## Curves on a Klein Bottle

Perhaps you’ve seen a Klein bottle before.

But perhaps you’ve not counted the curves on the Klein bottle before. I’m talking about unoriented essential simple closed curves, considered up to isotopy. How many are there?

On the torus (the orientable double cover of the Klein bottle) there are infinitely many such curves, parametrized by the rational numbers and 1/0. However, on the Klein bottle there are…. not so many. I’m not going to give a proof here, and I’ll spoil the fun after these two pictures and the jump.

Turns out there are four.

The red is orientation preserving and non-separating.

The green is orientation preserving and separating.

The blue and yellow are each orientation reversing and non-separating.

Let’s put them with the foliations shown earlier.

Let’s animate some of these things too.

Here’s another take on the one above.

The curves are shown like ribbons since I was also thinking about the twisted interval bundle over the Klein bottle.

This makes the blue and yellow into Mobius bands. And thus we can see this space as also the twisted circle bundle over the Mobius band.

The Flickr set has many more pictures.

you are right in saying that there are 4 isotopy classes of simple closed curves on K, but a circle bundle over the Mobius band it would be a three dimensional manifold, right? what a certain class of mathematicians know is that the Kleinbottle is the twisted circle bundle but over the circle… anyway, you have a super nice work!

juanmarqz said this on October 24, 2010 at 1:40 am |

juan, I think he’s saying the interval bundle over K is also a twisted circle bundle over the mobius band.

jw said this on October 24, 2010 at 11:37 pm |

ok, my fault, even so it is not clear which of the two twisted I-bundles over K is. But also is not evident that this construction isn’t the trivial one: KxI. Remember that there are three I-bundles over K…

juanmarqz said this on October 25, 2010 at 2:52 am |

Wow…. Very nice post… :D

Christian said this on October 26, 2010 at 7:59 pm |

A proof is given by considering the two types of double covers of the Klein bottle. One is the torus and the other is again the Klein bottle. The homology classes of simple-closed curves on the Klein bottle lift to the homology classes of simple closed curves or pairs of them in each double cover. A little algebra then gives you that the only nonzero homology classes possible in the Klein bottle are the four classes of the curves you have drawn above.

A more general statement about nonorientable manifolds of any dimension is in a 1979 paper by William Meeks, “Representing codimension-one homology classes on closed nonorientable manifolds by submanifolds” (Illinois Journal of Mathematics).

Dan said this on July 30, 2011 at 4:59 pm |