November 25, 2006
Knotted minimal surfaces
All knots can be spanned by an oriented nonsingular minimal surface, and my particular favorite is the threefoil knot surface. While playing around with surface evolver I got the idea of attaching a catenoid handle to it, producing this rather neat surface.
An obvious generalisation is to add a catenoid to the top too, producing a three-ended minimal surface.
I wonder how the knotted end of the surface continues. It seems rather likely that it will self-intersect and behave badly, although the "blades" look somewhat helicoidal. I don't know any reliable method of extending a meshed minimal surface onwards (lots of numerical instability issues).
Here is a rendering of the surface.
ADDENDUM: An obvious extension is to splice two knot ends together using the catenoid. The result is a nicely messy surface. There are of course two variants, depending on the parity of the knots. The width of the neck between them depends on their distance like in an ordinary catenoid; beyond a certain point the minmal surface pinches off.
Here is a rendering of a fairly "thick-necked" version of the surface.