Ropes have an even number of ends

Abstract

On 13-01-2024 the annual wintersymposium of the Koninlijk Wiskundig Genootschap (KWG) was held in the academiegebouw in Utrecht. The symposium had the theme ``inzichtelijk abstract''. Thomas Rot gave a lecture on his favourite theorem from topology. This article is a written account of this lecture. Audience comprised mostly of high school teachers and that is also the target audience of this article. The slides (in Dutch), which contain more pictures, are available~[8].

Figures (17)

• Figure 1: An even number of points is the boundary of some ropes, while an odd number of points is not the boundary of some ropes.
• Figure 2: How to quickly determine if you can escape a maze as on the left? Draw any curve to the outside and determine if the number of intersections with the maze is even or odd. If it is even, you can escape, if it is odd you can't. In the maze above you can escape from the blue point, as the number of intersections is four (even), but you cannot escape from the turquoise point, as the number of intersections is three (odd).
• Figure 3: Counting intersections should only be done when the curves meet transversely. On the left a transverse intersection is depicted, while on the right a non-transverse intersection is depicted: the two curves are tangent to each other. A somewhat deep fact in differential topology is that generically curves only have transverse intersections. By an arbitrary small perturbation of the curve all intersections will be transverse, and transversality is preserved under small enough perturbations.
• Figure 4: Can we escape this very simple maze on the left? Yes: the curve in blue intersects the maze twice, which is even. Why is this number even? Imagine deforming the blue curve to the red curve as in the middle picture. The red curve is an actual escape route. We can keep track of the intersections with the maze during this deformation. This is depicted on the right: In brown the intersections of all the curves with the maze are drawn. These are ropes, and the ends of these ropes are the intersections of the original curve with the maze. This shows that the intersection is even if we can escape. A slightly more complicated argument shows that this condition is also sufficient. To avoid a flood of complaints I need to be precise: we assume that the maze is a connected closed simple curve.
• Figure 5: The number of intersections of a closed surface and a closed curve in three dimensions is even. The proof is analogous of the proof described in Figure \ref{['fig:homotopy1']}: deform the curve to lie outside the surface and keep track where the family of curves intersect the surface. This traces out ropes, whose ends are the original intersections, which must therefore be even. Note that one of the ropes is a closed curve itself, which did not occur in Figure \ref{['fig:homotopy1']}.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 2.1
• Theorem 3.1
• Theorem 3.2
• Corollary 3.3
• Theorem 5.1
• Theorem 5.2
• Theorem 6.1
• Theorem 6.2
• Theorem 6.3