Dehn fillings of 3-manifolds and non-persistent tori

Abstract

We prove that if M is a connected, compact, orientable, irreducible 3-manifold with incompressible torus boundary and r is a planar boundary slope in ∂M, then either M contains an essential non-persistent torus with boundary slope r, or a closed essential torus which compresses in the Dehn filling M(r), or M(r) is `small' (meaning M(r) is a manifold of the form L#L′, where each factor L,L′ is either S3,S1×S2, or a lens space). We also give an example of an infinite family of hyperbolic manifolds with torus boundary, each of which contains two essential non-persistent punctured tori of distinct boundary slopes associated to two reducible Dehn fillings. The main result may also be applied to give a condition under which a knot in S3 with a reducible surgery must be cabled. We further study the reducible surgeries on knots in S3, and show that each prime factor in such a surgery is, with at most one exception, a lens space.