Saddle tangencies and the distance of Heegaard splittings

Abstract

We give another proof of a theorem of Scharlemann and Tomova and of a theorem of Hartshorn. The two theorems together say the following. Let $M$ be a compact orientable irreducible 3–manifold and $P$ a Heegaard surface of $M$. Suppose $Q$ is either an incompressible surface or a strongly irreducible Heegaard surface in $M$. Then either the Hempel distance $d\left(P\right)\le 2genus\left(Q\right)$ or $P$ is isotopic to $Q$. This theorem can be naturally extended to bicompressible but weakly incompressible surfaces.