Alan Schoen’s H’-T surface comes in a natural 1-parameter family that respects the order 6 rotational symmetry of the surface. Using the same trick we used to deform the Schwarz H surfaces, we can turn H’-T sidewise and exploit that it then still has order 2 rotational symmetries about vertical axes with torus quotients to find a 2-parameter deformation.
To the left is a side view that was a top view before. The hexagonal pattern is still visible but has been squeezed. The same surface from a more generic perspective is shown on the right.
This deformation comes at a price: The divisor of the squared Gauss map on the torus quotient looks complicated. The torus quotient of the genus 4 surface is in fact rhombic, and we are using here a double cover of it to be able to work with simpler rectangular tori. The parameters satisfy a+a’=.5, b+b’=.5, c+c’=.5 so that the translation by (1+τ)/2 preserves the divisor. Moreover, Abel’s theorem tells us that we want a+.5=b+c. There still is a period condition to solve, leaving us with a 2-parameter family that I don’t think made it into the literature.
Above we see two new nodal limits of this family. These together with results of Martin Traizet allow to prove the existence of this family, at least close to the nodal limits.