Annular and boundary reducing Dehn fillings

Let M be a simple 3-manifold, i.e. one that contains no essential sphere, disk, annulus or torus, with a torus boundary component ∂0M. One is interested in obtaining upper bounds for the distance (intersection number) Δ(α, β) between slopes α, β on ∂0M such that Dehn filling M along α, β produces manifolds M(α), M(β) that are not simple. There are ten cases, according to whether M(α) (M(β)) contains an essential sphere, disk, annulus or torus. Here we show that if M(α) contains an essential annulus and M(β) contains an essential disk then Δ(α, β)⩽2. This completes the determination of upper bounds for Δ(α, β) in all ten cases.