\(\partial\)-reducible Dehn surgery and annular Dehn surgery (Q1292744)

Let \(M\) be a compact, orientable, irreducible, \(\partial\)-irreducible, anannular 3-manifold with a torus, \(T\), on its boundary. Surgeries filling in \(T\) with a solid torus are defined by giving a simple curve \(r\) on \(T\) (not homologically trivial on \(T)\) and attaching a solid torus \(S\) so that \(r\) bounds a disc in \(S\). Denote the resulting manifold \(M(r)\). The authors prove that if \(M(r_1)\) is \(\partial\)-reducible and \(M(r_2)\) contains an essential annulus then the (minimal) intersection number of (the isotopy classes) of \(r_1\) and \(r_2\) is \(<4\).