Nabil Sayari - The reducibilty of surgered 3-manifolds and great Scharlemann cycles

We are interested in the following question: can a reducible manifold K(r) be obtained by a Dehn surgery on a nontrivial knot K? It is conjectured that only pq-Dehn surgery on nontrivial (p,q)-cable knots gives a reducible manifold. In the talk we consider this question using the techniques that Gordon and Luecke developped for the knot complement problem. The idea is to analyse the intersection of two planar surfaces P and Q properly embedded in the exterior of the knot K. This gives two planar graphs G_P and G_Q. When K(r) is reducible, G_Q must contain a Scharlemann cycle. It follows that K(r) contains a lens space as a connected summand. We use a slight generalisation of this construction: we study the intersection of a system of punctured spheres with a level surface, Q, where the completed spheres, , give a decomposition of K(r) into m prime factors. We find that the graph G_Q contains at least m-1 different Scharlemann cycles.