Louis H. Kauffman, Sofia Lambropoulou
This paper is a short introduction to and statement of the main theorems of our paper "Virtual Braids and the L-Move", JKTR, Vol. 15, No. 6 (2006), pp. 773-811. See also arxiv:Math.GT/0507035.
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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.
The virtual braid groups are generalizations of the classical braid groups. This paper gives an elementary proof that the classical braid group injects into the virtual braid group over the same number of strands.
Virtual knot theory has experienced a lot of nice features that did not appear in classical knot theory, e.g., parity and picture-valued invariants. In the present paper we use virtual knot theory effects to construct new representations of classical (pure) braids.
The aim of the present note is to show that the natural map from classical braids to virtual braids is an inclusion; this proof does not use any complete invariants of classical braids; it is based on the projection from virutal braids to classical braids (similar to the one given in \cite{Projection}); this projection is the idenitiy map on the set of virtual braids. The projection is well defined not only for the virtual braid group but also for the quotient group of the virtual (pure) braid group by the so-called virtualization move. The idea of this projection is closely related to the not…
For $n \geq 2$ we describe an $O(l^3n)$-time algorithm that determines if a length $l$ virtual braid word in the standard presentation of the virtual braid group ${\mathcal VB}_n$ represents the trivial virtual braid.
Congruence Subgroups of the Virtual Braid Group
通过扩展整系数Burau表示，将辫群的同余子群概念推广到虚拟辫群，证明其二级同余子群为纯虚拟辫群，恢复Arnol'd结果的虚拟类比。
The notion of a virtual knot introduced by L. Kauffman induces the notion of a virtual braid. It is closely related with a welded braid of R. Fenn, R. Rimanyi and C. Rourke. Alexander's and Markov's theorems for virtual knots and braids are proved. Similar results for welded knots and braids are also proved.
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.
This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this approach is to take pure virtual braids as primary. The generators of the pure virtual braid group are abstract solutions to the algebraic Yang-Baxter equation. This point of view illuminates representations of the virtual braid groups and pure virtual braid groups via solutions…
In this paper we prove a Markov Theorem for virtual braids and for some analogs of this structure. The virtual braid group is the natural companion in the category of virtual knots, just as the Artin braid group is the natural companion to classical knots and links. In this paper we follow the L--move methods to prove the Virtual Markov Theorem. One benefit of this approach is a fully local algebraic formulation of the Theorem.
We study the algebraic structures of the virtual singular braid monoid, $VSB_n$, and the virtual singular pure braid monoid, $VSP_n$. The monoid $VSB_n$ is the splittable extension of $VSP_n$ by the symmetric group $S_n$. We also construct a representation of $VSB_n$.
The L-move for classical braids extends naturally to trivalent braids. We follow the L-move approach to the Markov Theorem, to prove a one-move Markov-type theorem for trivalent braids. We also reformulate this L-Move Markov theorem and prove a more algebraic Markov-type theorem for trivalent braids. Along the way, we provide a proof of the Alexander's theorem analogue for spatial trivalent graphs and trivalent braids.
We prove Alexander- and Markov-type theorems for virtual spatial trivalent graphs and virtual trivalent braids. We provide two versions for the Markov-type theorem: one uses an algebraic approach similar to the case of classical braids and the other one is based on L-moves.
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。
We prove that under fairly general conditions an iterated exchange move gives infinitely many non-conjugate braids. As a consequence, every knot has infinitely many conjugacy classes of n-braid representations if and only if it has one admitting an exchange move.
Virtual Braids and Cluster Algebras
本文基于Hikami-Inoue表示，利用簇代数为虚拟辫群构造了新表示，证明“禁用关系”在其像中不成立，并扩展至平坦辫群与平坦虚拟辫群。
We show that if a link $L$ has a closed $n$-braid representative admitting non-degenerate exchange move, an exchange move that does not obviously preserve the conjugacy class, $L$ has infinitely many non-conjugate closed $n$-braid representatives.
Warping labeling for twisted knots and twisted virtual braids
本文引入扭曲纽结的翘曲度与“翘曲标号”，并将其推广到扭曲虚拟辫子，得到在除R2外所有R移动下不变的函数及基于$\mathbb{Z}_2$的不变量。
In the context of finite type invariants, Stanford introduced a family of equivalence relations on knots defined by the lower central series of the pure braid groups and characterized the finite type invariants in terms of the structure of the braid groups. It is known that this equivalence and Ohyama's equivalence defined by a local move are equivalent. On the other hand, in the virtual knot theory, the concept of Ohyama's equivalence was extended by Goussarov-Polyak-Viro, which called an $n$-equivalence. In this paper we extend Stanford's equivalence to virtual knots and virtual string links…
In this survey paper we present the $L$--moves between braids and how they can adapt and serve for establishing and proving braid equivalence theorems for various diagrammatic settings, such as for classical knots, for knots in knot complements, in c.c.o. 3--manifolds and in handlebodies, as well as for virtual knots, for flat virtuals, for welded knots and for singular knots. The $L$--moves are local and they provide a uniform ground for formulating and proving braid equivalence theorems for any diagrammatic setting where the notion of braid and diagrammatic isotopy is defined, the statements…
We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi--direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi--direct product of free groups. From these results we obtain a normal form of words in the virtual braid group. We introduce the concept of a universal braid group. This group contains the classical braid group and has as its quotient groups the singular braid group, virtual braid group, welded braid group, and classical braid group.
Two virtual link diagrams are homotopic if one may be transformed into the other by a sequence of virtual Reidemeister moves, classical Reidemeister moves, and self crossing changes. We recall the pure virtual braid group. We then describe the set of pure virtual braids that are homotopic to the identity braid.
Representations of flat virtual braids by automorphisms of free group
本文构造了平虚拟辫群 $FVB_n$ 在秩为 $2n$ 的自由群自同构下的表示，研究了其代数性质，建立了 $n=2$ 时的忠实性条件，并证明 $n \geq 3$ 时核含秩二自由群。
We define Dynnikov coordinates on virtual braid groups. We prove that they are faithful invariants of virtual 2-braids, and present evidence that they are also very powerful invariants for general virtual braids.
This article is dedicate to cabling on virtual braids. This construction gives a new generating set for the virtual pure braid group $VP_n$. Consequently we describe $VP_4$ as HNN-extension. As an application to classical braids, we find a new presentation of the Artin pure braid group $P_4$ in terms of the cabled generators.
The nontrivial kernel of Manturov-Nikonov map from classical braids to virtual braids
本文证明当k≥6时，Manturov-Nikonov映射不忠实，因Burau核是M-N核的子群。
Subgroup separability of surface braid groups and virtual braid groups
该工作考察曲面辫子群与虚拟（奇异）辫子群的子群可分性（LERF）及相关性质。
We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) or G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For virtual knots and graphs these invariants provide new graphical information that allows one to prove minimality theorems and to construct new invariants for free knots (unoriented and unlabeled Gauss codes taken up to abstract Reidemeister moves). A novel feature of this approach is that some knots are of sufficient complexity that they evaluate themselves in the…
Virtual braids are a combinatorial generalization of braids. We present abstract braids as equivalence classes of braid diagrams on a surface, joining two distinguished boundary components. They are identified up to isotopy, compatibility, stability and Reidemeister moves. We show that virtual braids are in a bijective correspondence with abstract braids. Finally we demonstrate that for any abstract braid, its representative of minimal genus is unique up to compatibility and Reidemeister moves. The genus of such a representative is thus an invariant for virtual braids. We also give a complete …
In the present paper we give a new method for converting virtual knots and links to virtual braids. Indeed the braiding method given in this paper is quite general, and applies to all the categories in which braiding can be accomplished. We give a unifying topological interpretation of virtuals and flats (virtual strings) and their isotopies via ribbon surfaces and abstract link diagrams. We also give reduced presentations for the virtual braid group, the flat virtual braid group, the welded braid group and several other categories of braids. The paper includes a discussion of the topological …
In this paper, we make use of the relations between the braid and mapping class groups of a compact, connected, non-orientable surface N without boundary and those of its orientable double covering S to study embeddings of these groups and their (virtual) cohomological dimensions. We first generalise results of Birman and Chillingworth and of Gonçalves and Guaschi to show that the mapping class group MCG(N ; k) of N relative to a k-point subset embeds in the mapping class group MCG(S; 2k) of S relative to a 2k-point subset. We then compute the cohomological dimension of the braid groups of all…
In this paper we discuss algebraic, combinatorial and topological properties of singular virtual braids. On the algebraic side we state the relations between classical and virtual singular objects, in addition we discuss a Birman-like conjecture for the virtual case. On the topological and combinatorial side, we prove that there is a bijection between singular abstract braids, horizontal Gauss diagrams and singular virtual braids, in particular using horizontal Gauss diagrams we obtain a presentation of the singular pure virtual braid monoid.
Let $UVB_n$ and $UVP_n$ be the unrestricted virtual braid group and the unrestricted virtual pure braid group on n strands respectively. We study the groups $UVB_n$ and $UVP_n$, and our main results are as follows: for $n\geq 5$, we give a complete description, up to conjugation, to all possible homomorphisms from $UVB_n$ to the symmetric group $S_n$. For $n\geq 3$, we characterise all possible images of $UVB_n$, under a group homomorphism, to any finite group $G$. For $n\geq 5$, we prove that $UVP_n$ is a characteristic subgroup of $UVB_n$. In addition, we determine the automorphism group of …
We introduce a recoupling theory for virtual braided trees. This recoupling theory can be utilized to incorporate swap gates into anyonic models of quantum computation.
We show a simple and easily implementable solution to the word problem for virtual braid groups.
We define virtual braid groups of type B and construct a morphism from such a group to the group of isomorphism classes of some invertible complexes of bimodules up to homotopy.
Virtual knots arise in the study of Gauss diagrams and Vassiliev invariants of usual knots. Virtual braids correspond naturally to virtual knots. We consider the group of virtual braids on n strings VB_n and its Burau representation, in particular we study their homological properties. We prove that the plus-construction of the classifying space of the virtual braid group on the infinite number of strings is an infinite loop space which is equivalent to a product of Q(S^0), S^1 and an infinite loop space Y. Connections with the K-functor of the integers are discussed.
On virtual singular braid groups
本文研究了虚拟奇异辫群的代数性质，引入了基于单词指数和的数值不变量，确定了到对称群的同态，并给出了核的半直积分解。
In this article, we give a necessary and sufficient condition for embedding a finite index subgroup of Artin's braid group into the mapping class group of a connected orientable surface.
Algebraic structures among virtual singular braids
该研究证明n股虚拟奇异辫子幺半群可嵌入群VSGₙ，给出其半直积结构与VSPGₙ的表示及子群分解，为虚拟奇异辫子的代数结构分析提供基础。
A 2-dimensional braid over an oriented surface-knot $F$ is presented by a graph called a chart on a surface diagram of $F$. We consider 2-dimensional braids obtained by an addition of 1-handles equipped with chart loops. We introduce moves of 1-handles with chart loops, called 1-handle moves, and we investigate how much we can simplify a 2-dimensional braid by using 1-handle moves. Further, we show that an addition of 1-handles with chart loops is an unbraiding operation.
In order to obtain a Markov theorem without stabilization, Birman and Menasco introduced the notion of exchange related braids. In this paper I study the way the Fiedler polynomial distinguishes conjugacy classes of some particular braided knots. I introduce the Kauffman bracket in the solid torus. Its Taylor expansion give finite type invariants similar to the Fiedler polynomial. I investigate how these invariants distinguish exchange related braids.
Orbits by the up-down action of braid diagrams
本文研究了虚拟或经典辫图在 ${\mathbb Z}$ 直积上的上-下作用，确定了所有元组的轨道及不可约辫图的轨道，并给出了辫图闭包允许上-下着色的条件。
In this article, we investigate various properties of the pure virtual braid group PV_3. From its canonical presentation, we obtain a free product decomposition of PV_3. As a consequence, we show that PV_3 is residually torsion free nilpotent, which implies that the set of finite type invariants in the sense of Goussarov-Polyak-Viro is complete for virtual pure braids with three strands. Moreover we prove that the presentation of PV_3 is aspherical. Finally we determine the cohomology ring and the associated graded Lie algebra of PV_3.
Virtual braid groups, virtual twin groups and crystallographic groups
本文证明虚拟辫子群与虚拟孪生群模去对应纯群换位子群后的商为晶体群，刻画其有限阶元素与共轭类，并实现 Bieberbach 群及无限虚拟循环群。
The monoid structure of singular twisted virtual braids
本文研究奇异扭曲虚拟辫子幺半群$STVB_n$的子幺半群结构，确定$STVP_n$为$STVB_n$到$S_n$满同态的核及其生成元与关系，构造类似满同态并推广表示，证明$STVB_n$可嵌入群。
Twisted braids
本文推广虚拟结理论，证明扭曲链环可表示为扭曲辫的闭包，并给出扭曲辫群和平坦扭曲辫群的简化表示，类比经典Alexander和Markov定理。
There is a well known injective homomorphism $ϕ:{\mathcal {B}}_n \rightarrow {\rm Aut}(F_n)$ from the classical braid group ${\mathcal {B}}_n$ into the automorphism group of the free group $F_n$, first described by Artin. This homomorphism induces an action of ${\mathcal {B}}_n$ on $F_n$ that can be recovered by considering the braid group as the mapping class group of $H_n$ (an upper half plane with $n$ punctures) acting naturally on the fundamental group of $H_n$. Kauffman introduced virtual links as an extension of the classical notion of a link in ${\mathbb {R}}^3$. As in the classical cas…
This paper explores the problem of unknotting closed braids and classical knots in mathematical knot theory. We apply evolutionary computation methods to learn sequences of moves that simplify knot diagrams, and show that this can be effective both when the evolution is carried out for individual knots and when a generic sequence of moves is evolved for a set of knots.
Virtual planar braid groups and permutations
本文完整刻画了虚拟孪生群到对称群的同态，由此确定其自同构群结构，并证明孪生群嵌入虚拟孪生群，类比辫群结果。
We investigate the resonance varieties, lower central series ranks, and Chen ranks of the pure virtual braid groups and their upper-triangular subgroups. As an application, we give a complete answer to the 1-formality question for this class of groups. In the process, we explore various connections between the Alexander-type invariants of a finitely generated group and several of the graded Lie algebras associated to it, and discuss possible extensions of the resonance-Chen ranks formula in this context.
Universal virtual braid groups
本文引入了通用虚拟编织群UV_n(c)，提供了一种统一的代数框架，用于虚拟编织结构，证明其具有残余有限性、线性、解字和共轭问题等性质，并分类了子群分离性和Howson性质。
We study b-arc foliation change and exchange move of open book foliations which generalize the corresponding operations in braid foliation theory. We also define a bypass move as an analogue of Honda's bypass attachment operation. As applications, we study how open book foliations change under a stabilization of the open book. We also generalize Birman-Menasco's split/composite braid theorem: Closed braid representatives of a split (resp. composite) link in a certain open book can be converted to a split (resp. composite) closed braid by applying exchange moves finitely many times.
Plat closures of spherical braids in $\mathbb{R}P^3$
本文定义了球面辫子的平闭包以构造$\mathbb{R}P^3$中的链环，证明了所有链环均可如此实现，并给出分支数公式及一组类马尔可夫移动，其完备性待研究。
We classify 3-braids up to (2,2)-move equivalence and in particular we show how to adjust the Harikae-Nakanishi-Uchida conjecture so it holds for closed 3-braids. As important steps to classify 3-braids up to (2,2)-move equivalence we prove the conjecture for 2-algebraic links and classify (2,2)-equivalence classes for links up to nine crossings. We analyze the behavior of Kei (involutive quandle) associated to a link under (2,2)-moves. We construct Burnside Kei, Q(m,n), and ask the question, motivated by classical Burnside question: for which values of m and n, is Q(m,n) finite?
A Markov theorem for plat closure of surface braids in Dunwoody and periodic Takahashi manifolds
本文针对Dunwoody与周期Takahashi流形中的链环，利用Heegaard分解和辫群plat闭包表示，提出算法确定并显式计算plat滑动等价移动，从而建立相应的Markov定理。
Khovanov and Rozansky's categorification of the HOMFLY-PT polynomial is invariant under braidlike isotopies for any link diagram and Markov moves for braid closures. To define HOMFLY-PT homology, they required a link to be presented as a braid closure, because they did not prove invariance under the other oriented Reidemeister moves. In this text we prove that the Reidemeister IIb move fails in HOMFLY-PT homology by using virtual crossing filtrations of the author and Rozansky. The decategorification of HOMFLY-PT homology for general link diagrams gives a deformed version of the HOMFLY-PT poly…
Virtual Thompson's group
本文受虚拟辫群推广自辫群且可生成虚拟链环、Thompson群$F$可构造有向链环的研究启发，定义了Thompson群$F$的虚拟版本并证明其可构造任意虚拟链环。
To a closed braid in a solid torus we associate a trace graph in a thickened torus in such a way that closed braids are isotopic if and only if their trace graphs can be related by trihedral and tetraherdal moves. For closed braids with a fixed number of strands, we recognize trace graphs up to isotopy and trihedral moves in polynomial time with respect to the braid length.
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