We describe the construction of a minimal presentation for the group of planar pure braids $\overline{P}_n$ on $n$ strands. The generators of this presentation are dual to the generators of the cohomology ring of $\overline{P}_n$ found by Y.Baryshnikov while the relations lie in the commutator subgroup of $\overline{P}_n$. We also construct an explicit (although non-minimal) presentation for the pure cactus group.
About presentations of braid groups and their generalizations
本文综述了辫子理论中一些较新的概念和结果，它们推广了辫子理论中的经典概念和结果。在这些概念中有各种部分辫子的逆幺半群。我们还观察到不同的介绍从标准的Artin介绍概括的辫子。也就是说，我们认为介绍了少量的发电机，Sergiescu图介绍和Birman-Ko-Lee介绍。V. V. Chaynikov关于奇异辫幺半群在Birman-Ko-Lee生成元中的字和共轭问题的工作也被描述。
The group of planar (or flat) pure braids on $n$ strands, also known as the pure twin group, is the fundamental group of the configuration space $F_{n,3}(\mathbb{R})$ of $n$ labelled points in $\mathbb{R}$ no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.
In this article we prove theorem on Lifting for the set of virtual pure braid groups. This theorem says that if we know presentation of virtual pure braid group $VP_4$, then we can find presentation of $VP_n$ for arbitrary $n > 4$. Using this theorem we find the set of generators and defining relations for simplicial group $T_*$ which was defined in the previuos article of the authors. We find a decomposition of the Artin pure braid group $P_n$ in semi-direct product of free groups in the cabled generators.
We describe presentations of braid groups of type $A$ arising from coloured quivers of mutation type $A$. We show that these can be interpreted geometrically as generalised triangulations of regular polygons.
We show that the Steinberg group $\text{St}(C_2,{\mathbb Z})$ associated with the Lie type $C_2$ and with integer coefficients can be realized as a quotient of the braid group $B_6$ by one relation. As an application we give a new braid-like presentation of the symplectic modular group $\text{Sp}_4({\mathbb Z})$.
We obtain new presentations for the imprimitive complex reflection groups of type $(de,e,r)$ and their braid groups $B(de,e,r)$ for $d,r \ge 2$. Diagrams for these presentations are proposed. The presentations have much in common with Coxeter presentations of real reflection groups. They are positive and homogeneous, and give rise to quasi-Garside structures. Diagram automorphisms correspond to group automorphisms. The new presentation shows how the braid group $B(de,e,r)$ is a semidirect product of the braid group of affine type $\widetilde A_{r-1}$ and an infinite cyclic group. Elements of $…
The aim of this article is to prove that the kernel of the map from the pure braid group $PB_{n},n\ge 4$ to the group $G_{n}^{3}$ consists of full twist braids and their exponents. The proof consists of two parts. The first part which deals with $n=4$ relies on the crucial tool in this construction having its own interest is the {\em realisability projection} saying that if two {\em realisable} $G_{4}^{3}$-elements are equivalent then they are equivalent by a sequence of realisable ones. The second part (an easy one) uses induction on $n$.
We generalize presentations of the fundamental group of discriminant complements and arrive at a class of presentations associated naturally with words in the free monoid of the alphabet $σ_1,\dots,σ_{n-1}$. Our study addresses invariance properties of these presentations and the presented groups under various operations on the words. In particular we prove that the group does only depend on the corresponding element in the positive braid monoid, and under mild hypotheses only on its conjugacy class in the braid group.
In this paper we show that every finite spatial graph is a connected sum of a planar graph, which is a forest, i.e. disjoint union of finite number of trees and a tangle. As a consequence we get that any finite spatial graph is a connected sum of a planar graph and a braid. Using these decompositions it is not difficult to find a set of generators and defining relations for the fundamental group of compliment of a spatial graph in 3-space $\mathbb{R}^3$.
Présentation des groupes de tresses purs et de certaines de leurs extensions
这篇论文可以追溯到1999年，但从未发表过。其主要部分包括在联合论文[Digne-Gomi，Presentation of pure辫群，J.纽结理论及其分支10（2001）609- 623]中。第2节和第6节没有包括在内。它们独立于论文的其余部分。在第二节中，我们定义了一个态射，从一个与Coxeter群W相关联的辫群B到ZT（其中T是反射集）与W的半直积。这种态射将映射N从W提升到Dyer在[Reflection subgroups of Coxeter systems，J. Algebra 135（1990）57- 73]中定义的（Z/2 Z）T，并且它对纯辫群P的限制与阿贝尔化态射相同。在第六节中，我们证明了我们的态射给出了W的扩张B/（P，P）由P/（P，P）所关联的上圈。最近，Beck [Abelianization of subgroups of reflection groups and their辫群：an application to cohomology]研究了这一扩展。136（2011）273- 293]和Marin [Reflection groups acting on their hyperplanes J. Algebra 322（2009）2848- 2860]和[晶体群和来自复反射群的平坦流形，arXiv 1509.06213]。
The submonoid of the $3$-strand braid group $\mathcal{B}_3$ generated by $σ_1$ and $σ_1 σ_2$ is known to yield an exotic Garside structure on $\mathcal{B}_3$. We introduce and study an infinite family $(M_n)_{n\geq 1}$ of Garside monoids generalizing this exotic Garside structure, i.e., such that $M_2$ is isomorphic to the above monoid. The corresponding Garside group $G(M_n)$ is isomorphic to the $(n,n+1)$-torus knot group-which is isomorphic to $\mathcal{B}_3$ for $n=2$ and to the braid group of the exceptional complex reflection group $G_{12}$ for $n=3$. This yields a new Garside structure …
A planar pure braid consists of $n$ descending smooth arcs, each connecting a point on one horizontal line $\ell_{1}$ to a point on a horizontal line $\ell_{2}$, which is required to be directly below the first point. Two arcs are allowed to cross, but no threefold intersections are allowed. The set $Γ_{n}$ of all planar pure braids on $n$ strands is a group with respect to a natural stacking operation. We show that $Γ_{n}$ is always a diagram group, in the sense of Guba and Sapir. A number of consequences follow, including biautomaticity and bi-orderability of the groups $Γ_{n}$. Moreover, ea…
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.
This paper upbuilds the theoretical framework of orbit braids in $M\times I$ by making use of the orbit configuration space $F_G(M,n)$, which enriches the theory of ordinary braids, where $M$ is a connected topological manifold of dimension at least 2 with an effective action of a finite group $G$ and the action of $G$ on $I$ is trivial. Main points of our work include as follows. We introduce the orbit braid group $\mathcal{B}_n^{orb}(M,G)$, and show that it is isomorphic to a group with an additional endowed operation (called the extended fundamental group of $F_G(M,n)$), formed by the homot…
