I have tried numerous times to come up with a surface that could proudly call itself the Diamond Surface with a Handle, in vain. It turns out this is easier with *two* handles.

The identifications are by horizontal translations and a vertical translation together with a diagonal translation by a half period.

The divisor of the squared Gauss map on the quotient torus is shown above. The parameters a and b need to satisfy a+b=1/2, and there is a 1-dimensional period problem to solve. The translation structures of G dh and 1/G dh are show below.

The two limits consist of noded planes at one end, and doubly periodic Karcher-Scherk surfaces at the other.

In between, the surfaces become surprisingly tall.

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Mathematica Notebook

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