Toroidal Dehn surgeries on knots in lens spaces

Abstract

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 V_r to M along their boundaries in such a way that r bounds a disc in V_r. 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.