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

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by ai’s on the left, the obtained elements are well defined in the free groups. Besides, these elements 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 . 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 .

2.4Virtual braids. Inclusion of classical braids into virtual braids

      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].

2.4.1Definitions of virtual braids

      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.

2.4.2Invariants of virtual braids

      In this section, we are going to present an invariant of virtual braids proposed by the first-named author in [Manturov, 2008] and show that the classical braid group is a subgroup of the virtual one. For an elementary proof of this fact see [Manturov, 2016b]. More precisely, we give a generalisation of the complete braid invariant described before for the case of virtual braids. The new “virtual invariant” is quite strong. The question of whether the invariant is complete was answered negatively by O. Chterental [Chterental, 2015]. The completeness of the multi-variable extension of the invariant (see [Manturov, 2003]) is unknown.

      Thus the main question is the word problem for the virtual braid group: how to recognise whether two different (regular) virtual braid diagrams β₁ and β₂ represent the same braid B.

      Remark 2.2. The recognition problem for virtual braids was solved by O. Chterental [Chterental, 2017].

      Given two braid diagrams one can apply the virtual braid group relations to one of those diagrams without getting the other diagram and one does not know whether he has to stop and say that they are not isomorphic or he has to continue.

      A partial answer to this question is the construction of a virtual braid group invariant; i.e., a function on virtual braid diagrams (or braid words) that is invariant under all virtual braid group relations. In this case, if for an invariant f we have f(β₁) ≠ f(β₂), then β₁ and β₂ represent two different braids.

      Here we give a generalisation of the complete classical braid group invariant for the case of virtual braids.

      Let G be the free group in generators a₁ . . . , an, t. Let Ei be the quotient set of right residue classes {ai}\G for i = 1, . . . , n.

      Definition

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