The classification of singly periodic minimal surfaces with genus zero and Scherk-type ends

Given an integer $k\geq 2$, let ${\mathcal S}(k)$ be the space of complete embedded singly periodic minimal surfaces in $\mathbb{R}^3$, which in the quotient have genus zero and $2k$ Scherk-type ends. Surfaces in ${\mathcal S}(k)$ can be proven to be proper, a condition under which the asymptotic geometry of the surfaces is well known. It is also known that ${\mathcal S}(2)$ consists of the $1$-parameter family of singly periodic Scherk minimal surfaces. We prove that for each $k\geq 3$, there exists a natural one-to-one correspondence between ${\mathcal S}(k)$ and the space of convex unitary nonspecial polygons through the map which assigns to each $M\in {\mathcal S}(k)$ the polygon whose edges are the flux vectors at the ends of $M$ (a special polygon is a parallelogram with two sides of length $1$ and two sides of length $k-1$). As consequence, ${\mathcal S}(k)$ reduces to the saddle towers constructed by Karcher.