Читать книгу Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory - Vassily Olegovich Manturov - Страница 21

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