Greg Kuperberg
Several authors have recently studied virtual knots and links because they admit invariants arising from R-matrices. We prove that every virtual link is uniquely represented by a link L in S X I, a thickened, compact, oriented surface S, such that the link complement (S X I) - L has no essential vertical cylinder.
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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.
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.
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…
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.
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。
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.
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。
The virtual genus of a virtual satellite link is equal to that of its companion.
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
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 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…
We prove that there is an algorithm to decide whehter two virtual links are equivalent or not
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.
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。
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.
The notion of chckerboard colorability for virtual links and abstract links is introduced. We study the Jones polynomials of virtual links and abstruct links. It is proved that a certain property of the Jones polynomials of classical links is valid for virtual links which admit checkerboard colorings.
We introduce two kinds of structures, called v-structures and t-structures, on biquandles. These structures are used for colorings of diagrams of virtual links and twisted links such that the numbers of colorings are invariants. Given a biquandle or a quandle, we give a method of constructing a biquandle with these structures. Using the numbers of colorings, we show that Bourgoin's twofoil and non-orientable virtual $m$-foils do not represent virtual links.
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。
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。
In the present paper we bring together minimality conditions proposed in previous two papers and present some new minimality conditions for classical and virtual knots and links.
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…
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…
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 show that all two-bridge knot and link complements are virtually fibered. We also show that spherical Montesinos knot and link complements are virtually fibered. This is accomplished by showing that such knot complements are finitely covered by great circle link complements.
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.
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…
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。
For an oriented virtual link, L.H. Kauffman defined the f-polynomial (Jones polynomial). The supporting genus of a virtual link diagram is the minimal genus of a surface in which the diagram can be embedded. In this paper we show that the span of the f-polynomial of an alternating virtual link L is determined by the number of crossings of any alternating diagram of L and the supporting genus of the diagram. It is a generalization of Kauffman-Murasugi-Thistlethwaite's theorem. We also prove a similar result for a virtual link diagram that is obtained from an alternating virtual link diagram by …
We construct explicitly the Khovanov homology theory for virtual links with arbitrary coefficients by using the twisted coefficients method. This method also works for constructing Khovanov homology for ``non-oriented virtual knots'' in the sense of Viro, in particular, for knots in ${\bf R}P^{3}$.
In the present paper, we construct the Khovanov homology theory for virtual links. Besides the direct approach with Z_{2} coefficients we also describe the Khovanov homology for framed links and the Khovanov homology using ``double cover''. The latter two approaches are based on the notion of ``atom''.
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.
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.
We address the question of detecting minimal virtual diagrams with respect to the number of virtual crossings. This problem is closely connected to the problem of detecting the minimal number of additional intersection points for a generic immersion of a singular link in $R^{2}$. We tackle this problem by the so-called $ξ$-polynomial whose leading (lowest) degree naturally estimates the virtual crossing number. Several sufficient conditions for minimality together with infinite series of new examples are given. We also state several open questions about $M$-diagrams, which are minimal accordin…
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…
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 …
The Jones polynomial and Khovanov homology of a classical link are invariants that depend upon an initial choice of orientation for the link. In this paper, we give a Khovanov homology theory for unoriented virtual links. The graded Euler characteristic of this homology is proportional to a similarly-defined unoriented Jones polynomial for virtual links, which is a new invariant in the category of non-classical virtual links. The unoriented Jones polynomial continues to satisfy an important property of the usual one: for classical or even virtual links, the unoriented Jones polynomial evaluate…
In the present paper, we construct the generalized Kuperberg bracket for two-component links with one component fibred. We consider a new geometrical complexity for such links and establish minimality of diagrams in a strong sense.
This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. The paper sets up the background virtual knot theory, defines rotational virtual knot theory, studies an extension of the bracket polynomial and the Manturov parity bracket for rotationals. We give examples of links that are not detected by the bracket polynomial but are detected by the extended parity bracket. Then the general frameworks for oriented and unoriented quantum …
Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer generators.
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…
This paper contains general formulae for the reduced relative Tutte, Kauffman bracket and Jones polynomials of families of virtual knots and links given in Conway notation and discussion of a counterexample to the Z-move conjecture of Fenn, Kauffman and Manturov.
We extend the Kamada-Miyazawa polynomial to virtual singular links, which is valued in $\mathbb{Z}[A^2, A^{-2}, h]$. The decomposition of the resulting polynomial into two components, one in $\mathbb{Z}[A^2, A^{-2}]$ and the other in $\mathbb{Z}[A^2, A^{-2}]h$ yields the decomposition of the Kauffman-Jones polynomial of virtual singular links into two components, one in $\mathbb{Z}[A^2, A^{-2}]$ and the other in $\mathbb{Z}[A^2, A^{-2}]A^2$, where both components are invariants for virtual singular links.
In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author. The relation between this new polynomial invariant and the affine index polynomial is discussed. In the second part we introduce a polynomial invariant for long flat virtual knots. In the third part we define a polynomial invariant for 2-component virtual links. This polynomial invariant can be regarded …
Algebraic invariants of multi-virtual links
本文研究多虚拟链环的代数不变量，确定生成移动集，引入算子quandle并构造相关不变量，对小型多虚拟纽结分类，还构造了无限族不等价单经典交叉多虚拟纽结。
By adding or removing appropriate structures to Gauss diagram, one can create useful objects related to virtual links. In this paper few objects of this kind are studied: twisted virtual links generalizing virtual links; signed chord diagrams staying halfway between twisted virtual links and Kauffman bracket / Khovanov homology; alternatable virtual links intermediate between virtual and classical links. The most profound role here belongs to a structure that we dare to call orientation of chord diagram. Khovanov homology is generalized to oriented signed chord diagrams and links in oriented t…
Singular twisted links and singular twisted virtual braids
本文引入奇异扭曲纽结理论，证明了奇异扭曲链的Alexander定理与Markov定理，并定义了奇异扭曲虚拟辫的幺半群结构及其约化表示。
Twisted virtual braids and twisted links
本文研究扭曲虚拟辫子，证明扭曲链环的 Alexander 型与 Markov 型定理，并给出扭曲虚拟辫子群的群表示及约化表示。
Geometric interpretations of some virtual knot invariants are given in terms of invariants of links in $\mathbb{S}^3$. Alexander polynomials of almost classical knots are shown to be specializations of the multi-variable Alexander polynomial of certain two-component boundary links of the form $J \sqcup K$ with $J$ a fibered knot. The index of a crossing, a common ingredient in the construction of virtual knot invariants, is related to the Milnor triple linking number of certain three-component links $J \sqcup K_1 \sqcup K_2$ with $J$ a connected sum of trefoils or figure-eights. Our main techn…
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