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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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, . . . , σn−1 (for classical crossings) and ζ1, . . . , ζn−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 figure (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.

      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 2.17. A virtual n-system is a set of elements figuree1 ∈ E1, e2 ∈ E2, . . . , en ∈ Enfigure.

      The aim of this subsection is to construct an invariant map f (non-homomorphic) from the set of all virtual n-strand braids to the set of virtual n-systems.

      Let β be a braid word. Let us construct the corresponding virtual n-system f(β) step-by-step. Namely, we shall reconstruct the function f(βψ) from the function f(β), where ψ is σi or figure or ζi.

      First, let us take n residue classes of the unit element of G: figuree, e, . . . , efigure. This means that we have defined

      Now, let us read the word β. If the first letter is ζi, then all words but ei,ei+1 in the n-systems stay the same, ei becomes equal to t and ei+1 becomes t−1 (here and in the sequel, we mean, of course, residue classes, e.g. [t] and [t−1]. But we write just t and t−1 for the sake of simplicity).

      Now, if the first letter of our braid word is σi, then all classes but ei+1 stay the same, and ei+1 becomes figure. Finally, if the first letter is figure, then the only changing element is ei: it becomes ai+1.

      The procedure for each next letter (generator) is the following. Denote the index of this letter (the generator or its inverse) by i. Assume that the left strand of this crossing originates from the point (p, 1), and the right one originates from the point (q, 1). Let ep = P, eq = Q, where P, Q are some words representing the corresponding residue classes. After the crossing all residue classes but ep, eq should stay the same.

      Then if the letter is ζi then ep becomes P · t, and eq becomes Q · t−1. If the letter is σi, then ep stays the same, and eq becomes figure. Finally, if the letter is figure, then eq stays the same, ep becomes PQ−1aqQ. Note that this operation is well defined.

      Actually, if we take the words figure instead of the words P, Q, then we get: in the first case

      and in the second case we obtain

      In the third case we obtain

      Thus, we have defined the map f from the set of all virtual braid diagrams to the set of virtual n-systems.

      Theorem 2.5. The function f, defined above, is a braid invariant. Namely, if β1 and β2 represent

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