Incompressible planar surfaces in 3-manifolds (Q760051)

Let M be an orientable 3-manifold and T a torus component of \(\partial M\). If (P,\(\partial P)\subset (M,T)\) is a compact surface then all on T non-contractible components of \(\partial P\) represent the same isotopy class of simple closed non-oriented curves on T. This class is called the boundary-slope of P. Let \({\mathcal P}\) be the set of boundary-slopes of incompressible, boundary-incompressible planar surfaces (P,\(\partial P)\subset (M,T)\). The authors prove that if r,s\(\in {\mathcal P}\) then \(| r\cdot s| \leq 4\) where \(r\cdot s\) denotes the integral (homological) intersection number of r and s. In particular, card(\({\mathcal P})\leq 6\). Another corollary is that, for any knot K in \(S^ 3\), at most six Dehn surgeries on K can yield a reducible 3-manifold. The authors also give several instructive examples of incompressible planar surfaces and their boundary-slopes.