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Given an integer k 2, let S(k) be the space of complete embedded singly periodic minimal surfaces in R3, which in the quotient have genus zero and 2k Scherk-type ends. It is well-known that S(2) consists of the 1-parameter family of singly periodic Scherk minimal surfaces. We prove that for each k 3, there exists a natural one-to-one correspondence between S(k) and the space of convex unitary nonspecial polygons through the map which assigns to each M 2S (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, S(k) reduces to the saddle towers constructed by Karcher (5).

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