Читать книгу Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory - Vassily Olegovich Manturov - Страница 26

Virtual braids have a purely combinatorial definition. Namely, one takes virtual braid diagrams and factorises them by virtual Reidemeister moves (all moves with the exception of the first classical and the virtual moves; the latter moves do not occur).

Definition 2.15. A virtual braid diagram on n strands is a graph lying in [1, n] × [0, 1] ⊂ ² with vertices of valency one (there should be exactly 2n such vertices with coordinates (i, 0) and (i, 1) for i = 1, . . . , n) and a finite number of vertices of valency four. The graph is a union of n smooth curves without horizontal tangent lines connecting a point on the line {y = 1} with those on the line {y = 0}; their intersection makes crossings (four-valent vertices). Each crossing should be either endowed with a structure of over-or undercrossing (as in the case of classical braids) or marked as a virtual one (by encircling it).

Definition 2.16. A virtual braid is an equivalence class of virtual braid diagrams by planar isotopies and all virtual Reidemeister moves (see Figs. 3.2 and 3.13) except the first classical move and the first virtual move.

A virtual braid diagram is called regular if any two different crossings have different ordinates.

Remark 2.1. We shall also treat braid words and braids familiarly, saying, e.g. “a strand of a braid word” and meaning “a strand of the corresponding braid”.

Let us describe the construction of the word by a given regular virtual braid diagram as follows. Let us walk along the axis Oy from the point (0, 1) to the point (0, 0) and watch all those levels z = t ∈ [0, 1] having crossings. Each such crossing permutes strands #i and #(i + 1) for some i = 1, . . . , n − 1. If the crossing is virtual, then we write the letter ζi, if not, we write σi if overcrossing is the “northeast-southwest” strand, and otherwise.

Thus, we have got a braid word by a given regular virtual braid diagram; see Fig. 2.10. Let us describe this presentation of virtual braids formally.

Like classical braids, virtual braids form a group (with respect to juxtaposition and rescaling the vertical coordinate). The generators of this group are:

σ₁, . . . , σ_n−1 (for classical crossings) and ζ₁, . . . , ζ_n−1 (for virtual crossings).

The inverse elements for the σ’s are defined as in the classical case. Obviously, for each i = 1, . . . , n − 1 we have (this follows from the second virtual Reidemeister move).

Fig. 2.10A virtual braid diagram and the corresponding braid word

One can show that the following set of relations [Vershinin, 2001] is sufficient to generate this group:

(1)(Braid group relations):

(2)(Permutation group relations):

(3)(Mixed relations):

The proof of this fact is left to the reader.