Deform the boundary of the Möbius band in the proper way

Question

The boundary of a Möbius band is an unknot in $\mathbb{R}^3$, so we can deform it via an ambient isotopy to the standard circle in a plane. In this way, how does the Möbius band look like (i.e. how the standard circle bounds a Möbius band in $\mathbb{R}^3$)? I can hardly imagine it. Could someone visualize it?

Answer 1

Werner Boy's Surface (with a Hole)

A really nice way to represent it is Boy's Surface with a hole poked in it.

Boy's surface is an immersion of the 2-dimensional projective plane $P$ into Euclidean $\Bbb R^3$. It's not an embedding because it has self-intersections and a triple point. But it's smooth everywhere and has no pinches or creases or cusps.

Now $P=M+D$ where $M$ is a Möbius band and $D$ is a disk, i.e. gluing $D$ and $M$ together gives $P$. The other way round, poking a hole (removing $D$) from $P$ will yield $M$. Notice that we can pick to remove a disc that does not intersect with the remaining $M$.

The page above has images of a sculpture in Oberwolfach, which is specially nice because it is minimizing Willmore energy using Bryant-Kusner-parametrization. Removing the top dome would leave a Möbius band with a flat circle as its boundary.

And there are also nice animations, cf. youtube: Boy's surface.

Great David Hilbert conjectured that no such immersion exists, but Werner Boy proved him wrong.

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Möbius Snail

A different representation with circular boundary is the "Sudanese Möbius strip" which has no self-intersections. An image and description are here.

Adding a disc in order to complete it to the projective plane will introduce self-intersections and creases, though, i.e. places where the surface is not smooth. More renderings are here.

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Möbius Wheel (from Boy's Surface)

Returning to Boy's surface, here are some renderings of a Möbius strip with (almost) circular boundary: W1, W2, W3, W4. These renderings were created by changing the Bryant-Kusner parametrization in such a way that only a part is rendered (which effectively pokes a hole) and by pulling the boundary of the resulting Möbius strip to the equatorial plane. The resulting shape has $D_{2\cdot 3}$ dihedral symmetry with a triple-point in the center. There is a detailed description.

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All images from Wikipedia / Wikimedia Commons.