Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory. Vassily Olegovich Manturov

Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory

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.

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 of A.

Let n be an integer. Consider the free product G of n copies of the group with generators a₁, . . . , an. Denote by Ei the right residue classes in G by {ai}; i.e., g₁, g₂ ∈ G represent the same element of Ei if and only if for some k.

Definition 2.13. An n-system is a set of elements e₁ ∈ E₁, . . . , en, ∈ En;

An ordered n-system is an n-system together with a permutation from Sn.

Proposition 2.1. There exists an injective map from equivalence classes of admissible systems of curves to ordered n-systems.

Proof. Since the permutation for equivalent admissible systems of curves is the same, we can fix the permutation s ∈ Sn and consider only equivalence classes of admissible systems of curves with permutation s (i.e., with all lower points fixed depending on the upper points in accordance with s). Thus we only have to show that there exists an injective map from the set of admissible systems of n curves with fixed lower points to n-systems.

To complete the proof of the proposition, it suffices to prove the following.

Lemma 2.4. Equivalence classes of curves with fixed points (i, 1) and (j, 0) are in one-to-one correspondence with Ei.

Proof. Denote by Pn. Obviously, π₁(Pn) ≅ G. Consider a small circle C centred at (i, 1) for some i with the lowest point X on it. Let ρ be a curve with endpoints (i, 1) and (j, 0). Without loss of generality, assume that ρ intersects C in a finite number of points. Let Q be the first such point that one meets while walking along ρ from (i, 1) to (j, 0). Thus we obtain a curve ρ′ coming from C to (j, 0). Now, let us construct an element of π₁(Pn, X). First it comes from X to Q along C clockwise. Then it goes along ρ until (j, 0). After this, it goes along Ox to the point (i, 0). Then it goes vertically upwards till the intersection with C in X. Denote the constructed element by W(ρ).

If we deform ρ outside C, then we obtain a continuous deformation of the loop, thus W(ρ) stays the same as an element of the fundamental group. The deformations of ρ inside C might change W(ρ) by multiplying it by ai on the left side. So, we have constructed a map from equivalence classes of curves with fixed points (i, 1) and (j, 0) to Ei.

The inverse map can be easily constructed as follows. Let W be an element of π₁(Pn, X). Consider a loop L representing W. Now consider the curve L′ that first goes from

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