A Trefoil and a Klein bottle
The trefoil is a fibered knot. Here’s the trefoil with a fiber.
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.
Let’s look at this from a few other angles.
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.
And from the side:
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.