Читать книгу Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory - Vassily Olegovich Manturov - Страница 23

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 can be readily checked; see Fig. 2.8.

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 σiσ_i+1σi → σ_i+1σiσ_i+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(Aσiσ_i+1σi), and in the lower part in Fig. 2.9 we show that of f(Aσ_i+1σiσ_i+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 ∈ [1, 2] the set of curves g(A, Ti) is a braid, thus g(A, Ti) generates the desired braid isotopy.

Thus, the function g(A) ≡ g(A, T) is well defined.

Now we are ready to prove the main theorem. First, let us prove the following lemma.

Lemma 2.2. Let A, A′ be two equivalent admissible systems of n curves. Then g(A) = g(A′).

Proof. Let At, t ∈ [0, 1], be a homotopy from A to A′. For each t ∈ [0, 1], At is a system of curves (possibly, not admissible). For each curve {a_i,t, i = 1,. . . , n, t ∈ [0, 1]} choose points X_i,t and Y_i,t, such that the interval from the upper point (upper interval) of the curve to X_i,t and the interval from Y_i,t to the lower point (lower interval) do not contain intersection points. Denote the remaining part of the curve (the middle interval) between X_i,t and Y_i,t by S_i,t. Now, let us parametrise all curves for all t by parameters {s_i,t ∈ [0, 1], i = 1, . . . , n} in the following way: for each t, the upper point of each curve has parameter s = 1, and the lower point has parameter s = 0. Besides, we require that for i < j and for each x ∈ S_i,t, y ∈ S_j,t we have s_i,t(x) < S_j,t(y). This is possible because we can vary parametrisations of upper and lower intervals on [0, 1]; for instance, we parametrise the middle interval of the j-th strand by a parameter in .

It is obvious that for t = 0 and t = 1 these parametrisations are admissible for both A and A′. For each t ∈ [0, 1] the parametrisation s generates a braid Bt in ³: we just take the parameter s_i,t for the strand a_i,t as the third coordinate. The strands do not intersect each other because parameters for different intervals cannot be equal to each other.

Thus the system of braids Bt induces a braid isotopy between B₀ = g(A) and B₁ = g(A′).

So, the function g is well defined on the equivalence classes of admissible systems of curves.

Now, to complete the proof of the main theorem, we need only to prove the following lemma.

Lemma 2.3. For any braid B, we have g(f(B)) = B.

Proof. Indeed, let us place B in a small neighbourhood of the “inclined plane” P in such a way that the ends of B are (i, 0, 0) and (j, 1, 1), i, j = 1. . . ,n.

Consider f(B) that lies in Oxy. It is an admissible system of curves for B. So, there exists an admissible parametrisation that restores B from f(B). By Lemma 2.1, each admissible parametrisation of f(B) generates B. So, g(f(B))= B.