The surfaces here are doubly periodic variations of the translation invariant catenoid (aka the fence of catenoids).
The Enneper-Weierstrass data are a touch more complicated than for the plane with catenoidal ends, but there are still no period problems to be solved. Other arrangements of the catenoids are possible, too:
These surfaces are akin to some classical triply periodic surfaces of Hermann Amandus Schwarz and Alan Schoen. It would be interesting to know whether this is more than superficial. For instance, do the nodal limits with neck sizes shrinking to zero satisfy the same Traizet balance equations?