Dehn fillings of 3-manifolds and non-persistent tori (Q1807595)

The main result of this paper is that a compact, connected, orientable, irreducible 3-manifold \(M\) with incompressible torus boundary with a planar boundary slope \(r\) either contains essential tori of special type or the manifold \(M(r)\) obtained by Dehn filling is small (is a connected sum of two pieces of the form \(S^3\), \(S^1\times S^2\), or a lens space). This result may be applied to give a condition under which a knot in \(S^3\) is cabled. It also shows that if \(M\) is the exterior of a knot in \(S^3\) with \(r\) a slope of \(\partial M\) and \(M(r)\) is reducible, then at most one of its prime factors is not a lens space.