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
2.3The curve algorithm for braids recognition
Below, we shall give a proof of the completeness of the invariant for the braid group elements invented by Artin, see [Gaifullin and Manturov, 2002].
The invariant to be constructed has a simple algebraic description as a map (non-homeomorphic) from the braid group Br(n) to the n copies of the free group in n generators.
Several generalisations of this invariant, such as the spherical and cylindrical braid group invariants, are also complete. The key point of such a completeness is that these invariants originate from several curves, and the braid can be uniquely restored from these curves.
Moreover, this approach finally led to the algorithmic recognition of virtual braids due to Oleg Chterental [Chterental, 2015].
Fig. 2.5Decomposing a pure braid
2.3.1Construction of the invariant
Let us begin with the definition of notions that we are going to use, and let us introduce the notation.
Definition 2.10. By an admissible system of n curves we mean a family of n non-intersecting non-self-intersecting curves in the upper half plane {y ≥ 0} of the plane Oxy such that each curve connects a point having ordinate zero with a point having ordinate one and the abscissas of all curve ends are integers from 1 to n. All points (i, 1), where i = 1, . . . , n, are called upper points, and all points (i, 0), i = 1, . . . , n, are called lower points.
Definition 2.11. Two admissible systems of n curves A and A′ are equivalent if there exists a homotopy between A and A′ in the class of curves with fixed endpoints lying in the upper half plane such that no interior point of any