Producing essential 2-spheres

Let M be an orientable 3-manifold M whose boundary is a torus, and which does not contain an essential 2-sphere. The goal is to minimize the number of slopes on the boundary of M which produce essential 2-spheres by Dehn filling, via their minimal geometric intersection number. Earlier papers in this direction are [Topology 35 (2) (1996) 395–409] and [Topology Appl. 43 (1992) 213–218]. In 1996, Gordon and Luecke proved in [Topology 35 (2) (1996) 395–409] that the slopes on the boundary of M intersect exactly once. They proved this using the representations of types which come from the intersection of planar graphs. aper gives another proof of this result. It uses the combinatorics of intersecting planar graphs, yet avoids using the representations of all types, and a topological argument [J. Knot Theory Ramifications 7 (5) (1998) 549–569]. The combinatorial aspects of this paper are very basic.