Using complexity bounds to study positive Heegaard diagrams of genus two

The complexity of a Heegaard splitting is the minimal intersection number of two essential simple closed curves which bound disks on either side of the splitting. In order to study the complexity of a splitting, we discuss symmetries and other properties of positive genus two Heegaard diagrams. The complementary regions in such a diagram are either octagonal or square, and we are able to find upper and lower bounds on the complexity of the splitting represented by the diagram in terms of the number of complementary squares of each of nine types. We are then able to give obstructions to a manifold being Seifert fibered in terms of this data, in addition to showing that manifolds with diagrams of a particular type are Seifert fibered. We also discuss manifolds with a Heegaard splitting of complexity two or less, which are Seifert fibered. We show how to compute the orbit space and the Seifert invariants for these manifolds