Fig. 2.2Unity. Operations in the braid group
Definition 2.4. The Artin m-strand braid group1 is the group of braids with the operation defined above. The braid group is denoted by Br(m).
The second definition of the braid group considered in the present chapter is an algebraic definition.
Definition 2.5. The m-strand braid group is the group given by the presentation with (m − 1) generators σ1, . . . , σm−1 and the following relations
for |i − j| ≥ 2 and
for 1 ≤ i ≤ m − 2.
These relations are called Artin’s relations.
Definition 2.6. Words in the alphabet of σ’s and σ−1’s will be referred to as braid words.
In fact, those two definitions give the same object:
Theorem 2.1. The two definitions of the braid group Br(m) given above are equivalent.
Proof. In order to prove the equivalence of the definitions, let us introduce the notion of the planar braid diagram.
To see what it is, let us project a braid on the plane Oxz.
In the general case we obtain a diagram that can be described as follows.
Definition 2.7. A braid diagram (for the case of m strands) is a graph lying inside the rectangle [1, m] × [0, 1] endowed with the following structure and having the following properties:
(1)Points (i, 0) and (i, 1), i = 1, . . . , m, are vertices of valency one; there are no other graph vertices on the lines {z = 0} and {z = 1}.
(2)All other graph vertices (crossings) have valency four.
(3)Unicursal curves; i.e., lines consisting of edges of the graph, passing from an edge to the opposite one, go from vertices with ordinate one and come to vertices with ordinate zero and descend monotonously.
(4)Each vertex of valency four is endowed with an over and undercrossing structure.
Obviously, all isotopy classes of geometrical braids can be represented by their planar diagrams. Moreover, after a small perturbation, all crossings of the braid can be set to have different z-coordinates.
It is easy to see that each element of the geometrical braid group can be decomposed into a product of the following generators σi’s: the element σi for i = 1, . . . , m − 1 consists of m − 2 segments connecting (k, 1) and (k, 0), k ≠ i, k ≠ i + 1, and two segments (i, 0) − (i + 1, 1), (i + 1, 0) − (i, 1), where the latter goes over the first one; see Fig. 2.3.
Different braid diagrams can generate the same braid. Thus we obtain some relations in σ1, . . . , σm−1.
Let us suppose that we have two equal geometrical braids B1 and B2. Let us represent the process of isotopy from B1 to B2 in terms of their planar diagrams. Each interval of this isotopy either does not change the disposition of their vertex ordinates, or in this interval at least two crossings have (in a moment) the same ordinate; in the latter case the diagram becomes irregular.
Fig. 2.3Generators of the braid group
We are interested in those moments where the algebraic description of our braid changes. We see that there are only three possible cases (all others can be reduced to these ones). The first case gives us the relation σiσj = σjσi, |i − j| ≥ 2 (this relation is called far commutativity), or an equivalent relation
Obviously, each of the latter two relations can be obtained from the first one. This simple observation is left to the reader as an exercise. This completes the proof of the theorem.
2.2The stable braid group and the pure braid group
For natural numbers m < n, there exists a natural embedding Br(m) ⊂ Br(n): a braid from Br(m) can be treated as a braid from Br(n) where the last (n − m) strands are vertical and unlinked (separated) with the others.
Definition 2.8. The stable braid group Br is the limit of groups Br(n) as n → ∞ with respect to these embeddings.
With each braid one can associate its permutation which takes an element k to l if the strand starting with the k-th upper point ends at the l-th lower point.
Definition 2.9. A braid is said to be pure if its permutation is identical. Obviously, pure braids generate a subgroup PBn ⊂ Brn.
An interesting problem is to find an explicit finite presentation of the pure braid group on n strands.
Here we shall present some concrete generators (according to [Artin, 1947]). A presentation of this group can be found in e.g. [Makanina, 1992].
There exists an algebraic Reidemeister–Schreier method that allows us to construct a presentation of a finite–index subgroup having a presentation of a finitely defined group, see e.g. [Crowell and Fox, 1963].
The following theorem holds.
Theorem 2.2. The group PB(m) is generated by braids
(see Fig. 2.4).
Fig. 2.4Generator bij of the pure braid group
