A Categorical Structure for the Virtual Braid Group

This article gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. This point of view makes many relationships between the virtual braid group and the pure virtual braid group apparent, and makes representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang–Baxter Equation equally transparent. In this categorical framework, the virtual braid group has nothing to do with the plane and nothing to do with virtual crossings. It is a natural group associated with the structure of algebraic braiding.     

Linear Representations of the Braid Groups of Some Manifolds

In this paper we prove that the braid group B_n(S²) of 2-sphere, mapping class group M(0,n) of the n-punctured 2-sphere and the braid group B₃(P²) of the projective plane are linear. Partially supported by the Russian Foundation for Basic Research (grant number 02-01-01118).Mathematics Subject Classifications (2000) 20F28, 20F36, 20G35.     

Equivariant Mapping Class Group and Orbit Braid Group

Motivated by the work of Birman about the relationship between mapping class groups and braid groups, the authors discuss the relationship between the orbit braid group and the equivariant mapping class group on the closed surface M with a free and proper group action in this paper. Their construction is based on the exact sequence given by the fibration F₀^GM → F(M/G, n). The conclusion is closely connected with the braid group of the quotient space. Comparing with the situation without the group action, there is a big difference when the quotient space is T².     

Normalizers of Some Classes of Subgroups in Braid Groups

In this paper, the normalizers of some classes of subgroups of the braid group B _n+1 are studied.     

A New Algorithm for Solving the Word Problem in Braid Groups

One of the most interesting questions about a group is whether its word problem can be solved and how. The word problem in the braid group is of particular interest to topologists, algebraists, and geometers, and is the target of intensive current research. We look at the braid group from a topological point of view (rather than a geometric one). The braid group is defined by the action of diffeomorphisms on the fundamental group of a punctured disk. We exploit the topological definition in order to give a new approach for solving its word problem. Our algorithm, although not better in complexity, is faster in comparison with known algorithms for short braid words, and it is almost independent of the number of strings in the braids. Moreover, the algorithm is based on a new computer presentation of the elements of the fundamental group of a punctured disk. This presentation can be used also for other algorithms.     

Braid groups are linear

The braid group can be defined as the mapping class group of the -punctured disk. A group is said to be linear if it admits a faithful representation into a group of matrices over . Recently Daan Krammer has shown that a certain representation of the braid groups is faithful for the case . In this paper, we show that it is faithful for all .

     

Braid pictures for Artin groups

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 , and and the affine diagrams , , and as subgroups of the braid groups of various simple orbifolds. The cases , , and are new. In each case the Artin group is a normal subgroup with abelian quotient; in all cases except the quotient is finite. We also illustrate the value of our braid calculus by giving a picture-proof of the basic properties of the Garside element of an Artin group of type .

     

The Braid Group and Linear Rigidity

We introduce the BC-operation (short for braid-companion) on tuples of matrices. It corresponds to Katz's middle convolution operation on local systems, and generalizes it to fields K of arbitrary characteristic. It is based on the Burau–Gassner representation of the braid group. We expect many applications, in Galois theory (for finite K) as well as in Katz's original set-up of local systems and linear differential equations (where K = ).     

基于辫群的代理签名方案

Artin辫群是可以有限表示的无限非交换群。近年来被认为是公钥加密的一种重要来源；代理签名[1]是一种具有特殊功能的数字签名．本文，我们首次提出基于辫群（主要利用辫群中的难解问题；共轭问题，多共轭问题，求根的问题）的代理签名方案．     

Center of some solvable groups with one defining relation

Necessary and sufficient conditions for the center of a metabelian group with one defining relation to be nontrivial are found. The center of such a group is described. The center of a group of the formF/Ng ^F is studied under certain conditions. By means of a new technique, the recent result of A. F. Krasnikov and the author on the center of a group of the above form is sharpened.Translated fromMatematickeskie Zametki, Vol. 64, No. 6, pp. 925–931, December, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01948.     

Strong Positivity in Right-Invariant Order on Braid Groups and Quasipositivity

Dehornoy constructed a right invariant order on the braid group B _n uniquely defined by the condition 1{\text{ if }}\beta _0 ,\beta _1$$ " align="middle" border="0"> are words in . A braid is called strongly positive if 1$$ " align="middle" border="0"> for any . In the present paper it is proved that the braid is strongly positive if the word does not contain . We also provide a geometric proof of the result by Burckel and Laver that the standard generators of a braid group are strongly positive. Finally, we discuss relations between the right invariant order and quasipositivity.     

Centralisers in Surface Braid Groups

Abstract

Let Σ be an orientable surface. We generalise Fenn–Rolfsen–Zhu's results on centralisers of singular braids on the disk to singular braids on Σ. As a corollary, we derive a simple and geometric proof of the fact that the word problem is solvable in the monoid of singular braids on n strands on Σ.     

Embedding right-angled Artin groups into graph braid groups

We construct an embedding of any right-angled Artin group G(Δ) defined by a graph Δ into a graph braid group. The number of strands required for the braid group is equal to the chromatic number of Δ. This construction yields an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid group.      

Garside Structure for Singular Braid Monoid in Birman,Ko, Lee Generators

We consider a Birman, Ko, and Lee (BKL) presentation (defined by Vershinin, 2003 Vershinin , V. V. ( 2003 ). On the singular braid monoid . http://www.arXiv.org.math. GR/6309339 . [CSA]  [Google Scholar]) for the semigroup of singular braids SB _n . We prove the embedding property for the monoid of positive singular braids and give a solution to the word and conjugacy problems in BKL generators.     

Quasi-isometrically embedded subgroups of braid and diffeomorphism groups

We show that a large class of right-angled Artin groups (in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group of area preserving diffeomorphisms of the disk fixing the boundary (with respect to the -norm metric); this extends results of Benaim and Gambaudo who gave quasi-isometric embeddings of and for all . As a consequence we are also able to embed a variety of Gromov hyperbolic groups quasi-isometrically in pure braid groups and in the group . Examples include hyperbolic surface groups, some HNN-extensions of these along cyclic subgroups and the fundamental group of a certain closed hyperbolic 3-manifold.

     

On amenability of two-generator groups

A discrete group G is amenable if there exists a finitely additive probability measure on G which is invariant under left translations and is defined on all subsets of G. It is proved that if the group is generated by two elements and is amenable then there are words being relators whose most of the consecutive pairs of the letters belong to a certain four-element set of pairs. This fact is applied to reproving non-amenability of a braid group. The same group provides an example showing that such type of condition is not su?cient for amenabilty.     

A p-adic model of DNA sequence and genetic code

Using basic properties of p-adic numbers, we consider a simple new approach to describe main aspects of DNA sequence and the genetic code. In our investigation central role plays an ultrametric p-adic information space whose basic elements are nucleotides, codons and genes. We show that a 5-adicmodel is appropriate for DNA sequence. This 5-adicmodel, combined with 2-adic distance, is also suitable for the genetic code and for amore advanced employment in genomics. We find that genetic code degeneracy is related to the p-adic distance between codons. The text was submitted by the authors in English. This paper is a slight modification of an article available in the electronic archive form arXiv:qbio. GN/0607018v1 (July 2006). Since that time some other papers on this subject have appeared, e.g. [1], [2].