The singly periodic genus-one helicoid

We prove the existence of a complete, embedded, singly periodic minimal sur face, whose quotient by vertical translations has genus one and two ends. T he existence of this surface was announced in our paper in Bulletin of the AMS, 29(1):77-81, 1993, Its ends in the quotient are asymptotic to one full turn of the helicoid, and, like the helicoid, it contains a vertical line. Module vertical translations, it has two parallel horizontal lines crossin g the vertical axis. The nontrivial symmetries of the surface, module verti cal translations, consist of: 180 degrees-rotation about the vertical line; 180 degrees rotation about the horizontal lines (the same symmetry); and t heir composition.