Dehn fillings on a two torus boundary components 3-manifold (Q1860764)

Let \(X\) be a 3-manifold with two incompressible torus boundary components: \(\partial X= T_1\cup T_2\). Suppose further that \(P\) and \(Q\) are planar surfaces properly embedded in \(X\). By analysing the possible intersections of \(P\) and \(Q\) in \(X\), the author attempts to extend to two torus boundary components the important analysis carried out by Gordon and Luecke for one torus boundary component. That is, let \(\alpha_i\) be a curve parallel to the curves of intersection of \(P\) with \(T_i\) and let \(\beta_i\) be a curve parallel to the curves of intersection of \(Q\) with \(T_i\). Then one seeks upper bounds on the geometric intersection numbers of \(\alpha_i\) with \(\beta_i\) in \(T_i\). Subject to certain nontriviality conditions, the author obtains such upper bounds for at least one of the pairs \((\alpha_i,\beta_1)\) (distance between boundary slopes).