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

      Proof. We have to demonstrate that the function f defined on virtual braid diagrams is invariant under all virtual braid group relations. It suffices to prove that, for the words β₁ = βγ₁ and β₂ = βγ₂ where γ₁ = γ₂ is a relation we have proved, we can also prove f(β₁) = f(β₂). During the proof of the theorem, we shall call it the A-statement.

      Indeed, having proved this claim, we also have f(β₁δ) = f(β₂δ) for arbitrary δ because the invariant f(β₁δ) (as well as f(β₂δ)) is constructed step-by-step; i.e., knowing the value f(β₁) and the braid word δ, we easily obtain the value of f(β₁δ). Hence, for braid words β, δ and for each braid group relation γ₁ = γ₂ we prove that f(βγ₁δ) = f(βγ₂δ). This completes the proof of the theorem.

      Let us return to the A-statement.

      To prove the A-statement, we must consider all virtual braid group relations. The commutation relation σiσj = σjσi for “far” i, j is obvious: all four strands involved in this relation are different, so the order of applying the operation does not affect on the final result. The same can be said about the other commutation relations, involving one σ and one ζ or two ζ’s.

      Now let us consider the relation which is pretty simple too.

      Actually, let us consider a braid word β, and let the word β₁ be defined as for some i. Let f(β) = (P₁, . . . , Pn), . Let p and q be the numbers of strands coming to the crossing from the left side and from the right side. Obviously, for j ≠ p, q we have . Besides, .

      Now let us consider the case (obviously, the case is quite analogous to this one).

      As before, denote f(β) by (. . . Pi . . .), and f(β₁) by , and the corresponding strand numbers by p and q. Again, we have: for j ≠ p, q : . Moreover, by definition of f(since the p-th strand makes an overcrossing twice), and .

      Now let us check the invariance under the third Reidemeister move. Let β be a braid word, β₁ = βζiζ_i+1ζi and β₂ = βζ_i+1,ζiζ_i+1. Let p, q, r be the global numbers of strands occupying positions n, n + 1, n + 2 at the bottom of β.

      Denote f(β) by (P₁, . . . , Pn), f(β₁) by , and f(β₂) by . Obviously, ∀i ≠ p, q, r we have . Direct calculations show that and .

      Now, let us consider the mixed move by using the same notation: β₁ = βζiζ_i+1σi, β₂ = σ_i+1ζiζ_i+1. As before, for all j ≠ p, q, r. Now, direct calculation shows that

      and

      Finally, consider the “classical” case ; the notation is the same. Again . Besides this, since the p-th strand forms two overcrossings in both cases then . Then,

      and

      As we see, the final results coincide and this completes the proof of the theorem.

      Thus, we have proved that f is a virtual braid invariant; i.e., for a given braid B the value of f does not depend on the diagram representing B. So, we can write simply f(B).

      Remark 2.3. In fact, we can think of f as a function valued not in (E₁, . . . , En), but in n copies of G: all these invariants were proved for the general case of (G, . . . , G). The present construction of (E₁, . . . , En) is considered for the sake of simplicity.

      Classical braids (i.e., braids without virtual crossings) can be considered up to two equivalences: classical (modulo only classical moves) and virtual (modulo all moves). Now, we prove that they are the same. This fact is not new. It follows from [Fenn, Rimanyi and Rourke, 1997]. An elementary proof was given in [Manturov, 2016b].

      Theorem 2.6. Two virtually equal classical braids B₁ and B₂ are classically equal.

      Proof. Since B₁ is virtually equal to B₂, we have f(B₁) = f(B₂). Now, taking into account that f is a complete invariant on the set of classical braids, we have B₁ = B₂ (in the classical sense).

      As in the case of virtual knots, in the case of virtual braids there exists a forbidden move, namely, . Now, we are going to show that it cannot be represented by a finite sequence of the virtual braid group relations.

      Theorem 2.7. A forbidden move (relation) cannot be represented by a finite sequence of legal moves (relations).