The construction of spherical k-Noids more general than the Jorge-Meeks k-Noids is a purely algebraic problem: One takes a sphere, chooses points representing the ends, writes down general rational functions for the Enneper-Weierstraß data which have the correct distribution of poles and zeroes, and then determines all parameter values so that all residues are real. However, the interplay between conformal data (e.g. the location of the points, for instance) and geometric (e.g. the direction of the limiting normals at the ends) is very subtle and little understood.

Symmetric placement of the points helps, like here at the vertices of a pyramid over a regular base. The image above shows that after fixing the conformal type there still is some freedom left. The right case is the most regular case, sometimes called the tetrahedroid, which also appears among the antiprismatic k-Noids.

Resources

Mathematica Notebook and CDF

PoVRay Sources