Finite Dehn surgery on knots

Abstract

Let K be a knot with a closed tubular neighbourhood N(K) in a connected orientable closed 3-manifold W , such that the exterior of K, M = W − intN(K), is irreducible. We consider the problem of which Dehn surgeries on K, or equivalently, which Dehn fillings on M , can produce 3-manifolds with finite fundamental group. For convenience, a surgery is called a G-surgery if the resultant 3-manifold has fundamental group G. If G is cyclic or finite, the surgery is also called a cyclic surgery or a finite surgery. Similar terminology will be used for Dehn fillings. The manifold obtained by the Dehn filling on M along ∂M with slope r is denoted by M(r). Let ∆(r1, r2) denote the minimal geometric intersection number (the distance) between two slopes r1 and r2 on ∂M . According to [T1], M belongs to one of the following three mutually exclusive categories: (I) M is a Seifert fibred space admitting no essential torus. (II) M is a hyperbolic manifold (i.e. intM admits a complete hyperbolic metric of finite volume). (III) M contains an essential torus. It is a remarkable result, the so-called cyclic surgery theorem [CGLS], that if M is not Seifert fibred, then all cyclic surgery slopes of K have mutual distance no larger than 1, and consequently, there are at most 3 cyclic surgeries on K. In this paper, we consider finite Dehn surgery on K and we prove, for instance, that if M is not a manifold of type (I) and is not a union along a torus of the twisted I-bundle over the Klein bottle and a cabled space, then there are at most 6 finite and cyclic surgeries on K of maximal mutual distance 5. Henceforth, we shall use finite/cyclic to mean either finite or (infinite) cyclic. It turns out to be convenient to consider the three cases mentioned above separately. In case (I) it is well-known that one can completely classify the finite/cyclic surgeries on K. Considering the torus knots for instance, one sees that there exist infinitely many knots whose exteriors are of type (I), each of which admit an infinity of distinct finite (cyclic or non-cyclic) surgery slopes. Our contributions deal with the cases (II) and (III). For the former we obtain