A Karcher JE saddle tower is an embedded surface in three-dimensional Euclidean space 
belonging to a doubly periodic family of minimal
surfaces that Karcher obtained from Scherk's
minimal surfaces. The initials JE refer to a Jacobi
elliptic function occurring in the data for its Enneper-Weierstrass
parameterization (Karcher and Palais 1999).
Each member and its conjugate are embedded surfaces parametrized by four-punctured rectangular tori. Their Gauss maps are degree-2 elliptic functions. In the symmetric cases, their zeros and poles occur at half-periods, and the diagonal of a rectangular fundamental domain joins a zero to a pole (Karcher 1988, 1989).
The translational part of the symmetry group is isomorphic to 
and is generated by vertical and horizontal translations. The horizontal one is parallel to a straight
line contained in the minimal
surface, and a rotation through 
about the same axis is
another symmetry (Karcher and Palais 1999).
An example appeared on the cover of the June/July 1999 issue of Notices of the American Mathematical Society (Karcher and Palais 1999).
