These surfaces can be interpreted as symmetrizations of the Double Enneper surface, or as a variation of the Jorge-Meeks k-Noids with the catenoidal ends being replaced by Enneper ends.
k=3 
k=5 
k=12
These surface probably don’t appear in the literature. They are not embedded, have high degree of the Gauß map, and their construction doesn’t pose any interesting challenges. The Enneper-Weierstraß representation is a straightforward modification of the one for the Jorge Meeks k-Noids:
r=0.9 
r=1.1
There even is an additional parameter r that squeezes the surface. I have, unfortunately, a weakness for the baroque ornaments these Enneper ends provide.
Resources
Mathematica Notebook
PoVRay Sources
Minimal Surfaces 