Heegaard Splittings with (disk, Essential Surface) Pairs That Intersect in One Point

We consider a Heegaard splitting M = H1 ∪S H2 of a 3-manifold M having an essential disk D in H1 and an essential surface F in H2 with |D ∩ F | = 1. From H1∪SH2, we obtain another Heegaard splitting H′ 1∪S′ H′ 2 by removing a neighborhood of F from H2 and attaching it to H1. As an application, by using a theorem due to Casson and Gordon, we give examples of 3-manifolds admitting two Heegaard splittings of distinct genera, where one of them is a strongly irreducible non-minimal genus splitting and it is obtained from the other by the above construction. We also show that all Heegaard splittings of a Seifert fibered space are related via the above construction.