2.3The curve algorithm for braids recognition
Below, we shall give a proof of the completeness of the invariant for the braid group elements invented by Artin, see [Gaifullin and Manturov, 2002].
The invariant to be constructed has a simple algebraic description as a map (non-homeomorphic) from the braid group Br(n) to the n copies of the free group in n generators.
Several generalisations of this invariant, such as the spherical and cylindrical braid group invariants, are also complete. The key point of such a completeness is that these invariants originate from several curves, and the braid can be uniquely restored from these curves.
Moreover, this approach finally led to the algorithmic recognition of virtual braids due to Oleg Chterental [Chterental, 2015].
Fig. 2.5Decomposing a pure braid
2.3.1Construction of the invariant
Let us begin with the definition of notions that we are going to use, and let us introduce the notation.
Definition 2.10. By an admissible system of n curves we mean a family of n non-intersecting non-self-intersecting curves in the upper half plane {y ≥ 0} of the plane Oxy such that each curve connects a point having ordinate zero with a point having ordinate one and the abscissas of all curve ends are integers from 1 to n. All points (i, 1), where i = 1, . . . , n, are called upper points, and all points (i, 0), i = 1, . . . , n, are called lower points.
Definition 2.11. Two admissible systems of n curves A and A′ are equivalent if there exists a homotopy between A and A′ in the class of curves with fixed endpoints lying in the upper half plane such that no interior point of any curve can coincide with any upper or lower point during the homotopy.
Analogously, the equivalence is defined for one curve (possibly, self-intersecting) with fixed upper and lower points: during the homotopy in the upper half plane no interior point of the curve can coincide with an upper or lower point.
In the sequel, admissible systems will be considered up to equivalence.
Let β be a braid diagram on the plane connecting the set of lower points {(1, 0), . . . , (n, 0)} with the set of upper points {(1, 1), . . . , (n, 1)}. Consider the topmost crossing C of the diagram β and push the lower branch along the upper branch to the upper point of it as shown in Fig. 2.6.
Naturally, this move spoils the braid diagram: the result, shown in Fig. 2.6 is not a braid diagram. The advantage of this “diagram” is that we have a smaller number of crossings.
Fig. 2.6Pushing the upper crossing
Fig. 2.7Pushing the next crossing
Now, let us do the same with the next crossing. Namely, let us push the lower branch along the upper branch towards the end. If the upper branch is deformed during the first move, we push the lower branch along the deformed branch (see Fig. 2.7).
Reiterating this procedure for all crossings (until the lowest one), we get an admissible system of curves. Denote its equivalence class by f(β).
Theorem 2.3. The function f is a braid invariant; i.e., for two diagrams β, β′ of the same braid we have f(β) = f(β′).
Proof. Having two braid diagrams, we can write the corresponding braid-words, and denote them by the same letters β, β′. We must prove that the admissible system of curves is invariant under braid isotopies. As we shall see, this statement is very simple from the algebraic point of view, but here it is useful for our purposes to consider it using curves techniques.
Fig. 2.8Invariance of f under the second Reidemeister move
The invariance under the commutation relations σiσj = σjσi, |i − j| ≥ 2, is obvious: the order of pushing two “far” branches does not change the result.
The invariance under
In the leftmost part of Fig. 2.8, the dotted line indicates the arbitrary behaviour for the upper part of the braid diagram. The rightmost part of Fig. 2.8 corresponds to the system of curves without
Finally, the invariance under the transformation σiσi+1σi → σi+1σiσi+1 is shown in Fig. 2.9. In the upper part (over the horizontal line) in Fig. 2.9 we demonstrate the behaviour of f(Aσiσi+1σi), and in the lower part in Fig. 2.9 we show that of f(Aσi+1σiσi+1) for an arbitrary braid A. In the middle-upper part, a part of the curve is shown by a dotted line. By removing it, we get the upper-right picture which is just the same as the lower-right picture.
Note that the behaviour of the diagram in the upper part A of the braid diagram is arbitrary. For the sake of simplicity it is pictured by three straight lines.
Thus we have proved that
This completes the proof of the theorem.
In fact, the following statement holds.
Theorem 2.4. The function f is a complete invariant.
In order to prove Theorem 2.4, we should be able to restore the braid from its admissible system of curves.
Fig. 2.9Invariance of f under the third Reidemeister move
In the sequel, we shall deal with braids whose end points are (i,
