arXiv research

Introduces a reduction system for Artin-Tits groups, improving algorithms and proving periodicity results.

An approach by J.Wu describes homotopy groups $π_{n}(S^2)$ of the standard 2-sphere as isotopy classes of spherical $n+1$--strand Brunnian braids is investigated in the case $n=3$ for applications.

Classifies braid elements in elliptic fibrations up to conjugacy.

We give a finite presentation for the braid twist group of a decorated surface. If the decorated surface arises from a triangulated marked surface without punctures, we obtain a finite presentation for the spherical twist group of the associated 3-Calabi-Yau triangulated category. The motivation/application is that the…

We study the rational permutation braids, that is the elements of an Artin-Tits group of spherical type which can be written $x^{-1} y$ where $x$ and $y$ are prefixes of the Garside element of the braid monoid. We give a geometric characterization of these braids in type $A_n$ and $B_n$ and then show that in spherical …

We show that, in an Artin-Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to all Artin-Tits groups of spherical type a result that was previously known for bra…

The paper defines plat closures for spherical braids and shows links in $\mathbb{R}P^3$ can be realized this way.

The cycling operation is a special kind of conjugation that can be applied to elements in Artin's braid groups, in order to reduce their length. It is a key ingredient of the usual solutions to the conjugacy problem in braid groups. In their seminal paper on braid-cryptography, Ko, Lee et al. proposed the {\it cycling …

The goal of this mostly expository paper is to present several candidates for hyperbolic structures on irreducible Artin-Tits groups of spherical type and to elucidate some relations between them. Most constructions are algebraic analogues of previously known hyperbolic structures on Artin braid groups coming from natu…

We consider Thompson's groups from the perspective of mapping class groups of surfaces of infinite type. This point of view leads us to the braided Thompson groups, which are extensions of Thompson's groups by infinite (spherical) braid groups. We will outline the main features of these groups and some applications to …

We define the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams A_n, B_n=C_n and D_n and the affine diagrams tilde{A}_n, tilde{B}_n, tilde{C}_n and tilde{D}_n as subgroups of the braid gr…

We investigate the rigidity and asymptotic properties of quantum SU(2) representations of mapping class groups. In the spherical braid group case the trivial representation is not isolated in the family of quantum SU(2) representations. In particular, they may be used to give an explicit check that spherical braid grou…

Uniform proof of property R_infinity for specific Artin-Tits groups.

Study on braid monodromy of Lefschetz fibrations, proving infinite index subgroup.

We are interested in the 3-Calabi-Yau categories $\mathcal{D}$ arising from quivers with potential associated to a triangulated marked surface $\mathbf{S}$ (without punctures). We prove that the spherical twist group ST of $\mathcal{D}$ is isomorphic to a subgroup (generated by braid twists) of the mapping class group …

The paper defines parabolic subgroups for complex braid groups and proves they form a lattice.

An element in Artin's braid group B_n is said to be periodic if some power of it lies in the center of B_n. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in B_n are exponential in the braid index n for the special case of periodic braids. We overcome this difficult…

Paper classifies Heegaard Floer minimal knots in sutured manifolds.

We give a conjectural classification of virtually cocompactly cubulated Artin-Tits groups (i.e. having a finite index subgroup acting geometrically on a CAT(0) cube complex), which we prove for all Artin-Tits groups of spherical type, FC type or two-dimensional type. A particular case is that for $n \geq 4$, the $n$-st…

For every compact surface $S$ of finite type (possibly with boundary components but without punctures), we show that when $n$ is sufficiently large there is no lift $σ$ of the surface braid group $B_n(S)$ to $\operatorname{Diff}(S,n)$, the group of $C^1$ diffeomorphisms preserving $n$ marked points and restricting to t…

Symplectic Torelli groups of positive rational surfaces are trivial or sphere braid groups.

Given a group $G$ and a subset $X \subset G$, an element $g \in G$ is called quasi-positive if it is equal to a product of conjugates of elements in the semigroup generated by $X$. This notion is important in the context of braid groups, where it has been shown that the closure of quasi-positive braids coincides with t…

New model for Calabi-Yau-$\mathbb{X}$ categories using decorated marked surfaces.

A family of TQFTs parametrised by G-crossed braided spherical fusion categories has been defined recently as a state sum model and as a Hamiltonian lattice model. Concrete calculations of the resulting manifold invariants are scarce because of the combinatorial complexity of triangulations, if nothing else. Handle deco…

Study shows mapping class group dimension for surfaces with punctures.

The minimal standardizer of a curve system on a punctured disk is the minimal braid that transforms it into a system formed only by round curves. We give an algorithm to compute it in a geometrical way. Then, we generalize this problem algebraically to parabolic subgroups of Artin-Tits groups of spherical type and we s…

A graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with convex geodesic bico…

Reduces conjecture for Artin groups to simpler cases.

We prove that the exponential growth rate of the regular language of penetration sequences is smaller than the growth rate of the regular language of normal form words, if the acceptor of the regular language of normal form words is strongly connected. Moreover, we show that the latter property is satisfied for all irr…

Classic braids embed in virtual braids.

Spider category comparison proves equivalence to Sikora's quotient category.

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…

We construct a state-sum type invariant of smooth closed oriented $4$-manifolds out of a $G$-crossed braided spherical fusion category ($G$-BSFC) for $G$ a finite group. The construction can be extended to obtain a $(3+1)$-dimensional topological quantum field theory (TQFT). The invariant of $4$-manifolds generalizes s…

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

The paper finds minimal generating sets and abelianizes the quasitoric braid group.

Study virtual braid groups, proving a key subgroup result.

We prove that an Artin-Tits group of type $\tilde C$ is the group of fractions of a Garside monoid, analogous to the known dual monoids associated with Artin-Tits groups of spherical type and obtained by the "generated group" method. This answers, in this particular case, a general question on Artin-Tits groups, gives …

Paper explores relations between braid groups and their quotients.

Paper proves homotopy braid group properties over integers and three strands.

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 $k$-free braid groups $G_n^k$. 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.

The paper studies orbifold braid groups and their properties.

We give formulae for the first homology of the $n$-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 $n$-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.

New family of braided Thompson groups introduced using recursive braids.

Study on braid groups' congruence subgroups and their crystallographic quotients.

Paper refines braidoid equivalence for spherical knotoids.