By Richard Schoen’s theorem, it is not possible to make a genus one catenoid, but one can make a genus one surface with Enneper ends. In the most symmetric case, one still has a 1-parameter family.
It might be amusing to study the limits for higher genus analogues. I also wonder what happens to the handle when one rotates the two Enneper ends as in the twisted version of the Double Enneper surface.
Resources
PoVRay Sources
Related Surfaces
- Double Enneper
- Fence of Catenoids
- Torus with Catenoid and Enneper End
- Dihedralizations
Minimal Surfaces 