Curves on a Klein Bottle
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.