There is a fascinating video entitled, "Who cares about topology? (Inscribed rectangle problem)". It beautifully demonstrates a proof that on every closed curve in a plane, there is at least one set of four points that lie on the corners of a rectangle. The "punch line" of the proof is a demonstration that a particular surface generated from pairs of points on the curve has the topology of a Mobius strip, and that the edge of the Mobius strip maps to the curve, which requires the surface to intersect itself.
In the proof presented in the video, a Mobius strip is formed because one of the two edges of a cut needs to have its orientation reversed before the cut can be healed. That requires a half-twist, which makes a Mobius strip. It makes very good sense.
It seems obvious that any odd number of half-twists would have the same result. It also seems obvious that a knotted Mobius strip, or a knotted (2N+1) Mobius strip also would be an equally valid construction. My question has two parts: A) Is this "equally valid construction" idea correct (in the context of the proof); and B) If A is correct, does the infinity of such alternative constructions prove that there are an infinite number of solutions to the problem?
It's probably obvious that I'm not a topologist. My intuition says that the answer to B is probably "no", but I don't know if or why it would be "no".