New virtual braid group structure and presentation found.
problem Understanding the structure of virtual pure braid groups.
method Cabling construction and HNN-extension approach.
result New presentation of and groups.
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48 results for “pure virtual braid group”
Study virtual braid groups, proving a key subgroup result.
problem Understanding congruence subgroups in virtual braid groups.
method Using an extension of the integral Burau representation.
result Proved level 2 congruence subgroup of virtual braid group is pure virtual braid group.
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…
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…
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.
The paper explores various braid-like groups and their properties.
problem Investigating the pure braid groups and their relatives.
method Examining resonance varieties, lower central series ranks, Chen ranks, residual and formality properties.
result Discussed natural homomorphisms and methods to distinguish braid-like groups.
The paper characterizes crystallographic groups derived from virtual braid and twin groups.
problem Characterizing crystallographic groups from virtual braid and twin groups.
method Analyzing quotients of virtual braid and twin groups by their commutator subgroups.
result The quotients of virtual braid and twin groups by their commutator subgroups are crystallographic groups.
Study on pure virtual braids, their formality, and related ranks.
problem Formality and ranks of pure virtual braid groups.
method Investigation of resonance varieties, lower central series ranks, Chen ranks, and Alexander-type invariants.
result Complete answer to 1-formality question for pure virtual braid groups.
Classifies orbits of Hurwitz actions on dihedral quandles.
problem Classifying orbits of Hurwitz actions on dihedral quandles.
method Introduced three computable invariants to classify orbits.
result Complete classification of orbits under Hurwitz action.
Paper extends Stanford's equivalence to virtual knots and proves equivalence to Goussarov-Polyak-Viro's n-equivalence.
problem Characterizing finite type invariants in virtual knot theory.
method Using the lower central series of the pure virtual braid group to define an L_n-equivalence.
result L_n-equivalence on virtual string links is equivalent to Goussarov-Polyak-Viro's n-equivalence.
Commutator subgroups and crystallographic quotients of virtual extensions of symmetric groupsmath.GR
Study of commutator subgroups and crystallographic quotients of virtual groups.
problem Investigate commutator subgroups and crystallographic quotients of virtual groups.
method Derived explicit finite presentations and proved crystallographic properties.
result Explicit finite presentations of commutator subgroups and crystallographic quotients.
On the pure virtual braid group math.GT
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…
The article calculates the minimal model dimensions for classifying spaces of surface braid groups.
problem Classifying spaces for families of virtually abelian subgroups of surface braid groups.
method Analyzes the pure and full braid groups of surfaces with at least one boundary component or one puncture, proving analogous results for amenable subgroups.
result Minimal dimensions of models for classifying spaces are equal to the virtual cohomological dimension plus the rank of virtually abelian subgroups.
Virtual singular braids embed in a group with normal form.
problem Embedding virtual singular braids into algebraic structures.
method Presented a semi-direct product structure and provided a normal form.
result Virtual singular braid group is a semi-direct product of and .
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…
Study on algebraic structures of virtual singular braid monoid and its splittable extension.
problem Algebraic structures of virtual singular braid monoid and its splittable extension.
method Exploration of algebraic structures, construction of representation.
result Monoid is the splittable extension of by the symmetric group .
The paper defines and analyzes configuration Lie groupoids and orbifold braid groups.
problem Understanding the structure and properties of orbifold braid groups.
method Definitions and proofs of fibration theorems, short exact sequences, and poly-virtually free structures.
result The pure orbifold braid groups have poly-virtually free structure, generalizing classical braid groups.
The paper studies submonoids of singular twisted virtual braids and their properties.
problem Examining submonoids within singular twisted virtual braid monoid.
method Established epimorphisms, identified generators and defining relations, constructed other epimorphisms.
result Identified generators and defining relations for and other submonoids.
The paper solves the conjugacy problem in a specific braid group quotient and finds infinite virtually cyclic subgroups.
problem Solving the conjugacy problem in the Artin braid group quotient .
method Using systems of equations over the integers derived from the action of on the abelianization of . Also, explicitly realizing infinite virtually cyclic subgroups.
result Explicitly realized infinite virtually cyclic subgroups in .
Classification of the virtually cyclic subgroups of the pure braid groups of the projective planemath.GR
We classify the (finite and infinite) virtually cyclic subgroups of the pure braid groups of the projective plane. The maximal finite subgroups of are isomorphic to the quaternion group of order 8 if , and to if . Further, for all , up to isomorphism, the foll…
Unified algebraic framework for virtual braid structures with strong structural consequences.
problem Unified algebraic framework for virtual braid structures with various types of crossings.
method Introducing the universal virtual braid group and proving its properties.
result Strong structural consequences including residual finiteness, linearity, and solvability of conjugacy problems.
Study of unrestricted virtual braid groups and their properties.
problem Characterize and describe homomorphisms of unrestricted virtual braid groups.
method Analyzing homomorphisms to symmetric groups and finite groups, characterizing images, proving characteristic subgroups, determining automorphism groups, and studying residual properties.
result Complete description of homomorphisms from to for .
Paper studies pure virtual twin groups and their automorphisms.
problem Understanding the structure and automorphisms of pure virtual twin groups.
method Analyzes stable isotopy classes of immersed circles on surfaces, extending classical knot theory.
result Proves is an irreducible right-angled Artin group with trivial center and gives its precise presentation.
New braid representations using virtual knot theory.
problem No classical features in virtual knot theory.
method Construct new braid representations using virtual knot theory.
result New representations of classical braids.
The paper proves a conjecture about braid groups of surfaces and their virtual cohomological dimensions.
problem Investigating the braid groups of surfaces and their homomorphisms.
method Analyzing the inclusion of configuration spaces and induced homomorphisms.
result Proves the conjecture for S^2 and RP^2, determining their virtual cohomological dimensions.
Paper explores algebraic, topological, and combinatorial properties of singular virtual braids.
problem Understanding singular virtual braids and their properties.
method Algebraic relations, topological and combinatorial bijections, presentations.
result A bijection between singular abstract braids and singular virtual braids, leading to a presentation of the singular pure virtual braid monoid.
New groups and subgroups classified with representations and properties.
problem Classifying representations of new groups and subgroups.
method Introduced new groups and subgroups, classified representations into GL_n(C), investigated properties.
result Classified homogeneous 2-local representations of M_kVT_n into eight distinct types.
Classic braids embed in virtual braids.
problem Embedding classic braids in virtual braids.
method Elementary proof using generalizations of braid groups.
result Classical braid group injects into virtual braid group.
Paper defines doodles on closed surfaces, unifying classical and virtual theories.
problem Classifying doodles on closed surfaces, especially non-orientable ones.
method Introducing twisted virtual doodles, defining twin groups, and proving Alexander- and Markov-type theorems.
result Unified theory of doodles, showing trivial center and residually finite properties.
Study on groups of unrestricted virtual braids and their properties.
problem Understanding the structure and properties of virtual braids and links.
method Exploration of unrestricted virtual braids, fused links, flat virtual braids, and virtual Gauss braids.
result Definition and study of groups of flat virtual braids and virtual Gauss braids, including their linearity.
If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this paper we give a sufficient criterion (called the …
The paper examines subgroup separability for surface and virtual braid groups.
problem Subgroup separability of surface and virtual braid groups.
method Study of subgroup separability (LERF) properties.
result Properties of subgroup separability for surface and virtual braid groups are explored.
Virtual knot theory, introduced by Kauffman, is a generalization of classical knot theory of interest because its finite-type invariant theory is potentially a topological interpretation of Etingof and Kazhdan's theory of quantization of Lie bi-algebras. Classical knots inject into virtual knots, and flat virtual knots…
Study on virtual singular braid groups with algebraic properties and homomorphisms.
problem Algebraic properties and homomorphisms of virtual singular braid groups.
method Numerical invariants, homomorphisms, semi-direct product decompositions, presentations, and quotients.
result Determined all group homomorphisms from to and obtained corresponding semi-direct product decompositions.
The paper constructs representations of flat virtual braids by free group automorphisms.
problem Representing flat virtual braids by automorphisms of free groups.
method Construction of representations of flat virtual braid groups by automorphisms of free groups of rank .
result Established conditions of faithfulness and properties of the kernel for .
Virtual Thompson's groupmath.GT
New virtual version of Thompson's group created to handle virtual knots.
problem Creating a mathematical framework for virtual knots.
method Defining a virtual version of Thompson's group and proving its properties.
result Any virtual link can be constructed from an element of the new virtual Thompson's group.
Virtual Braidsmath.GT
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.
Polynomial-time algorithm for virtual braid triviality.
problem Determining if virtual braid words represent the trivial braid.
method An -time algorithm for .
result Determines triviality of virtual braid words efficiently.
The paper extends knot theory to twisted virtual braids and links.
problem Generalizing knot theory to include twists.
method Introduced twisted virtual braids and proved theorems for twisted links.
result The Alexander and Markov theorems were extended to twisted links.
Paper constructs representations for virtual braids and flat braids.
problem Calculating hyperbolic volumes of knot complements.
method Cluster algebra approach for virtual braid group.
result Forbidden relations do not hold in virtual braid group representation.
Surface framed braidsmath.GT
In this paper we introduce the framed pure braid group on strands of an oriented surface, a topological generalisation of the pure braid group . We give different equivalents definitions for framed pure braid groups and we study exact sequences relating these groups with other generalisations of , usually…
Virtual twin groups map to symmetric groups, revealing automorphism structure.
problem Understanding homomorphisms between virtual twin groups and symmetric groups.
method Using irreducible right-angled Coxeter groups and right-angled Artin groups.
result A complete description of homomorphisms between virtual twin groups and symmetric groups, including the structure of the automorphism group of .
We find finite presentations for the automorphism group of the Artin pure braid group and the automorphism group of the pure braid group associated to the full monomial group.
Study automorphisms of pure braid groups on sphere homotopy groups.
problem Understanding automorphisms' effect on sphere homotopy groups.
method Examined Delta-group structure, proved invariance of cycle and boundary groups, computed action for few strands.
result Induced action of all automorphisms of pure braid groups on sphere homotopy groups.
New proof shows homomorphisms from pure braid groups to hyperbolic groups have cyclic images or factor through forgetful maps.
problem Characterizing homomorphisms from pure braid groups to hyperbolic groups.
method Extending and proving a new rigidity result for pure braid groups, focusing on homomorphisms to hyperbolic groups.
result Homomorphisms from pure braid groups to hyperbolic groups either have cyclic images or factor through a forgetful map.
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 …
Corrects earlier work on surface orbifold pure braid groups.
problem Proving a four-term exact sequence for surface orbifold pure braid groups.
method Analyzes surface orbifold pure braid groups for all genus ≥ 1, 2D orientable orbifolds with cone points.
result Proves a four-term exact sequence for surface orbifold pure braid groups.