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 {ai,t, i = 1,. . . , n, t ∈ [0, 1]} choose points Xi,t and Yi,t, such that the interval from the upper point (upper interval) of the curve to Xi,t and the interval from Yi,t to the lower point (lower interval) do not contain intersection points. Denote the remaining part of the curve (the middle interval) between Xi,t and Yi,t by Si,t. Now, let us parametrise all curves for all t by parameters {si,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 ∈ Si,t, y ∈ Sj,t we have si,t(x) < Sj,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
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.
2.3.2Algebraic description of the invariant
The general situation in the construction of a complete invariant is the following: one constructs a new object that is in one-to-one correspondence with the described object. However, the new object might also be badly recognisable.
Now, we shall describe our invariant algebraically. It turns out that the final result is very easy to recognise. Namely, the problem is reduced to the recognition problem of elements in a free group. So, there exists an injective map from the braid group to the (n copies of) the free group with n generators that is not homomorphic.
Each braid B generates a permutation. This permutation can be uniquely restored from any admissible system of curves corresponding to B. Indeed, for an admissible system A of curves, the corresponding permutation maps i to j, where j is the ordinate of the strand with the upper point (i, 1). Denote this permutation by p(A). It is obvious that p(A) is invariant under equivalence
