The Plateau construction that works for the translation invariant Fischer-Koch Surfaces can be modified to create screw-motion invariant surfaces by changing the angle between the horizontal lines as shown above.
This leads for every odd k to a 1-parameter family of surfaces with 2k helicoidal ends and genus one in the quotient by a suitable screw motion. They limit in two different parking garage structures with k helicoids. In one case, all helicoids have the same handedness, and they are placed at the vertices of a regular (k-1)-gon and its center. In the other limit case, helicoids with the same handedness are placed at the vertices of a regular (k+1)-gon, and an additional helicoid with opposite handedness is located at its center.
The first case shows that these surfaces can be interpreted as screw-motion invariant Karcher-Scherk surfaces with an additional helicoid along the screw motion axis.
Below are two animations showing this deformation.
- Screw-motion invariant Karcher-Scherk surfaces
- Translation invariant Fischer-Koch surfaces
- Screw motion invariant Fischer-Koch-Freese surfaces
- Translation invariant helicoid with handles.