Just as one can add handles to the Chen-Gackstatter surface to obtain the higher genus Chen-Gackstatter-Thayer surfaces, one can add handles to the torus with one catenoid and one Enneper end:
The flexibility of the Enneper end makes this a 1-parameter family.
For the higher genus Chen-Gackstatter-Thayer surfaces, there is numerical evidence that with increasing genus they will converge to the singly periodic Scherk surface. Will these surfaces here have the same limit, or will the Enneper end persist? Can the limit be controlled by choosing the size of the Enneper end suitably?
Minimal Surfaces 