The challenge was to create a minimal, knotted Möbius strip of finite total curvature. The lifting method for a well chosen planar projection of a trefoil knot gives another version of the trefoil knot that is suitable for the Björling-Schwarz formula and gives finite total curvature surfaces.

Below is an untwisted version of the corresponding Björling surface next to a 5 times twisted cylinder.

For twist 1/2, one obtains the desired Möbius strip. I have colored it with a periodic rainbow palette, and below are front and back of the same surface, so to speak.

Finally, two versions with faster spinning normals:

Resources

Mathematica Notebook

PoVRay Sources