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 case of 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 Saddle Towers — with a vertical straight line ( Karcher JE case), respectively a planar symmetry line ( JD 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.
- [K2] Construction of Minimal Surfaces, H Karcher , in “Surveys in Geometry”, Univ. of Tokyo, 1989, and Lecture Notes No. 12, SFB 256, Bonn, 1989, pp. 1 — 96.

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