Morgan says, "After a while, the audience is sure it's just the helicoid, when suddenly the unexpected hole appears.
It appeared that the new sort of helicoid could have any number of tunnels and still qualify as the same sort of minimal surface as the basic helicoid and the plane.
GENUS ONE In the rubbery world of topology, it's possible to imagine creating a helicoid by carefully deforming and stretching the surface of a punctured sphere rather than by expanding and twirling a fiat soap film.
Putting a tunnel in the helicoid is equivalent to adding a handle--just like the one that sprouts from a coffee mug--to a punctured sphere.
Meeks III of the University of Massachusetts and Harold Rosenberg of the Universite Denis Diderot in Paris proved that a complete, embedded minimal surface that was topologically a punctured sphere with no handles had to be either the basic helicoid or the plane.
The new proof by Weber, Hoffman, and Wolf establishes that a helicoid with one handle doesn't told back on itself.
To do so, they took advantage of a particular property of the helicoid.
The process of untwisting pushes the handles vertically farther and farther apart, until finally, at the limit, there's a helicoid with just one handle.