A closed path on this space is a set of paths with , and , where is some permutation of . Drawing the graphs of all these paths in 3 space, what we see is strands between the and planes, possibly tangled, with composition 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 homomorphism , where is the symmetric group on letters. For , we get an element of from map sending . This works because of our requirement on the points that the braids start and end, and since our homotopies fix basepoints. The kernel of 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 . Hence, we have a short exact sequence
We can also describe braid groups in more generality. Let be a manifold. The configuration space of ordered points on is defined to be . The group acts on by permuting coordinates, and the corresponding quotient space is called the configuration space of unordered points on . In the case that , we obtain the regular and pure braid groups as and respectively.
The group can be given the following presentation. The presentation was given in Artin’s first paper  on the braid group. Label the braids through as before. Let be the braid that twists strands and , with passing beneath . Then the generate , and the only relations needed are
The pure braid group has a presentation with
that is, wraps the ith strand around the jth strand, and defining relations
- 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.
|Date of creation||2013-03-22 13:51:51|
|Last modified on||2013-03-22 13:51:51|
|Last modified by||bwebste (988)|
|Synonym||Artin’s braid group|
|Defines||pure braid group|