braid group

Let Cn be the space of unordered n-tuples of distinct points in the complex plane. The braid groupMathworldPlanetmath Bn is the fundamental groupMathworldPlanetmathPlanetmath of Cn.

A closed path γ on this space is a set of n paths γi:[0,1] with γi(t)γj(t), and γi(1)=γσ(i)(0), where σ is some permutationMathworldPlanetmath of {1,,n}. Drawing the graphs of all these paths in 3 space, what we see is n strands between the z=0 and z=1 planes, possibly tangled, with compositionMathworldPlanetmathPlanetmath given by stacking these braids on top of each other. Homotopy corresponds to isotopy of the braid, homotopies of the strands such that none of them cross. This is the origin of the name “braid group”

The braid group determines a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ϕ:BnSn, where Sn is the symmetric groupMathworldPlanetmathPlanetmath on n letters. For γBn, we get an element of Sn from map sending iγi(1). This works because of our requirement on the points that the braids start and end, and since our homotopies fix basepoints. The kernel of ϕ consists of the braids that bring each strand to its original order. This kernel gives us the pure braid group on n strands, and is denoted by Pn. Hence, we have a short exact sequenceMathworldPlanetmath


We can also describe braid groups in more generality. Let M be a manifoldMathworldPlanetmath. The configuration space of n ordered points on M is defined to be Fn(M)={(a1,,an)Mnaiajforij}. The group Sn acts on Fn(M) by permuting coordinates, and the corresponding quotient space Cn(M)=Fn(M)/Sn is called the configuration space of n unordered points on M. In the case that M=, we obtain the regularPlanetmathPlanetmathPlanetmath and pure braid groups as π1(Cn(M)) and π1(Fn(M)) respectively.

The group Bn can be given the following presentationMathworldPlanetmathPlanetmath. The presentation was given in Artin’s first paper [1] on the braid group. Label the braids 1 through n as before. Let σi be the braid that twists strands i and i+1, with i passing beneath i+1. Then the σi generate Bn, and the only relationsMathworldPlanetmathPlanetmath needed are

σiσj=σjσifor |i-j|2, 1i,jn-1σiσi+1σi=σi+1σiσi+1for  1in-2

The pure braid group has a presentation with

generatorsaij=σj-1σj-2σi+1σi2σi+1-1σj-2-1σj-1-1 for 1i<jn

that is, aij wraps the ith strand around the jth strand, and defining relations

ars-1aijars={aijif i<r<s<j or r<s<i<jarjaijarj-1if r<i=s<jarjasjaijasj-1arj-1if i=r<s<jarjasjarj-1asj-1aijasjarjasj-1arj-1if r<i<s<j


  • 1 E. Artin Theorie der Zöpfe. Abh. Math. Sem. Univ. Hamburg 4(1925), 42-72.
  • 2 V.L. Hansen Braids and Coverings. London Mathematical Society Student Texts 18. Cambridge University Press. 1989.
Title braid group
Canonical name BraidGroup
Date of creation 2013-03-22 13:51:51
Last modified on 2013-03-22 13:51:51
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 15
Author bwebste (988)
Entry type Topic
Classification msc 20F36
Synonym Artin’s braid group
Related topic Tangle
Defines pure braid group
Defines braid
Defines configuration space