Combinatorial Methods in Dehn Surgery

This is an expository paper, in which we give a summary of some of the joint work of John Luecke and the author on Dehn surgery. We consider the situation where we have two Dehn llings M() and M() on a given 3-manifold M, each containing a surface that is either essential or a Heegaard surface. We show how a combinatorial analysis of the graphs of intersection of the two corresponding punctured surfaces in M enables one to nd faces of these graphs which give useful topological information about M() and M(), and hence, in certain cases, good upper bounds on the intersection number (;) of the two lling slopes.