Here’s a neat trick that Charles Frohman related to me a number of years ago. I think it’s in this paper , though I wasn’t able to find it in there. The trick is beautifully simple, but there don’t seem to be that many applications of it. In fact, as far as I know the only application is in Frohman’s paper. Here’s the Lemma: Let be a Heegaard surface in a 3-manifold (other than the 3-sphere) and assume there is aball such that the intersection of and is not planar. Then is reducible. (Recall that a Heegaard surface is reducible if there is a sphere that intersects it in a single essential loop.)
To prove this, we first recall that a loop in a handlebody is a core if the complement in of a neighborhood of is a compression body. The first step of proving the Lemma is to show that having a non-planar intersection implies that the ball contains a core of one of the handlebodies of the Heegaard splitting. This is not too difficult if you look at the intersection of the ball with each of the handlebodies, but it’s technical enough that I’ll leave the details to the interested reader.
Step two is much easier: Because is not the 3-sphere, the complement of is not a ball. If we remove a neighborhood of from then what’s left of will not be a ball either, so the sphere will be an essential sphere in the complement . Because is a core of one of the handlbodies, the surface is also a Heegaard splitting for .
Haken’s Lemma states that every Heegaard splitting for a reducible 3-manifold is reducible, so is reducible as a Heegaard splitting of . But the reducing sphere for in is also a reducing sphere for in , and that does it.
Of course, there are stronger things that one can say about the intersection of a strongly irreducible Heegaard splitting with a ball, such as what Scharlemann proved in , but these require a lot more work, and a much stronger assumption. The beauty of Frohman’s trick is the simplicity of its assumptions and of its proof. Unfortunately, this situation doesn’t seem to come up very often in practice. Maybe this is more important as a bit of intution, something to keep in the back of your mind even if it doesn’t make it into your final proof.
 Frohman, Charles Minimal surfaces and Heegaard splittings of the three-torus. Pacific J. Math. 124 (1986), no. 1, 119–130.
 Scharlemann, Martin Local detection of strongly irreducible Heegaard splittings. Topology Appl. 90 (1998), no. 1-3, 135–147.