Related papers: Braid Groups are Linear

Given two nonzero complex parameters $l$ and $m$, we construct by the mean of knot theory a matrix representation of size $\chl$ of the BMW algebra of type $A_{n-1}$ with parameters $l$ and $m$ over the field $\Q(l,r)$, where $m=\unsurr-r$.…

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…

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…

Dehornoy showed that the Artin braid groups $B_n$ are left-orderable. This ordering is discrete, but we show that, for $n >2$ the Dehornoy ordering, when restricted to certain natural subgroups, becomes a dense ordering. Among subgroups…

This paper is the first part of a series of papers aimed at improving the classification by Formanek of the irreducible representations of Artin braid groups of small dimension. In this paper we classify all the irreducible complex…

Let X be a 2-sphere with n punctures. We classify all conjugacy classes of Zariski-dense representations $$\rho: \pi_1(X)\to SL_2(\mathbb{C})$$ with finite orbit under the mapping class group of X, such that the local monodromy at one or…

Here a loop braid representation is a monoidal functor $\mathsf{F}$ from the loop braid category $\mathsf{L}$ to a suitable target category, and is $N$-charge-conserving if that target is the category $\mathsf{Match}^N$ of charge-conserving…

Let S be a principally embedded sl_2 subalgebra in sl_n for n > 2. A special case of results of the third author and Gregg Zuckerman implies that there exists a positive integer b(n) such that for any finite-dimensional irreducible sl_n…

We classify an action of the $n$-strand braid group on the free group of rank $n$ which is similar to the Artin representation in the sense that the $i$-th generator $\sigma_{i}$ of $B_{n}$ acts so that it fixes all free generators $x_{j}$…

We consider finite-sheeted, regular, possibly branched covering spaces of compact surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides…

We show that the image of the pure braid group under the monodromy action on the homology of a cyclic covering of degree d of the projective line is an arithmetic group provided the number of branch points is sufficiently large compared to…

Burau representation of the Artin braid group remains as one of the very important representations for the braid group. Partly, because of its connections to the Alexander polynomial which is one of the first and most useful invariants for…

We show that the Lawrence-Krammer representation based on two parameters that was used by Bigelow and independently Krammer to show the linearity of the braid group is generically irreducible, but that when its parameters are specialized to…

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…

Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic from the braid group to the mapping class group. We prove here that this map is trivial in stable homology with…

A new method for deriving universal \v{R} matrices from braid group representation is discussed. In this case, universal \v{R} operators can be defined and expressed in terms of products of braid group generators. The advantage of this…

The braid group $B_n$ maps homomorphically into the Temperley-Lieb algebra $\TL_n$. It was shown by Zinno that the homomorphic images of simple elements arising from the dual presentation of the braid group $B_n$ form a basis for the vector…

When $\pi:\widetilde{\Sigma}\rightarrow D^2$ is a cover of the disc branched over $n$ marked points, the braid group $B_n$ acts on the disc by homeomorphisms fixing the marked points setwise. A braid $\beta$ \textit{lifts} if there is a…

We construct a linear representation of the group IA(F_n) of IA-automorphisms of a free group F_n, an extension of the Gassner representation of the pure braid group P_n. Although the problem of faithfulness of the Gassner representation is…

For every $n\geq 1$, the flat braid group $\mathrm{FB}_n$ is an analogue of the braid group $B_n$ that can be described as the fundamental group of the configuration space $$\left\{ \{x_1, \ldots, x_n \} \in \mathbb{R}^n / \mathrm{Sym}(n)…