Reducible Dehn Surgery and Annular Dehn Surgery

Let M be a compact, orientable, irreducible, ∂-irreducible, anannular 3manifold with one component T of ∂M a torus. A slope r on T is a T isotopy class of essential, unoriented, simple closed curves on T , and the distance between two slopes r1 and r2, denoted by 4(r1, r2), is the minimal geometric intersection number among all the curves representing the slopes. For a slope r on T , we denote by M(r) the surgered manifold obtained by attaching a solid torus J to M along T so that r bounds a disk in J . Now consider two distinct slopes r1, r2 on T . There are many results showing how constraints on the topology of M(r1) and M(r2) put constraints on 4(r1, r2). For example, C. Gordon and J. Luecke [5] have shown that if M(r1) and M(r2) are reducible, then 4(r1, r2) ≤ 1. C. Gordon [3] has shown that if M contains no essential torus, and M(ri) contains an essential torus, i = 1, 2, then 4(r1, r2) ≤ 8. Y-Q Wu [8] has shown that if M(r1) and M(r2) are ∂-reducible, then 4(r1, r2) ≤ 1. In this paper, we shall estimate 4(r1, r2) when M(r1) is reducible, and M(r2) contains an essential annulus. The main result is the following theorem: