These families are parametrized by 4-punctured rectangular
tori; they and their conjugates are embedded.
For the visual appearance of these surfaces it is particularly
important that the punctures are centers of polar coordinate lines.
Formulas are taken from [K1] or [K2]. The Gauss maps for these
surfaces are degree 2 elliptic functions. The cases shown are
particularly symmetric, the zeros and poles of the Gauss map
are half-period points and the punctures are there. In the
Karcher JD Saddle tower,
the diagonal of the rectangular fundamental domain joins the two
zeros, and in the Je-case it joins a zero and a pole of the Gauss map.
Under suitable choices of the modulus of the Torus these
surfaces look like a fence of Scherk
a vertical straight line (
Karcher JE case), respectively a planar symmetry
case), separating these towers. The conjugate surfaces
look qualitatively the same in the JD-cases and like a
checkerboard array of horizontal handles between vertical
planes in the JE-cases.
[K1] H. Karcher, Embedded minimal surfaces derived from Scherk's examples, Manuscripta Math. 62 (1988) pp. 83--114.
For a discussion of techniques for creating minimal surfaces with various qualitative features by appropriate choices of Weierstrass data, see either [KWH], or pages 192--217 of [DHKW].
[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