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.