Virtual Crossing Number and the Arrow Polynomial
H. A. Dye, Louis H. Kauffman
我们引入了一个新的多项式不变量的虚拟节点和链接，并使用此不变量计算的虚拟交叉数和极小曲面亏格的下界。
We introduce a new polynomial invariant of virtual knots and links and use this invariant to compute a lower bound on the virtual crossing number and the minimal surface genus.
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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.
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。
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…
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…
Introduction to Virtual Knot Theory
本文介绍虚纽结理论，阐述了新的概念和构造，包括奇偶括号多项式、箭多项式、奇偶箭多项式和箭多项式的推广。本文是相对独立的，它描述了虚拟的纽结理论组合，并在加厚表面的纽结理论。箭头多项式（Dye和Kauffman）是琼斯多项式的自然推广，通过使用状态和中的图的定向结构获得。本文讨论了Vassily Manturov开创的宇称的应用，并用他的宇称括号多项式给出了Fenn，Kauffman和Manturov的一个猜想的反例。该文件给出了一个阐述的分类的箭头多项式由于染料，考夫曼和Manturov和它给出了一个例子（从许多发现亚伦Kaestner）的一对虚拟的结，没有区分的Khovanov同源性（模2），或箭头多项式，但区别于一个分类的箭头多项式。奇偶校验计算的其他示例被指示。例如，同一对通过奇偶括号多项式来区分。
On two categorifications of the arrow polynomial for virtual knots
给出了矢多项式的两个简化，矢多项式是Kauffman括号多项式在虚结点上的推广.箭头多项式将括号多项式扩展到无限多个变量，每个变量对应于从括号的定向状态求和中的每个循环计算的整数{ \it arrow number}。这些证明基于与这些箭头数相关的新的分级，并给出了与定向虚结和链接相关的同调论，通过Khovanov链复形上的额外结构。应用于虚拟节点和虚拟链环的虚拟交叉数和曲面亏格的估计。关键词：Jones多项式，括号多项式，扩展括号多项式，箭头多项式，Miyazawa多项式，Khovanov复形，Khovanov同调，Reidemeister移动，虚纽结理论，微分，偏微分，分次，点分次，向量分次。
Arrow ribbon graphs
我们在带状图上引入了一种新的结构，箭头结构。我们将Bollobás-Riordan多项式推广到具有这种结构的带状图。扩展多项式满足压缩-删除关系，并且自然地表现出带状图的部分对偶性。我们构造了一个由虚链路构成的箭头带状图，其扩展的Bollobás-Riordan多项式专门用于最近由H.Dye和L. Kauffman引入的虚链路的箭头多项式.这一结果将经典的Thistlethwaite定理推广到虚链路的箭头多项式。
Lower bounds on virtual crossing number and minimal surface genus
利用箭头多项式计算了虚纽结图的虚交叉数和极小曲面亏格的下界。特别是，我们专注于几个有趣的例子。
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。
Categorification of Biquandle Arrow Weight Invariants via Quivers
本文使用biquandle coloring quiver对biquandle arrow weight多项式不变量进行范畴化，为定向虚拟和经典纽结获得了新的无限族多项式不变量。
We develop a calculus for diagrams of knotted objects. We define Arrow presentations, which encode the crossing informations of a diagram into arrows in a way somewhat similar to Gauss diagrams, and more generally w-tree presentations, which can be seen as `higher order Gauss diagrams'. This Arrow calculus is used to develop an analogue of Habiro's clasper theory for welded knotted objects, which contain classical link diagrams as a subset. This provides a 'realization' of Polyak's algebra of arrow diagrams at the welded level, and leads to a characterization of finite type invariants of welde…
Let R f = Z[A $\pm$1 ] be the algebra of Laurent polynomials in the variable A and let R a = Z[A $\pm$1 , z 1 , z 2 ,. .. ] be the algebra of Laurent polynomials in the variable A and standard polynomials in the variables z 1 , z 2 ,. .. . For n $\ge$ 1 we denote by VB n the virtual braid group on n strands. We define two towers of algebras {VTL n (R f)} $\infty$ n=1 and {ATL n (R a)} $\infty$ n=1 in terms of diagrams. For each n $\ge$ 1 we determine presentations for both, VTL n (R f) and ATL n (R a). We determine sequences of homomorphisms {$ρ$ f n : R f [VB n ] $\rightarrow$ VTL n (R f)} $\…
Connected sum and crossing numbers of flat virtual knots
研究平坦虚拟纽结连通和对图与基点的依赖性，证明复合纽结的极小交叉图必为连通和图，且交叉数在连通和下具有超加性。
On Virtual Crossing Number Estimates For Virtual Links
我们解决的问题，检测最小的虚拟图相对于虚拟交叉点的数量。这个问题是密切相关的问题检测的最小数量的额外的交点的通用浸入的一个奇异的链接在$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。
Virtual Crossing Realization
我们研究虚拟的经典的初始和最终图的合痕序列，问这样的序列时，可以改变成一个经典的合痕序列取代虚拟交叉与经典的交叉。给出了一个不存在这种虚交叉实现的序列的例子。提出了虚合痕序列可实现条件的一个猜想，得到了虚合痕序列可实现的一个充分条件，并将该猜想用2-纽结和加厚曲面纽结表示.
In this paper we construct new invariants of knotoids including the odd writhe, the parity bracket polynomial, the affine index polynomial and the arrow polynomial, and give an introduction to the theory of virtual knotoids. The invariants in this paper are defined for classical knotoids in analogy to corresponding invariants of virtual knots. The affine index polynomial and the arrow polynomial provide bounds on the height (minimum crossing distance between endpoints) of a classical knotoid. We show that classical knotoids have symmetric affine index polynomials.
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…
In this note we give a new lower bound on the virtual crossing number via the writhe polynomial, which refines a result of B. Mellor. The proof is based on a new interpretation of the writhe polynomial. The characterization of the writhe polynomial is also discussed.
Biquandle Arrow Weight Enhacements
本文提出双括号箭头权重这一新的无限族不变量，通过为高斯图中箭头交点赋予阿贝尔群权重来增强双括号同态集不变量，并证明其非平凡且严格。
In the present paper, we develop a picture formalism which gives rise to an invariant that dominates several known invariants of classical and virtual knots: the Jones polynomial, the Kuperberg bracket, and the normalised arrow polynomial.
The virtual unknotting number of a virtual knot is the minimal number of crossing changes that makes the virtual knot to be the unknot, which is defined only for virtual knots virtually homotopic to the unknot. We focus on the virtual knot obtained from the standard (p,q)-torus knot diagram by replacing all crossings on one overstrand into virtual crossings and prove that its virtual unknotting number is equal to the unknotting number of the $(p,q)$-torus knot, i.e. it is (p-1)(q-1)/2.
A table of knotoids in $S^3$ up to seven crossings
本文对S³上最多七次交叉的结oid进行了完整分类，并探讨了其镜像性和旋转对称性，展示了此类分类在蛋白质纠缠中的应用价值。
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,…
A virtual link may be defined as an equivalence class of diagrams, or alternatively as a stable equivalence class of links in thickened surfaces. We prove that a minimal crossing virtual link diagram has minimal genus across representatives of the stable equivalence class. This is achieved by constructing a new parity theory for virtual links. As corollaries, we prove that the crossing, bridge, and ascending numbers of a classical link do not decrease when it is regarded as a virtual link. This extends corresponding results in the case of virtual knots due to Manturov and Chernov.
An Extended Bracket Polynomial for Virtual Knots and Links
本文定义了虚纽结和虚链环的一个新的不变量，我们称之为扩展括号多项式，并<K>对虚纽结或虚链环K用< >表示。这个不变量是K的定向图的括号状态的状态求和。每个状态在平面上被简化为一个虚拟的4-正则图，多项式取由这些简化图在环Q[A，A^{-1}]上生成的模中的值。这篇论文相对独立，有关于虚结和长虚结的背景资料。我们给出了许多应用扩展括号的例子，包括岸野图和平面岸野图的非平凡性和单交叉虚拟化的非经典性的新证明。本文有一节是关于用扩展括号状态和估计虚交叉数的。给出了具有任意最小嵌入亏格和任意高的虚拟交叉数与最小嵌入亏格之间的正差的虚拟节点的例子。引入了< >的简化<K>并表示为A[K]。这个简化的扩展括号，箭头多项式，是变量的无限集中的多项式。它非常强大（例如检测平面Kishino图）并且易于计算。本文描述了A[K]的一种计算机算法，并使用箭头多项式和扩展括号多项式来确定一些虚拟节点可以表示的最小亏格曲面。
Parity, Skein Polynomials and Categorification
本文研究了交叉宇称在琼斯多项式虚结点括号展开中的应用。此外，我们考虑的应用程序的奇偶校验的箭头多项式，以及为两个多项式的重复。我们提出了一些例子，通过我们的计算发现。我们提供的表计算这些不变量的虚拟结最多4个真正的交叉。
The virtual skein relation for the Jones polynomial of the virtual link diagram was introduced by N. Kamada, S. Nakabo, and S. Satoh. H. A. Dye, L. H. Kauffman, and Y. Miyazawa introduced multivariable polynomial, an invariant of virtual links, which is a refinement of Jones polynomial. In this paper, we give a skein relation for the multivariable polynomials among positive, negative, and virtual crossings with some restrictions. We apply this relation to study some properties of virtual links obtained by replacing a real crossing by a virtual crossing.
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
Biquandle Arrow Weight Quiver Representations
定义了一类与双箭袋、阿贝尔群、箭头权值等数据相关的无限族箭图表示值不变量，用于经典和虚拟纽结，并提取了四个新多项式不变量作为解范畴化。
Parity Biquandles
我们使用交叉宇称来构造一个广义的双量子点，我们称之为宇称双量子点。这些结构包括所有的双量子点作为标准示例，称为偶宇称双量子点。此外，我们发现所有的宇称倍对于一个特定的建设命名的z-奇偶亚历山大Biquandle，我们表明，相关的多项式产生一个下界的奇数交叉的数量，以及实际的交叉和虚拟交叉的虚拟结的总数。此外，我们将这种结构扩展到链接，以获得一个虚拟链路的组件之间的交叉点的数量的下界。
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。
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…
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。
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.
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…
The intersection polynomials of a long virtual knot I: Definitions and properties
本文提出十二个交多项式作为长虚拟结的不变量，扩展了虚拟结的交多项式，并研究了其在对称性、交叉变化和连接积下的性质及与闭合和导数的关系。
On tabulating virtual strings
虚弦可以被定义为曲面上的闭曲线模某些等价关系。Turaev定义了几个不变量的虚拟字符串，我们用它来产生一个表的虚拟字符串多达4个交叉。我们讨论了将列表扩展到5个过境点的进展情况。我们还提供了一个反例角上的声明。
We consider arrow diagrams of links in $S^3$ and define $k$-moves on such diagrams, for any $k\in\mathbb N$. We study the equivalence classes of links in $S^3$ up to $k$-moves. For $k=2$, we show that any two knots are equivalent, whereas it is not true for links. We show that the Jones polynomial at a $k$-th primitive root of unity is unchanged by a $k$-move, when $k$ is odd. It is multiplied by $-1$, when $k$ is even. It follows that, for any $k\ge 5$, there are infinitely many classes of knots modulo $k$-moves. We use these results to study the Hopf crossing number. In particular, we show t…
This paper provides the complete table of prime knot projections with their mirror images, without redundancy, up to eight double points systematically thorough a finite procedure by flypes. In this paper, we show how to tabulate the knot projections up to eight double points by listing tangles with at most four double points by an approach with respect to rational tangles of J. H. Conway. In other words, for a given prime knot projection of an alternating knot, we show how to enumerate possible projections of the alternating knot. Also to tabulate knot projections up to ambient isotopy, we in…
We introduce two sequences of two-variable polynomials $\{ L^n_K (t, \ell)\}_{n=1}^{\infty}$ and $\{ F^n_K (t, \ell)\}_{n=1}^{\infty}$, expressed in terms of index value of a crossing and $n$-dwrithe value of a virtual knot $K$, where $t$ and $\ell$ are variables. Basing on the fact that $n$-dwrithe is a flat virtual knot invariant we prove that $L^n_K$ and $F^n_K$ are virtual knot invariants containing Kauffman affine index polynomial as a particular case. Using $L^n_K$ we give sufficient conditions when virtual knot does not admit cosmetic crossing change.
Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by V.~Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such an arcs disjoint from crossings. In the paper we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knot…
We use the Bar-Natan Zh-correspondence to identify the generalized Alexander polynomial of a virtual knot with the Alexander polynomial of a two component welded link. We show that the Zh-map is functorial under concordance, and also that Satoh's Tube map (from welded links to ribbon knotted tori in $S^4$) is functorial under concordance. In addition, we extend classical results of Chen, Milnor, and Hillman on the lower central series of link groups to links in thickened surfaces. Our main result is that the generalized Alexander polynomial vanishes on any knot in a thickened surface which is …
A sequence of $F$-polynomials $\{ F^n_K (t, \ell)\}_{n=1}^{\infty}$ of virtual knots $K$ was defined by Kaur, Prabhakar, and Vesnin in 2018. These polynomials have been expressed in terms of index value of crossing and $n$-writhe of $K$. By the construction, $F$-polynomials are generalizations of the Kauffman's Affine Index Polynomial, and are invariants of virtual knot $K$. We present values of $F$-polynomials of oriented virtual knots having at most four classical crossings in a diagram.
A Self-Linking Invariant of Virtual Knots
在本文中，我们引入了一个新的不变量的虚拟节点和链接，是不平凡的无穷多个虚拟，但平凡的经典节点和链接。首先用括号多项式的一个相对变量表示不变量，然后用它的指数从这个多项式中提取不变量，特别是对于结点的情况。括号多项式的这种类似物将被表示为{K}（带有花括号），并被称为二元括号多项式。不变量组合学的关键是用相关图的2-着色来解释状态和。对于虚纽结，新的不变量J（K）是对图的奇数交叉点的扭曲的限制（如果一个交叉点连接纽结高斯码中的奇数交叉点，则该交叉点是奇数。对于经典纽结，奇交叉的集合是空的。）对于K为虚纽结，J（K）非零意味着K非平凡、非经典且不等价于其平面镜像。本文还considers推广的双重着色的状态的二元括号的情况下，三个和更多的颜色。讨论了图的着色与四色定理的关系。
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.
Seifert circles, crossing number and the braid index of generalized knots and links
本文引入广义交叉和广义Reidemeister型移动，研究与交叉类型无关的链环性质，证明了广义纽结和链环总交叉数与辫指数间的不等式，并适用于虚拟奇链环。
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。
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
Classical and Virtual Pseudodiagram Theory and New Bounds on Unknotting Numbers and Genus
伪图是缺少一些交叉信息的结的图。本文对R.花木然后，我们将这一理论扩展到虚拟结的领域，结的推广。特别是，我们调查有多少交叉信息必须知道得出结论，一个图是一个图的unknot（平凡化数）。我们还考虑了需要多少信息来识别一个非平凡的结，一个经典的结，或一个非经典的结。然后，我们应用伪图理论开发新的上界的unknotting数，虚拟unknotting数，和亏格。
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.
Let $n$ be a positive integer. The aim of this paper is to study two local moves $V(n)$ and $V^{n}$ on welded links, which are generalizations of the crossing virtualization. We show that the $V(n)$-move is an unknotting operation on welded knots for any $n$, and give a classification of welded links up to $V(n)$-moves. On the other hand, we give a necessary condition for which two welded links are equivalent up to $V^{n}$-moves. This leads to show that the $V^{n}$-move is not an unknotting operation on welded knots except for $n=1$. We also discuss relations among $V^{n}$-moves, associated co…
In this paper, we define the parity virtual Alexander polynomial following the work of BDGGHN [1] and Kaestner and Kauffman [10]. The properties of this invariant are explored and some examples are computed. In particular, the invariant demonstrates that many virtual knots can not be unknotted by crossing change on only odd crossings.
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.
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。
Virtual shadow modules and their link invariants
我们引入一个代数Z[X，S]，它与一个虚birack X和X-shadow S的对（X，S）相关联。我们使用Z[X，S]上的模来定义虚拟birack阴影计数不变量的增强，将birack阴影模不变量扩展到虚拟情况。我们对扭曲的虚拟情况重复这个构造。作为应用，我们证明了新的不变量可以检测方向反转，并且不受纽结群、Arrow多项式和Miyazawa多项式的影响，扭曲版本不受扭曲Jones多项式的影响.
Two-variable Parity Polynomial for Virtual Knotoids
提出虚拟结环的二元奇偶多项式不变量，基于经典交叉点的奇偶性分类，能区分奇扭数和仿射指标多项式无法区分的虚拟结环对，并证明其为一阶Vassiliev不变量。
The theory of welded and extended welded knots is a generalization of classical knot theory. Welded (resp. extended welded) knot diagrams include virtual crossings (resp. virtual crossings and wen marks) and are equivalent under an extended set of Reidemeister-type moves. We present a new class of invariants for welded and extended welded knots and links using a multi-skein relation, following Z. Yang's approach for virtual knots. Using this skein-theoretic approach, we find sufficient conditions on the coefficients to obtain invariance under the extended Reidemeister moves appropriate to weld…
Parity and Exotic Combinatorial Formulae for Finite-Type Invariants of Virtual Knots
本文给出了虚纽结不变量的高斯图公式的例子，这些公式在经典纽结情况下没有类似的。这些组合公式包含有关如何将子图嵌入虚拟结图的附加信息。额外的信息来自第二作者最近发现的奇偶性概念。对于平坦虚纽结的奇偶性，新的组合公式是Kauffman有限型不变量。然而，许多组合公式具有奇异性质。证明了对于任意n，存在一个n阶的整数值虚拟化不变组合公式（即它在改变一个箭头的方向但保持符号的映射下是稳定的）.因此，它不是Goussarov-Polyak-Viro有限型的。此外，每一个齐次Polyak-Viro组合公式允许分解成一个“偶数”部分和一个“奇数”部分。对于高斯宇称，公式的任何部分都不是GPV有限型时，它是在经典的纽结集上的非常数。此外，还得到了11个新的非GPV有限型的二阶非平凡组合公式.
We generalize the Wriggle polynomial, first introduced by L. Folwaczny and L. Kauffman, to the case of virtual tangles. This generalization naturally arises when considering the self-crossings of the tangle. We prove that the generalization (and, by corollary, the original polynomial) are Vassiliev invariants of order one for virtual knots, and study some simple properties related to the connected sum of tangles.
The parity hierarchy and new invariants of knots in thickened surfaces
在本文中，我们构造了一个不变量的虚拟纽结加厚球与g手柄，这个不变量是一个Laurent多项式在2g+3个变量。为此，我们使用修改的Wirtinger介绍的纽结群和宇称的概念介绍了V. O. Manturov。第四节利用V. O. Manturov提出的奇偶性层次概念对不变式进行了改进（构造不变模）。也就是说，我们区分奇数交叉和两种类型的偶数交叉;后两种类型取决于是否偶数交叉仍然是偶数/奇数后，所有奇数交叉的图被删除。不变量的构造也适用于虚拟节点。
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