Characteristic submanifold theory and toroidal Dehn filling

Abstract

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