Characteristic subsurfaces and Dehn filling

Let $M$ be a simple knot manifold. Using the characteristic submanifold theory and the combinatorics of graphs in surfaces, we develop a method for bounding the distance between the boundary slope of an essential surface in $M$ which is not a fiber or a semi-fiber, and the boundary slope of a certain type of singular surface. Applications include bounds on the distances between exceptional Dehn surgery slopes. It is shown that if the fundamental group of $M(\alpha)$ has no non-abelian free subgroup, and if $M(\beta)$ is a reducible manifold which is not homeomorphic to $S^1 \times S^2$ or $P^3 \char93 P^3$, then $\Delta(\alpha, \beta)\le 5$. Under the same condition on $M(\beta)$, it is shown that if $M(\alpha)$ is Seifert fibered, then $\Delta(\alpha, \beta)\le 6$. Moreover, in the latter situation, character variety techniques are used to characterize the topological types of $M(\alpha)$ and $M(\beta)$ in case the bound of $6$ is attained.