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(α) has no non-abelian free subgroup, and if M(β) is a reducible manifold which is not homeomorphic to S ¹ × S ² or P ³ # P ³, then Δ(α, β) ≤ 5. Under the same condition on M(β), it is shown that if M (α) is Seifert fibered, then Δ(α, β) ≤ 6. Moreover, in the latter situation, character variety techniques are used to characterize the topological types of M(α) and M(β) in case the bound of 6 is attained.