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 fillings $M(\alpha)$ and $M(\beta)$ 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 find faces of these graphs which give useful topological information about $M(\alpha)$ and $M(\beta)$, and hence, in certain cases, good upper bounds on the intersection number $\Delta(\alpha, \beta)$ of the two filling slopes.