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.
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48 results for “braid groups”
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…
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.
The paper finds minimal generating sets and abelianizes the quasitoric braid group.
problem Understanding the structure of quasitoric braids and their subgroup properties.
method Provided two minimal generating sets and determined the abelianization.
result Minimal generating sets and abelianization of the quasitoric braid group were determined.
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.
Paper explores relations between braid groups and their quotients.
problem Understanding relations between braid groups and their quotients.
method Recalling and introducing elements of congruence braid groups, establishing isomorphisms between crystallographic and congruence braid groups.
result Established isomorphisms between crystallographic braid groups and quotients of congruence braid groups.
On homotopy braidsmath.GR
Paper proves homotopy braid group properties over integers and three strands.
problem Understanding homotopy braid group properties.
method Proved linearity over integers and torsion freeness for three strands.
result Homotopy braid group on three strands is torsion free.
On braids and groups math.GT
In [V.O. Manturov, Non-reidemeister knot theory and its applications in dynamical systems, geometry, and topology, arxiv:1501.05208] the first named author gave the definition of -free braid groups . Here we establish connections between free braid groups, classical braid groups and free groups: we describe e…
Polynomials with distinct critical values have braid monodromy groups equal to braid groups.
problem Understanding the structure of braid monodromy groups of polynomials.
method Analyzing the critical values of polynomials to determine their braid monodromy groups.
result The braid monodromy group of a polynomial equals the braid group if the polynomial has distinct critical values.
Orbifold braid groupsmath.GR
The paper studies orbifold braid groups and their properties.
problem Understanding orbifold braid groups and their subgroups.
method Detailed study of orbifold braid groups, proving injectivity and triviality of centers.
result Most orbifold braid groups have trivial centers.
We give formulae for the first homology of the -braid group and the pure 2-braid group over a finite graph in terms of graph theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the -braid group over the graph is torsion-free and the conjectures about the first h…
We ask if any finite type generalized braid group is a subgroup of some classical Artin braid group. We define a natural map from a given finite type generalized braid group to a classical braid group and ask if this map is an injective homomorphism. We prove that this map is a homomorphism for the braid groups of type…
New representation for braid groups and surface braid groups, extending Lawrence-Krammer-Bigelow.
problem Constructing representations for braid groups and surface braid groups.
method Pro-nilpotent tower of representations, starting with the original LKB representation.
result 3-variable enrichment of the Lawrence-Krammer-Bigelow representation.
New family of braided Thompson groups introduced using recursive braids.
problem Dehornoy-Brin braided Thompson group generalization.
method Using recursive braids and strand diagrams to define new groups.
result New groups are finitely generated if is finitely generated.
Study on braid groups' congruence subgroups and their crystallographic quotients.
problem Understanding congruence subgroups and crystallographic quotients of braid groups.
method Investigation of lower central series of congruence braid groups related to .
result Quotients of congruence braid groups are almost crystallographic.
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…
In this paper we introduce distinct approaches to loop braid groups, a generalisation of braid groups, and unify all the definitions that have appeared so far in literature, with a complete proof of the equivalence of these definitions. These groups have in fact been an object of interest in different domains of mathem…
Magic braids can be made with leatherworking technique.
problem Determine which braids can be made with leatherworking technique.
method Explore their relation to several braid groups.
result Magic braids can be made with leatherworking technique.
Drinfel'd used associators to construct families of universal representations of braid groups. We consider semi-associators (i.e., we drop the pentagonal axiom and impose a normalization in degree one). We show that the process may be reversed, to obtain semi-associators from universal representations of 3-braids. We v…
Homomorphism from braid groups to Steinberg groups defined.
problem Understanding the relationship between braid groups and Steinberg groups.
method Construction of a homomorphism from braid groups to Steinberg groups.
result Description of the image and kernel of the homomorphism.
Homotopy braid groups are shown to be torsion-free.
problem The existence of non-abelian torsion-free quotients of the braid group.
method Diagrammatic theory of welded braids and Artin representation.
result Homotopy braid groups are torsion-free.
Study the relationship between orbit braid group and equivariant mapping class group on surfaces.
problem Understanding the relationship between mapping class groups and braid groups with group actions.
method Using the fibration and exact sequence.
result The conclusion is closely connected with the braid group of the quotient space.
New braid group representations into mapping class groups are derived from disk coverings.
problem Tackling the representation of braid groups into mapping class groups.
method Explicit expressions of braid group representations into automorphism groups of free groups using covering groupoids.
result Each generator of braid group inside mapping class group is a product of Dehn twists.
Completes results on complex braid group parabolic subgroups.
problem Proves properties of complex braid group parabolic subgroups.
method Uses Garside groupoid structure of B(G31) to extend results.
result Proves main theorems for B(G31) parabolic subgroups.
We give a complete classification of homomorphisms from the braid group on strands to the braid group on strands when is at least 5. We also classify endomorphisms of the braid group on 4 strands, as well as homomorphisms from the commutator subgroup of the braid group on strands to the braid group on …
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.
Study of wild mapping class groups and their cabled braids.
problem Understanding the structure of wild mapping class groups and their cabled versions.
method Define and study generalizations of pure -braid groups, establish product decompositions, and introduce fission trees.
result Obtain cabled versions of braid groups, related to braid operads.
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 , the welded pure braid groups , the virtual pure braid groups , as w…
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.
We give presentations of braid groups and pure braid groups on surfaces.
Two crossing homomorphisms on braid groups are shown to be equivalent.
problem Comparing two definitions of crossing homomorphisms on braid groups.
method Diagrammatic and algebraic definitions of crossing homomorphisms compared and computed for simple braids.
result Diagrammatic and algebraic crossing homomorphisms are equivalent.
Positive braids linked to knot invariants and geometric monodromy groups.
problem Understanding knot invariants and geometric monodromy groups for positive braids.
method Associate braid monodromy groups to positive braids, identify these groups with framed mapping class groups for knots, and use these to determine knot invariants.
result Geometric monodromy groups of irreducible singularities are determined by genus and Arf invariant of associated knots.
On Marked Braid Groupsmath.GT
In the present paper, we introduce -braids and, more generally, -braids for an arbitrary group . They form a natural group-theoretic counterpart of -knots, see \cite{reidmoves}. The underlying idea, used in the construction of these objects --- decoration of crossings with some additional informa…
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…
The study examines when braid groups of manifolds are Kähler.
problem When are braid groups of manifolds Kähler?
method The approach uses previous results on Kähler groups and homological properties of braid groups.
result With some exceptions, the pure braid group of a Riemann surface with at least 2 strands is never Kähler.
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.
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 .
When the lower central series stops: a comprehensive study for braid groups and their relativesmath.GT
Study on when the lower central series stops for various groups, including braid groups.
problem Understanding when the lower central series stops for different groups.
method Various techniques applied to braid groups and related groups.
result Complete computation of the lower central series for most groups studied.
Study of decorated surfaces with vortices and their group structures.
problem Understanding group structures of decorated surfaces with vortices.
method Proved isomorphism between cluster braid group, braid twist group, and fundamental group of moduli space.
result Finite presentations of isomorphic groups were given.
Article provides conditions for embedding braid groups into mapping class groups.
problem Embedding braid groups into mapping class groups.
method Necessary and sufficient condition for embedding finite index subgroups.
result Conditions for embedding braid groups into mapping class groups.
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.
Braid group proof shows 7-strand is CAT(0).
problem Proving the 7-strand braid group is CAT(0).
method Elaborating on Haettel, Kielak, and Schwer's argument.
result The 7-strand braid group is proven to be CAT(0).
New link groups are derived from torus necklaces, connecting braid groups to reflection groups.
problem Understanding the relationship between braid groups and reflection groups.
method Constructing torus necklaces and linking them to braid groups of -reflection groups.
result Link groups of torus necklaces are precisely braid groups of -reflection groups, with meridians as braid reflections.
We give a necessary and sufficient condition for a graph to have a right-angled Artin group as its braid group for braid index . In order to have the necessity part, graphs are organized into small classes so that one of homological or cohomological characteristics of right-angled Artin groups can be applied. Fi…
For every group genetic code with finite number of generating and at most with one defining relation we introduce the braid group of this genetic code. This construction includes the braid group of Euclidean plane, the braid groups of closed orientable surfaces, B type groups of Artin-Brieskorn, and allow us to study a…
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.
Generators found for a specific subgroup of braid groups.
problem Prove generators for the level congruence subgroups of braid groups.
method Proved using th powers of half-twists and the braid Torelli group.
result Solved Margalit's problem, generalizing previous work.
Twisted braidsmath.GT
Paper proves any twisted link can be described as a unique twisted braid.
problem Understanding twisted links and braids.
method Proved using Alexander and Markov Theorems for classical braids and links.
result Any twisted link can be described as the closure of a unique twisted braid.