A note on the Lawrence-Krammer-Bigelow representation

The Lawrence-Krammer representation

The Lawrence-Krammer representation of the braid groups recently came to prominence when it was shown to be faithful by myself and Krammer. It is an action of the braid group on a certain homology module $H_2(\tilde{C})$ over the ring of Laurent polynomials in $q$ and $t$. In this paper we describe some surfaces in $\tilde{C}$ representing elements of homology. We use these to give a new proof that $H_2(\tilde{C})$ is a free module. We also show that the $(n-2,2)$ representation of the Temperley-Lieb algebra is the image of a map to relative homology at $t=-q^{-1}$, clarifying work of Lawrence…

Braid groups and euclidean simplices

When Daan Krammer and Stephen Bigelow independently proved that braid groups are linear, they used the Lawrence-Krammer-Bigelow representation for generic values of its variables q and t. The t variable is closely connected to the traditional Garside structure of the braid group and plays a major role in Krammer's algebraic proof. The q variable, associated with the dual Garside structure of the braid group, has received less attention. In this article we give a geometric interpretation of the q portion of the LKB representation in terms of an action of the braid group on the space of non-dege…

Lawrence-Bigelow representations, bases and duality

We study homological representations of mapping class groups, including the braid groups. These arise from the twisted homology of certain configuration spaces, and come in many different flavours. Our goal is to give a unified general account of the fundamental relationships (non-degenerate pairings, embeddings, isomorphisms) between the many different flavours of homological representations. Our motivating examples are the Lawrence-Bigelow representations of the braid groups, which are of central importance in the study of the braid groups themselves, as well as their connections with quantu…

A family of representations of braid groups on surfaces

We propose a family of new representations of the braid groups on surfaces that extend linear representations of the braid groups on a disc such as the Burau representation and the Lawrence-Krammer-Bigelow representation.

Obstructions to trivializing a knot

The recent proof by Bigelow and Krammer that the braid groups are linear opens the possibility of applications to the study of knots and links. It was proved by the first author and Menasco that any closed braid representative of the unknot can be systematically simplified to a round planar circle by a sequence of exchange moves and reducing moves. In this paper we establish connections between the faithfulness of the Krammer-Lawrence representation and the problem of recognizing when the conjugacy class of a closed braid admits an exchange move or a reducing move.

Faithfulness of the Lawrence representation of braid groups

The Lawrence representation $L_{n,m}$ is a family of homological representation of the braid group $B_n$, which specializes to the reduced Burau and the Lawrence-Krammer representation when $m$ is 1 and 2. In this article we show that the Lawrence representation is faithful for $m \geq 2$.

Braid Groups are Linear

The braid groups B_n can be defined as the mapping class group of the n-punctured disc. The Lawrence-Krammer representation of the braid group B_n is the induced action on a certain twisted second homology of the space of unordered pairs of points in the n-punctured disc. Recently, Daan Krammer showed that this is a faithful representation in the case n=4. In this paper, we show that it is faithful for all n.

Faithful Linear Representations of the Braid Groups

We give an exposition of the work of Bigelow and Krammer who proved that the Artin braid groups are linear.

辫子群表示到奇异辫子幺半群的扩张

Extensions of braid group representations to the monoid of singular braids

本文研究辫子群表示向奇异辫子幺半群及奇异辫子群的扩张，定义线性表示间的缺陷，构造 LKBR 的扩张并计算其与 Burau 扩张外积的缺陷。

运动群与映射类群的拓扑表示——一种统一的函子构造

Topological representations of motion groups and mapping class groups -- a unified functorial construction

本文通过函子方法统一构造同调表示，引入定义在包含映射类群和运动群的范畴上的同调表示函子，统一了Lawrence-Bigelow等已知构造并得到新表示。

The mapping class group of a genus two surface is linear

In this paper we construct a faithful representation of the mapping class group of the genus two surface into a group of matrices over the complex numbers. Our starting point is the Lawrence-Krammer representation of the braid group B_n, which was shown to be faithful by Bigelow and Krammer. We obtain a faithful representation of the mapping class group of the n-punctured sphere by using the close relationship between this group and B_{n-1}. We then extend this to a faithful representation of the mapping class group of the genus two surface, using Birman and Hilden's result that this group is …

Embeddings of the braid groups of covering spaces, classification of the finite subgroups of the braid groups of the real projective plane, and linearity of braid groups of low-genus surfaces

Let M be a compact, connected surface, possibly with a finite set of points removed from its interior. Let d,n be positive integers, and let N be a d-fold covering space of M. We show that the covering map induces an embedding of the n-th braid group B_n(M) of M in the (dn)-th braid group B_{dn}(N) of N, and give several applications of this result. First, we classify the finite subgroups of the n-th braid group of the real projective plane, from which we deduce an alternative proof of the classification of the finite subgroups of the mapping class group of the n-punctured real projective plan…

Representations of braid groups

In this paper we survey some work on representations of $B_n$ given by the induced action on a homology module of some space. One of these, called the Lawrence-Krammer representation, recently came to prominence when it was shown to be faithful for all $n$. We will outline the methods used, applying them to a closely related representation for which the proof is slightly easier. The main tool is the Blanchfield pairing, a sesquilinear pairing between elements of relative homology. We discuss two other applications of the Blanchfield pairing, namely a proof that the Burau representation is not …

曲面构型上的海森堡同调

Heisenberg homology on surface configurations

受经典辫群表示启发，本文研究带边界可定向曲面上无序构型的同调，利用离散海森堡群表示定义非交换局部系统，构造了扭曲与非扭曲的映射类群表示，并证明其保持半双线性形式。

曲面辫群与映射类群表示的多项式性

Polynomiality of surface braid and mapping class group representations

本文研究了曲面辫群与映射类群的同调表示函子，通过函子短正合序列建立相干表示间的关系，证明了其多项式性，并应用于扭曲同调稳定性及表示核的理解。