Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory. Vassily Olegovich Manturov

Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory

Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory - Vassily Olegovich Manturov

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along L′, after this goes vertically downwards until (i, 0) and finally, horizontally until (j, 0). Obviously, W(L′) = W. It is easy to see that for different representatives L of W we obtain the same L′. Besides, for L1 = aiL2, the curves figure and figure are isotopie This completes the proof of the lemma.

      Thus, for a fixed permutation s, admissible systems of curves can be uniquely encoded by n-systems, which completes the proof of the proposition.

      Now, we see that this invariant is a quite simple object: elements of Ei can be easily compared.

      Let us describe the algebraic construction of the invariant f in more detail.

      Let β be a braid word, written as a product of generators figure, where each εj is either +1 or −1; 1 ≤ ij ≤ n − 1 and σ1, . . . , σn−1 are the standard generators of the braid group Br(n).

      We are going to construct the n-system step-by-step while writing the word β. First, let us write n empty words (in the alphabet a1, . . . , an). Let the first letter of β be σj. Then all words except for the word ej+1 should stay the same (i.e., empty), and the word ej+1 becomes figure. If the first crossing is negative; i.e., figure then all words except ej stay the same and ej converts to aj+1. While considering each next crossing, we do the following. Let the crossing be figure. Let p and q be the numbers of strands coming from the left side and from the right side respectively. If this crossing is positive; i.e., σj, then all words except eq stay the same, and eq becomes figure. If it is negative, then all crossings except ep stay the same, and ep becomes figure. After processing all the crossings, we get the desired n-system.

      Example 2.1. For the trivial braid written as figure the construction operation works as follows:

      

      A priori these words may be non-trivial; they must only represent trivial residue classes, say, figure.

      However, it is not the case.

      Proposition 2.2. For the trivial braid, the algebraic algorithm described above gives trivial words.

      Proof. Indeed, the algebraic number of occurrences of ai in the word ei equals zero. This can be easily proved by induction on the number of crossings. In the initial position all words are trivial. The induction step is obvious. Thus, the final word ei equals figure, where p = 0.

      From this approach, one can easily obtain the well known invariant (action) as follows. Instead of a set of n words e1, . . . , en, one can consider the words figure. Since ei’s are defined up to a multiplication by ai’s on the left, the obtained elements are well defined in the free groups. Besides, these elements figure are generators of the free group. This can be checked by a step-by-step confirmation. Thus, for each braid B we obtain a set Q(B) of generators for the braid group. So, the braid B defines a transformation of the free group figure. It is easy to see that for two braids, the transformation corresponding to the product equals the composition of transformation. Thus, one can speak about the action of the braid group on the free group. Since f is a complete invariant, this action has an empty kernel.

      Definition 2.14. This action is called the Hurwitz action of the braid group BRn on the free group figure.

      Just as classical knots can be obtained as closures of classical braids, virtual knots can be similarly obtained by closing virtual braids. Virtual braids were suggested by V. V. Vershinin, [Vershinin, 2001].

      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 figureСкачать книгу