Related papers: Commutator Subgroups of Singular Braid Groups

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…

In this paper we introduce the framed pure braid group on $n$ strands of an oriented surface, a topological generalisation of the pure braid group $P_n$. We give different equivalents definitions for framed pure braid groups and we study…

For a natural number $n$, denote by $B_n$ the braid group on $n$ strings and by $SM_n$ the singular braid monoid on $n$ strings. $SM_n$ is one of the most important extensions of $B_n$. In [13], Y. Mikhalchishina classified all homogeneous…

Let $n\ge 2$. Let $VB_n$ (resp. $VP_n$) be the virtual braid group (resp. the pure virtual braid group), and let $VT_n$ (resp. $PVT_n$) be the virtual twin group (resp. the pure virtual twin group). Let $\Pi$ be one of the following…

In the present paper we study structural aspects of certain quotients of braid groups and virtual braid groups. In particular, we construct and study linear representations $B_n\to {\rm GL}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm1}]\right)$,…

The n-strand braid group can be defined as the fundamental group of the configuration space of n unlabeled points in a closed disk based at a configuration where all n points lie in the boundary of the disk. Using this definition, the…

Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided…

The mixed braid groups are the subgroups of Artin braid groups whose elements preserve a given partition of the base points. We prove that the centralizer of any braid can be expressed in terms of semidirect and direct products of mixed…

In this work, we study the relationship between congruence subgroups $B_n[m]$ and $\mathcal{N}_n(\sigma_1^m)$ the normal closure of $\sigma_1^m$, where $\sigma_1$ is the classical generator of $B_n$. We characterize the conditions under…

Let B_n be the braid group on n strands, with n at least 4, and let Mod(S) be the extended mapping class group of the sphere with n+1 punctures. We show that the abstract commensurator of B_n is isomorphic to a semidirect product of Mod(S)…

The singularities of the representation variety of $B_3$ are studied, where $B_3$ is the knot group on 3 strands. Specifically, we determine which semisimple representations are smooth points of this variety.

Let VB$_n$ be the virtual braid group on $n$ strands and let $\mathfrak{S}_n$ be the symmetric group on $n$ letters. Let $n,m \in \mathbb{N}$ such that $n \ge 5$, $m \ge 2$ and $n \ge m$. We determine all possible homomorphisms from VB$_n$…

We study a novel type of braid groups on a closed orientable surface $\Sigma$. These are fundamental groups of certain manifolds that are hybrids between symmetric products and configuration spaces of points on $\Sigma$; a class of examples…

We consider normal subgroups $N$ of the braid group $B_n$ such that the quotient $B_n/N$ is an extension of the symmetric group by an abelian group. We show that, if $n\geq 4$, then there are exactly 8 commensurability classes of such…

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…

In the present paper, we introduce $\mathbb{Z}_2$-braids and, more generally, $G$-braids for an arbitrary group $G$. They form a natural group-theoretic counterpart of $G$-knots, see \cite{reidmoves}. The underlying idea, used in the…

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…

We study the representations of the commutator subgroup K_{n} of the braid group B_{n} into a finite group . This is done through a symbolic dynamical system. Some experimental results enable us to compute the number of subgroups of K_{n}…

In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $n>3$, $\Aut(P_n)$ is generated by the subgroup $\Aut_c(P_n)$ of central automorphisms of $P_n$, the subgroup $\Aut(B_n)$ of…

We give a complete classification of homomorphisms from the braid group on $n$ strands to the braid group on $2n$ strands when $n$ is at least 5. We also classify endomorphisms of the braid group on 4 strands, as well as homomorphisms from…