Characteristic submanifold theory and toroidal Dehn filling

The exceptional Dehn filling conjecture of the second author concerning the relationship between exceptional slopes αα and ββ on the boundary of a hyperbolic knot manifold MM has been verified in all cases other than small Seifert filling slopes. In this paper, we verify it when αα is a small Seifert filling slope and ββ is a toroidal filling slope in the generic case where MM admits no punctured-torus fiber or semi-fiber, and there is no incompressible torus in M(β)M(β) which intersects ∂M∂M in one or two components. Under these hypotheses we show that Δ(α,β)≤5Δ(α,β)≤5. Our proof is based on an analysis of the relationship between the topology of MM, the combinatorics of the intersection graph of an immersed disk or torus in M(α)M(α), and the two sequences of characteristic subsurfaces associated to an essential punctured torus properly embedded in MM.