Explore Topology

Stable Knots

Two of these three ropes are knotted in the same way. Find out which ones!

The knot puzzle has a scientific background: When deformed, the information about how knots are twisted is still preserved. Researchers seek to exploit these effects and develop physical systems such as quantum chips that are not easily disturbed by environmental influences.

Hairy Things!

Can you comb the ball and the doughnut without creating a single whirl and so that all hairs lie neatly next to each other? Try it out!


It is possible to comb the surface of the ball in such a way that only one whirl is created – but it will never be entirely free of whirls. The hairy doughnut, on the other hand, can be combed in such a way that no whirl whatsoever is created. The minimum number of whirls depends solely on the object’s topology, in this case on the number of holes (none for the sphere, one for the donut).

From the Perspective of an Ant

Image you were an ant walking on this “bottle”. What could such a walk look like?


The Klein bottle, named after the mathematician Felix Klein, is a topological curiosity: it is said to be a “non-orientable surface”. It has only one side, meaning that there is no “inside” and “outside” – as opposed to the surface of a sphere. The ant can reach every point of the Klein bottle on its walk, so it cannot be caught inside. The orientability of a surface is a topological property.

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