Here a braid with strands is thought of as pieces of string joining points at the top of the diagram with -points at the bottom.
(This is a picture of a -strand braid.)
We can transform / ‘isotope’ these braid diagrams just as we can transform knot diagrams, again using Reidemeister moves. The ‘isotopy’ classes of braid diagrams form a group in which the composition is obtained by putting one diagram above another.
The identity consists of vertical strings, so the inverse is obtained by turning a diagram upside down:
Let be the space of configurations of points in the complex plane, whose elements are those -tuples such that whenever . The symmetric group acts on by permuting coordinates. Let be the orbit space (the space of -element subsets of if one likes), and let be the image of under the quotient . We take as basepoint for , and as basepoint for .
Evidently a braid is represented by a path with . Such a path may be uniquely lifted through the covering projection to a path such that . The end of the path has the same underlying subset as but with coordinates permuted: . Thus the braid is exhibited by non-intersecting strands, each one connecting an to , and we have a map appearing as the quotient map of an exact sequence
which is part of a long exact homotopy sequence corresponding to the fibration .
Artin presentation of
The Artin braid group, , defined using strands is a group given by
generators: , ;
In terms of automorphisms on free groups
The braid group may be alternatively described as the mapping class group of a 2-disk with punctures (call it ). Meanwhile, the fundamental group (with basepoint on the boundary) is a free group on generators; the functoriality of implies we have an induced homomorphism
If an automorphism is isotopic to the identity, then of course is trivial, and so the homomorphism factors through the quotient , so we get a homomorphism
and this turns out to be an injection.
Explicitly, the generator used in the Artin presentation above is mapped to the automorphism on the free group on generators defined by
We will look at such groups for small values of .
By default, has no generators and no relations, so is trivial.
By default, has one generator and no relations, so is infinite cyclic.
(We will simplify notation writing , .)
This then has presentation
It is also the ‘trefoil group’, i.e., the fundamental group of the complement of a trefoil knot.
Simplifying notation as before, we have generators and relations
Surface braid groups
In terms of the geometric definition above, it is possible to consider configurations of points on surfaces other than the plane, which gives rise to the more general notion of a surface braid group. For example, the Hurwitz braid group (or sphere braid group) comes from considering configurations of points on the 2-sphere . Algebraically, the Hurwitz braid group has all of the generators and relations of the Artin braid group , plus one additional relation: