Let M be a compact, orientable 3-manifold with ∂M a torus. If r is a slope on ∂M (the isotopy class of an unoriented essential simple loop), then we can form the closed 3-manifold M(r) by gluing a solid torus Vr to M along their boundaries in such a way that r bounds a disc in Vr. We say that M(r) is obtained from M by r-Dehn filling.

Assume now that M contains no essential sphere, disc, torus or annulus. Then, by Thurston's Geometrization Theorem for Haken manifolds [T1, T2], M is hyperbolic, in the sense that int M has a complete hyperbolic structure of finite volume. Furthermore, M(r) is hyperbolic for all but finitely many r [T1, T2] and the precise nature of the set of exceptional slopes E(M)={r: M(r) is not hyperbolic} has been the subject of a considerable amount of investigation. The maximal observed value of e(M)=[mid ]E(M)[mid ] (the cardinality of E(M)) is 10, realized, apparently uniquely, by the exterior of the figure eight knot [T1].

Let Δ(r1, r2) denote as usual the minimal geometric intersection number of two slopes r1 and r2. If [Sscr ] is any set of slopes, then clearly any upper bound for Δ([Sscr ])=max{Δ(r1, r2): r1, r2∈[Sscr ]} gives one for [mid ][Sscr ][mid ]. For example, one can check (using [GLi, lemma 2·1]) that for 1[les ]Δ([Sscr ])[les ]10, the maximum values of [mid ][Sscr ][mid ] are as given in Table 1.

In particular, any upper bound for Δ(M)=Δ(E(M)) gives a corresponding bound for e(M). (The maximal observed value of Δ(M) is 8, realized by the figure eight knot exterior and the figure eight sister manifold [T1, HW].)

If M(r) is not hyperbolic, then it is either reducible (contains an essential sphere), toroidal (contains an essential torus), a small Seifert fibre space (one with base S2 and at most three singular fibres), or a counterexample to the Geometrization Conjecture [T1, T2]. A survey of the presently known upper bounds on the distances Δ(r1, r2) between various classes of exceptional slopes r1 and r2, and the maximal values realized by known examples, is given in [Go2]. (See also [Wu2] for a discussion of the additional cases that arise when M has more than one boundary component.) In the present note we prove the following theorem, which deals with one further pair of possibilities.