We can think of a mathematical knot as a knotted piece of string (or in our case, wire) with its free ends joined. Examples are shown below.
There is a remarkable theorem that every knot can be realized as the boundary of a surface. Moreover, Herbert Seifert produced a very simple algorithm for constructing an orientable surface with boundary from any given knot.

Below is a video that I made (sorry for the amateurish quality!) of knotted pieces of wired dipped into bubble mix. You can see that the knots are the boundaries of the bubble surface. The first example is a surface with a figure eight knot as its boundary. The second example is a bubble with a trefoil knot as its boundary.

It is interesting that neither of these surfaces is a Seifert surface, because they are both one-sided (I noticed that the first one was one sided after I published the video). In particular the bubble for the trefoil knot is a Möbius band (a cylinder with three half-twists). Silly aside: wouldn’t it be cool if you could blow these bubbles and produce Klein bottles (Klein bubbles?)? Ha ha ha.

I made a third knot (the 3-twisted double of the unknot, if you must know) but I could not make it the boundary of a bubble. Most of it was fine, but there was one location in which two bubble surfaces would cross. This knot can be the boundary of a surface, but apparently for the configuration I chose the surface is not a minimal surface.

That got me to thinking, is it possible to find a configuration of the knot so that the bubble is a surface? More generally, I have the following question.

True or false: any knot can be obtained as the boundary of a minimal surface.

Does anyone know the answer? I suspect that it is false, but maybe not.

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Kindergarten mathematics « Division by Zeroon October 12, 2009at 3:49 pm