructed. These invariants can be used to detect the non-trivialities of Kishino knot and flat Kishino knot.
Classification of virtual links by arc shift move
本文证明n-齐次真虚拟链环的弧移位操作可作为解结操作，探究其弧移位次数与奇拧数的关联，并确定了弧移位次数上界恰好为n的虚拟链环图序列Lₙ，助力n-分支虚拟链环的分类。
New presentations of a link and a virtual link are introduced and algebraic systems on links and virtual links are constructed respectively. Based on the algebraic systems, Reduction Crossing Algorithms for them are proposed which are used to reduce the number of crossings in a link and virtual link. For known unknots, one can transform them into a trivial knot in a polynomial time by applying corresponding algorithm. As special consequences, Goeritz's unknot and Thistlethwaite's unknot are unknotted. Moreover, an infinite family of knots $K_{G_{2k,2l}}$ are unknotted in $O(n^2)$ time where $n…
We extend the theory of hyperbolicity of links in the 3-sphere to tg-hyperbolicity of virtual links, using the fact that the theory of virtual links can be translated into the theory of links living in closed orientable thickened surfaces. When the boundary surfaces are taken to be totally geodesic, we obtain a tg-hyperbolic structure with a unique associated volume. We prove that all virtual alternating links are tg-hyperbolic. We further extend tg-hyperbolicity to several classes of non-alternating virtual links. We then consider bounds on volumes of virtual links and include a table for vol…
We succeed to generalize spun knots of classical 1-knots to the virtual 1-knot case by using the `spinning construction'. That, is, we prove the following: Let $Q$ be a spun knot of a virtual 1-knot $K$ by our method. The embedding type $Q$ in $S^4$ depends only on $K$. Furthermore we prove the following: The submanifolds, $Q$ and the embedded torus made from $K,$ defined by Satoh's method, in $S^4$ are isotopic. We succeed to generalize the above construction to the virtual 2-knot case. Note that Satoh's method says nothing about the virtual 2-knot case. Rourke's interpretation of Satoh's met…
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…
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。
Great circle links and virtually fibered knots
我们表明，所有的两桥结和链接互补几乎纤维化。我们还表明，球形Montesinos结和链接补充几乎纤维。这是通过证明这样的结补语被大循环链补语覆盖来实现的。
It is known that the Kauffman-Murasugi-Thislethwaite type inequality becomes an equality for any (possibly virtual) adequate link diagram. We refine this condition. As an application we obtain a criterion for virtual link diagram with exactly one virtual crossing to represent a properly virtual link.
Coverings, composites and cables of virtual strings
虚串可以定义为平面图在某种图移下的一个等价类。虚拟字符串与虚拟结相关，因为对虚拟结图的简单操作产生虚拟字符串的图。在本文中，我们考虑三个操作的虚拟字符串或虚拟字符串产生另一个虚拟字符串，即覆盖，组成和电缆。特别是，我们研究虚拟字符串不变的覆盖操作。我们还展示了一个复合虚拟字符串的基础矩阵是如何与其组件的基础矩阵，纠正Turaev的结果。最后，我们调查电缆下发生了什么，一些不变量定义的Turaev。
Virtual, Welded, and Ribbon Links in Arbitrary Dimensions
我们定义了一个推广的虚拟链接到任意尺寸的几何定义，由于卡特等人。我们表明，许多经典链接的伦型不变量扩展到虚拟链接的不变量。我们还定义了广义的虚拟链接图和高斯码来表示虚拟链接，并使用这样的图来构建一个组合biquandle不变量的虚拟$2$链接。在$2$ -链接的情况下，我们还探讨了福克斯米尔诺电影的虚拟情况下的概括。此外，我们还讨论了将焊接连接的概念扩展到更高维度的定义。对于在$4$维或更大的带状结，我们表明，结quandle是一个完整的分类不变的连接总和与平凡打结$S^1\times S^{n-1}$，所有涉及的同位素可以采取产生的带状结的稳定等价。
On Alternateness of the Jones-Kauffman Polynomial of Virtual Links
本文研究虚拟链环的Jones-Kauffman多项式的交错性，定义拟交错虚拟链环类，利用扭转技术构造无限族，并给出棋盘可着色链环的拟行列式的连通状态展开。
Invariants of virtual links and twisted links using affine indices
本文利用经典交叉点上的仿射指数，梳理仿射指数多项式与 n-writhe 的关系，并将其推广到可由切割系统计算的虚拟链环不变量，同时构造扭曲链环不变量。
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。
Khovanov homology for virtual links with arbitrary coefficients
利用扭曲系数方法，我们构造了任意系数虚链路的Khovanov同调论。该方法也适用于构建Viro意义上的"非定向虚拟结“的Khovanov同源性，特别是${\bf R}P^{3}$中的结。
What is a virtual link?
一些作者最近研究了虚拟结和链接，因为他们承认R-矩阵产生的不变量。我们证明了在S X I中每个虚链环都唯一地由链环L表示，S X I是一个加厚的、紧致的、有向的曲面S，使得链环补（S X I）- L没有本质的垂直柱面。
A virtual doodle is an equivalence class of virtual diagrams under an equivalence relation generated by flat version of classical Reidemesiter moves and virtual Reidemsiter moves such that Reidemeister moves of type 3 are forbidden. In this paper we discuss colorings of virtual diagrams using an algebra, called a doodle switch, and define an invariant of virtual doodles. Besides usual colorings of diagrams, we also introduce doubled colorings.
The Khovanov Complex for Virtual Links
本文构造了虚链环的Khovanov同调论。除了直接的方法与Z_{2}系数，我们还描述了Khovanov同源框链接和Khovanov同源使用"双覆盖“。后两种方法是基于"原子“的概念。
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.
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。
On Virtual Crossing Number Estimates For Virtual Links
我们解决的问题，检测最小的虚拟图相对于虚拟交叉点的数量。这个问题是密切相关的问题检测的最小数量的额外的交点的通用浸入的一个奇异的链接在$R^{2}$。我们解决这个问题，所谓的$ξ$ -多项式的领导（最低）度自然估计虚拟交叉数。给出了极小性的几个充分条件以及无穷级数的新例子。我们还陈述了几个关于$M$ -图的公开问题，根据我们的充分条件，这些图是极小的。
The arrow polynomial is an invariant of framed oriented virtual links that generalizes the virtual Kauffman bracket. In this paper we define the homological arrow polynomial, which generalizes the arrow polynomial to framed oriented virtual links with labeled components. The key observation is that, given a link in a thickened surface, the homology class of the link defines a functional on the surface's skein module, and by applying it to the image of the link in the skein module this gives a virtual link invariant. We give a graphical calculus for the homological arrow polynomial by taking th…
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