Positive diagrams for Seifert fibered spaces

Abstract

A Heegaard diagram for a 3-manifold M is a closed, oriented surface S together with a pair (X, Y) of compact 1-manifolds in S whose components serve as attaching curves for the 2-handles of the two sides of a Heegaard splitting for M. The diagram is positive if X and Y can be oriented so that the intersection number <X,Y>_p = +1 at each point p in their intersection. Every (compact, orientable) 3-manifold can be represented by a positive diagram, 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 with phg(M) never more than hg(M)+2. 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.