The prismatic k-Noids are obtained by placing the ends of a k-Noid at the vertices of a regular prism. This leaves a free parameter that controls the limiting normals of the ends. The image to the right below suggests another interpretation: Three catenoids, joined by a triple neck.

The most symmetric case is the cuboid to the left. To the right a 14-Noid for amusement. 

I should also mention that the handles connecting the catenoids can also grow inwards. This creates a little bit of a mess which is bearable in the extreme case of a 4-Noid. The left example is also on page 31b of Hermann Karcher’s Tokyo notes.

An interesting question remotely related to these surfaces is the problem to construct a bounded, embedded minimal surface contained in a ball. One could imagine using catenoidal necks connecting concentric spheres inwards and outwards. The spheres would be closer and closer to each other with increasing radius, requiring more and more smaller and smaller necks, so that the final surface would look from the outside just like a sphere. I would call this conjectured object the Death Star.

Resources

Mathematica Notebook

PoVRay Sources.