Longitudinal slope and Dehn fillings (Q1409329)

Let \(M\) be an irreducible 3-manifold with indecomposable torus boundary \(T\) and \(\gamma\) a slope on \(T\) bounding an incompressible surface of genus \(g\). It is assumed that there is a slope \(r\) which produces an essential 2-sphere by Dehn filling and \(q\) is the minimal geometric intersection number between the essential 2-sphere and the core of the Dehn filling. It is then shown that \(q= 2\) or the minimal geometric intersection number between \(\gamma\) and \(r\) is bounded by \(2g- 1\).