One can easily replace the annular ends of the translation-invariant Karcher surfaces by periodic Enneper ends. This begins with the case of two ends, when the Scherk surface is just a plane.

An additional parameter controls how fast the ends wiggle. Below are two different choices for 6 ends, with a top view of the left surface.

I don’t know whether these surfaces have any mathematical significance beyond being exercises in minimal surface design. But the wheel below would possibly make a decent tool for cutting herbs.

Scherk-Enneper
Resources

Mathematica Notebook and CDF

PoVRay Sources