the invariance under the third Reidemeister move. Let β be a braid word, β1 = βζiζi+1ζi and β2 = βζi+1,ζiζi+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
Now, let us consider the mixed move by using the same notation: β1 = βζiζi+1σi, β2 = σi+1ζiζi+1. As before,
and
Finally, consider the “classical” case
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,
Theorem 2.7. A forbidden move (relation) cannot be represented by a finite sequence of legal moves (relations).
Proof. Actually, let us calculate the values f(σ1σ2ζ1) and f(ζ2σ1σ2). In the first case we have:
In the second case we have:
As we see, the final results are not the same (i.e., they represent different virtual n-systems); thus, the forbidden move changes the virtual braid.
Remark 2.4. If we put t = 1, the results f(X) and f(Y) become the same. Thus that is the variable t that “feels” the forbidden move.
2.4.2.1A 2n-variable generalisation of the invariant
In the present section we define a stronger version of the f invariant described in the previous section. This generalised invariant was invented by V. O. Manturov soon after the paper [Chterental, 2015] was published; however, the definition remained unpublished since the invariant of (n + 1) variables itself was conjecturally complete.
We begin with the group
Definition 2.18. An extended virtual n-system is a set of elements
Now we construct an invariant
To be precise, consider a crossing corresponding to an i-th generator or its inverse: either σi, ζi or
