Boundary slopes of punctured tori in 3-manifolds

Let $M$ be an irreducible 3-manifold with a torus boundary component $T$, and suppose that $r,s$ are the boundary slopes on $T$ of essential punctured tori in $M$, with their boundaries on $T$. We show that the intersection number $\Delta(r,s)$ of $r$ and $s$ is at most $8$. Moreover, apart from exactly four explicit manifolds $M$, which contain pairs of essential punctured tori realizing $\Delta(r,s)=8,8,7$ and 6 respectively, we have $\Delta(r,s)\le 5$. It follows immediately that if $M$ is atoroidal, while the manifolds $M(r), M(s)$ obtained by $r$- and $s$-Dehn filling on $M$ are toroidal, then $\Delta(r,s)\le 8$, and $\Delta(r,s)\le 5$ unless $M$ is one of the four examples mentioned above.

Let $\mathcal{H}_0$ be the class of 3-manifolds $M$ such that $M$ 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 $M\in \mathcal{H}_0$, where $\partial M$ has a torus component $T$, and $\partial M-T\ne \varnothing$. Let $r,s$ be slopes on $T$ such that $M(r), M(s)\notin \mathcal{H}_0$. Then $\Delta(r,s)\le 5$. The exterior of the Whitehead sister link shows that this bound is best possible.