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, 0, 0) and (j, 1, 1) with all strands coming upwards with respect to the third projection coordinates. They obviously correspond to standard braids with upper points (j, 0, 1). This correspondence is obtained by moving neighbourhoods of upper points along Oy.
Consider a braid B and consider the plane P = {y = z} in Oxyz. Let us place B in a small neighbourhood of P in such a way that its strands connect points (i, 0, 0) and (j, 1, 1), i, j = 1, . . . , n. Both projections of this braid on Oxy and Oxz are braid diagrams. Denote the braid diagram on Oxy by β.
The next step now is to transform the projection on Oxy without changing the braid isotopy type; we shall just deform the braid in a small neighbourhood of a plane parallel to Oxy.
It turns out that one can change abscissas and ordinates of some intervals of strands of b in such a way that the projection of the transformed braid on Oxy constitutes an admissible system of curves for β.
Indeed, since the braid lies in a small neighbourhood of P, each crossing on Oxy corresponds to a crossing on Oxz. Thus, the procedure of pushing a branch along another branch in the plane parallel to Oxy deletes a crossing on Oxy, preserving that on Oxz.
Thus, we have described the geometric meaning of the invariant f.
Definition 2.12. By an admissible parametrisation (in the sequel, all para-metrisations are thought to be smooth) of an admissible system of curves we mean a set of parametrisations for all curves by parameters t1, . . . , tn such that at the upper points all ti are equal to one, and at the lower points ti are equal to zero.
Any admissible system A of n curves with an admissible parametrisation T generates a braid representative: each curve on the plane becomes a braid strand when we consider its parametrisation as the third coordinate. The corresponding braid has end points (i, 0, 0) and (j, 1, 1), where i, j = 1, . . . , n. Denote it by g(A, T).
Lemma 2.1. The result g(A, T) does not depend on T.
Proof. Indeed, let us consider two admissible parametrisations T1 and T2 of the same system A of curves. Let Ti, i ∈ [1, 2], be a continuous family of admissible parametrisations between T1 and T2, say, defined by the formula T = (i − 1)T1 + (2 − i)T2. For each i ∈ [1, 2], the curves from Ti do not intersect each other, and for each i