| We study complete, properly immersed singly periodic minimal surfaces in R3 that have genus one in the quotient and catenoid ends. A well-known example in this class is the fence of catenoids, which is translation invariant and has two ends in the quotient. While it is impossible to have translation invariant examples with fewer ends in the quotient, we find screw-motion invariant minimal surfaces with just one end, called one-noids. We use geometric clues to determine candidates for the Weierstrass data of our surfaces. We then adjust the parameters in these data using symmetry and intermediate value arguments to ensure that the surfaces are well-defined and satisfy all desired properties. |
