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( r 1 , r 2 ) denote as usual the minimal geometric intersection number of two slopes r 1 and r 2 . If [Sscr ] is any set of slopes, then clearly any upper bound for Δ([Sscr ])=max{Δ( r 1 , r 2 ): r 1 , r 2 ∈[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 S 2 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( r 1 , r 2 ) between various classes of exceptional slopes r 1 and r 2 , 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.