After the triply periodic genus one helicoid and the doubly periodic genus one helicoid, the next step is one of their singly periodic limits:
This is a singly periodic surface of genus 2 with 6 ends, with the same kind of skew handles as in our other surfaces. Since the singly periodic genus-one helicoid already designates a surface, this surface here currently lacks a name.
This surface desingularizes a configuration of three vertical planes: two are parallel, and the third is orthogonal. Curiously, the 6-ended Scherk surface doesn’t allow these particular end angles.
Here, the ends have been varied to 60º:
An open question is whether there is an embedded screw-motion deformation. Below is an attempt that proves nothing. Numerically, the periods are closed, and the growth rates of the helicoidal ends match numerically, but that’s all we know at the moment.
Supporting documents will follow, as usual.
