In 1996, David Hoffman, Hermann Karcher and Fusheng Wei proved the existence of an embedded, translation invariant minimal surface asymptotic to the helicoid that has genus 1 in the quotient. It played a crucial role in the construction of the genus one helicoid.
It has a pair of horizontal straight lines and contains the z-axis. Closing the horizontal period condition for this surface requires to find a rhombic torus that admits a meromorphic 1-form with just a double order pole, a double order zero, and purely real periods. There is a unique such torus, whose translation structure can be obtained by slitting the complex plane along the segment [-1,1] and identifying the top edge of [-1,0] (resp. [0,1]) with the bottom edge of [0,1] (resp. [-1,0]) using translations.
Above is an image of half of a translational fundamental piece that shows how the handle is formed.
Minimal Surfaces 