These singly periodic surfaces are parametrized (aa) by
rectangular tori; our lines extend polar coordinates around
the two punctures to the whole Torus. The surfaces look
like a fence of catenoids, joined by handles; they were made by
Karcher and Hoffman, responding to the suggestive skew 4-noids.
The morphing parameter aa is the modulus (a function of the
length ratio) of the rectangular Torus.
For a discussion of techniques for creating minimal surfaces with various qualitative features by appropriate choices of Weierstrass data, see:
[KWH] H. Karcher, F. Wei, and D. Hoffman, The genus one helicoid, and the minimal surfaces that led to its discovery, in “Global Analysis in Modern Mathematics, A Symposium in Honor of Richard Palais' Sixtieth Birthday”, K. Uhlenbeck Editor, Publish or Perish Press, 1993
[DHKW] U. Dierkes, S. Hildebrand, A. Kuster, and O. Wohlrab, Minimal Surfaces I, Grundlehren der math. Wiss. v. 295 Springer-Verlag, 1991