These surfaces show a “standard” deformation between singly and doubly periodic Scherk limits, albeit of higher genus. The divisor of the squared Gauss map is below.

Below are 8 copies of a translational fundamental piece near the two limits: The left limit consists of an experimental genus 3 Karcher-Scherk surface, the right of singly periodic Scherk surfaces with different frequencies.

Resources

Mathematica Notebook

PoVRay Sources