A Trefoil and a Klein bottle

The trefoil is a fibered knot. Here’s the trefoil with a fiber.

Trefoil

We can take any simple closed curve on a fiber, push it off to one side, around the knot exterior in the meridional direction, and then onto the other side of the fiber.

Picture 1

Picture 2

Picture 3

Picture 4

Picture 5

This can be done so that the curve traces out an annulus with embedded interior.
Picture 6

Let’s look at this from a few other angles.

Trefoil with monodromy sweep ghosted Trefoil with monodromy annulus
Trefoild with mondoromy sweep 2 Trefoil with monodromy annulus 2

So what happens when we do this three times?

Due to the symmetry of the trefoil and our chosen presentation, we can just rotate it all to find out.

Here are some views from above.

Trefoil Klein top ghosted

trefoil klein cross sections

And from the side:

Trefoil klein cross sections 2

Trefoil monodromy Klein 6

The trefoil has monodromy with the finite order of 6.  Halfway through, at three iterations, it reverses the orientations of simple closed curves in the knot’s exterior.  So the annulus from before gives rise to an immersed Klein bottle in the exterior of the trefoil.

Trefoil monodromy Klein 5

~ by Ken Baker on May 26, 2008.

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