Positive diagrams for Seifert fibered spaces

Every (compact, orientable) 3-manifold can be represented by a positive Heegaard diagram (all curve crossings in the same direction), but the argument for this suggests that the minimal genus, phg(M), for a positive diagram may be much larger than the minimal genus, hg(M), among all diagrams. This paper investigates this situation for the class of closed orientable Seifert manifolds over an orientable base. We show that phg(M)=hg(M) for most of these manifolds and phg(M)⩽hg(M)+2 always holds. The cases phg(M)>hg(M) are determined and occur when the minimal genus splitting is “horizontal”. The arguments provide an alternate proof distinguishing these from “vertical” splittings.