A k-Noid is a minimal surface with k catenoidal ends. So the catenoid is a 2-Noid (in fact, the only one). For each k there is a particularly symmetric k-Noid, described in 1983 by Luquesio P. Jorge and William H. Meeks III.

 

There are many more k-Noids, and their moduli have been extensively studied, but are not yet completely understood.

Resources:

Luquesio P. Jorge and William H. Meeks III: The topology of complete minimal surfaces of finite total Gaussian curvature; Topology 1983.

Mathematica Notebook

PoVRay Sources