homotopy class. Then there exists a homotopy st from s0 to s1 such that st is self transversal for every t and the number of double points (counted with multiplicities) of st does not depend on t.
Theorem 3.4. Let s0 be a curve in general position in the surface Σ. Then there is a homotopy st from so to a curve s1 which has the minimal number of double points in the homotopy class, and the number of double points in st does not increase with t.
A solution to Turaev’s problem follows immediately from these theorems: given two curves s0 and s1, one can homotop them to minimal curves
Remark 3.2. Theorem 3.3 is not valid for unions of curves. For example, one cannot permute the components in Fig. 3.20 without adding new intersection points. On the other side, Theorem 3.4 remains true.
Fig. 3.20Non-equivalent minimal pairs of curves
3.2.1The disc flow
In paper [Hass and Scott, 1994] a simple construction of flow of curves is given which homotops a finite set of curves in a surface in such a way that:
(1)The number of self-intersections does not increase for each curve,
(2)The number of intersections does not increase for each pair of curves,
(3)Each curve either disappears in finite time or becomes close to a geodesic,
(4)This flow can be extended continuously to k-parameter families of curves.
This algorithm can be programmed easily and admits a generalisation to higher dimensions.
Let γ be a piecewise smooth immersed curve in a Riemannian surface F. Let us cover F by convex discs D1, . . . , Dn whose radii are smaller than the injectivity radius. We choose discs to be in general position so that any point in F belongs to the boundary of no more than two discs, the boundaries of the discs intersect transversely the curve γ, and the discs of halved radii with the same centers cover still the surface F. Such a cover will be called well situated with respect to γ. Let us number the discs D1, D2, . . . cyclically so that Dn+i = Di for any i ≥ 1.
Roughly speaking, a disc flow will be defined as homotopy of each arc in γ ∩ D1 to the unique geodesics with the same ends, then the process repeats for each disc D2, D3, . . . .
We investigate properties and convergence of the flow, then we show that the number of intersection points does not increase.
Fig. 3.21Disc flow
We start with several combinatorial lemmas.
We call an embedded loop of the curve γ any embedded subarc of γ with coinciding ends which bounds an embedded disc, see Fig. 3.22 left. An embedded bigon is a pair of subarcs of γ with common ends that bound an embedded disc, see Fig. 3.22 right. An embedded loop (bigon) is called innermost if the correspondent embedded disc does not contain any other arcs of γ.
Fig. 3.22An embedded loop and an embedded bigon
Lemma 3.1. 1. Given a triangle ABC and several embedded curves crossing it, such that any two curves have at most one intersection point in the triangle and neither curve intersects the side BC, then on each side AB and AC there is an innermost triangle adjacent to it.
2. Given an innermost bigon with embedded curves crossing it, for each edge of the bigon there is an innermost triangle adjacent to it.
Proof. 1. The proof is by induction on the number k of the intersecting curves. For k =1 the statement is evident. Assume that the statement holds for k = n. Consider the case of k = n + 1 intersecting curves. Let D be the intersection point on the side AB closest to A, and E be the intersection point on the side AC which belongs to the same intersecting curve as D. Then the triangle EAD contains at most n intersecting curves and none of them intersects the side AD. By induction assumption there is an innermost triangle adjacent to the side EA, thus, adjacent to AC. The reasonings for the side AB are analogous.
2. Since the bigon is innermost, any two curves inside intersect in at most one point. Draw a curve intersecting the bigon near to one of its vertices and not intersecting other curves. This curve splits the bigon into two triangles one of which is innermost. By the first statement, there are innermost triangles adjacent to edges of the triangle of the splitting. These innermost triangles cannot be adjacent to the splitting curve, thus, we can remove the supplementary splitting curve.
Remark 3.3.
(1)The lemma does not require the intersecting curves to be in general position. There can be multiple intersection points.
(2)The second statement of the lemma remains valid if one takes an embedded loop, which does not contains any bigons, instead of the bigon. That is, there is an innermost triangle adjacent to the edge of such loop.
Lemma 3.2. A finite set of piecewise smooth transversal curves in a convex disc can be homotoped (with respect to the boundary) to a set of geodesics such that the number of self-intersection and intersection points of the curves do not increase during the homotopy.
Proof. The proof is based on induction on double point number (with multiplicity). We count here an intersection point of multiplicity k as
Assume that the number of double points is minimal. Then the curves do not have self-intersections and any two curves intersect in at most one point.
If there are closed curves, then take an innermost closed curve. This curve can be contracted to a point without intersecting other curves. Repeating this operation we homotop our set of curves to a set without closed curves. So we can suppose there are no closed curves in the set.
