The deformation family of this surface imperceptibly moves the small handle up and down. The divisor of the squared Gauss map is below.

Below are 8 copies of a translational fundamental piece near the two limits: They show the subtle differences. The left limit consists of genus 0/1 Karcher-Scherk surfaces joined end-to-end with catenoidal stitches, the right limit gets by without stitches but uses genus 1/2 Karcher-Scherk surfaces instead.

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Mathematica Notebook