We consider the question of existence of embedded doubly periodic minimal surfaces in $\bfR^3$ with Scherk-type ends, surfaces that topologically are Scherk's doubly periodic surface with handles added in various ways. We extend the existence results of H. Karcher and F. Wei to more cases, and we find other cases where existence does not hold.
"Embedded, doubly periodic minimal surfaces." Experiment. Math. 9 (2) 197 - 219, 2000.