Boundary slopes of punctured tori in 3-manifolds

Let be an irreducible 3-manifold with a torus boundary component , and suppose that are the boundary slopes on of essential punctured tori in , with their boundaries on . We show that the intersection number of and is at most . Moreover, apart from exactly four explicit manifolds , which contain pairs of essential punctured tori realizing and 6 respectively, we have . It follows immediately that if is atoroidal, while the manifolds obtained by - and -Dehn filling on are toroidal, then , and unless is one of the four examples mentioned above.

Let be the class of 3-manifolds such that is irreducible, atoroidal, and not a Seifert fibre space. By considering spheres, disks and annuli in addition to tori, we prove the following. Suppose that , where has a torus component , and . Let be slopes on such that . Then . The exterior of the Whitehead sister link shows that this bound is best possible.