Toroidal and annular Dehn fillings

Abstract

Suppose $M$ is a hyperbolic 3-manifold which admits two Dehn fillings $M(r_1)$ and $M(r_2)$ such that $M(r_1)$ contains an essential torus and $M(r_2)$ contains an essential annulus. It is known that $Δ= Δ(r_1, r_2) \leq 5$. We will show that if $Δ= 5$ then $M$ is the Whitehead sister link exterior, and if $Δ= 4$ then $M$ is the exterior of either the Whitehead link or the 2-bridge link associated to the rational number $3/10$. There are infinitely many examples with $Δ= 3$.