arXiv research

Classic braids embed in virtual braids.

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…

Study virtual braid groups, proving a key subgroup result.

The paper examines subgroup separability for surface and virtual braid groups.

The paper characterizes crystallographic groups derived from virtual braid and twin groups.

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.

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…

Study on virtual singular braid groups with algebraic properties and homomorphisms.

New virtual version of Thompson's group created to handle virtual knots.

The paper constructs representations of flat virtual braids by free group automorphisms.

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 strin…

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.

Virtual singular braids embed in a group with normal form.

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…

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 gene…

The paper extends knot theory to twisted virtual braids and links.

Paper constructs representations for virtual braids and flat braids.

The paper studies submonoids of singular twisted virtual braids and their properties.

Virtual twin groups map to symmetric groups, revealing automorphism structure.

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.

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.

Unified algebraic framework for virtual braid structures with strong structural consequences.

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 …

Classifies orbits of Hurwitz actions on dihedral quandles.

The paper solves the conjugacy problem in a specific braid group quotient and finds infinite virtually cyclic subgroups.

The article calculates the minimal model dimensions for classifying spaces of surface braid groups.

We show a simple and easily implementable solution to the word problem for virtual braid groups.

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-constructi…

Study of unrestricted virtual braid groups and their properties.

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.

Paper proves any twisted link can be described as a unique twisted braid.

New groups from strand diagrams show polycyclic subgroups are virtually abelian and undistorted.

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 paper defines and analyzes configuration Lie groupoids and orbifold braid groups.

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 def…

The paper defines new representations and groups related to virtual links.

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.

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 consid…

Article provides conditions for embedding braid groups into mapping class groups.

Study of commutator subgroups and crystallographic quotients of virtual groups.

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 Goussar…

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 g…

We classify the (finite and infinite) virtually cyclic subgroups of the pure braid groups $P_{n}(RP^2)$ of the projective plane. The maximal finite subgroups of $P_{n}(RP^2)$ are isomorphic to the quaternion group of order 8 if $n=3$, and to $\Z_{4}$ if $n\geq 4$. Further, for all $n\geq 3$, up to isomorphism, the foll…

In this mostly survey paper, we investigate the resonance varieties, the lower central series ranks, and the Chen ranks, as well as the residual and formality properties of several families of braid-like groups: the pure braid groups $P_n$, the welded pure braid groups $wP_n$, the virtual pure braid groups $vP_n$, as w…

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 …

New braid representations using virtual knot theory.

Study on when the lower central series stops for various groups, including braid groups.

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…