Exceptional Surgery on Knots

Let M be an irreducible, compact, connected, orientable 3-manifold whose boundary is a torus. We show that if M is hyperbolic, then it admits at most six finite/cyclic fillings of maximal distance 5. Further, the distance of a finite/cyclic filling to a cyclic filling is at most 2. If M has a nonboundary-parallel, incompressible torus and is not a generalized 1-iterated torus knot complement, then there are at most three finite/cyclic fillings of maximal distance 1. Further, if M has a non-boundary-parallel, incompressible torus and is not a generalized 1or 2-iterated torus knot complement and if M admits a cyclic filling of odd order, then M does not admit any other finite/cyclic filling. Relations between finite/cyclic fillings and other exceptional fillings are also discussed. Let Af be a compact, connected, orientable 3-manifold whose boundary is a torus. M is called irreducible if any embedded 2-sphere in M bounds an embedded 3-ball. Otherwise, M is called reducible. A slope on dM is a dAf-isotopy class r of essential, simple closed curves. A slope r is called a boundary slope if there is a properly embedded, orientable, incompressible, dincompressible surface F in M such that dF is a nonempty set of parallel simple closed curves on dM of slope r. The distance between two slopes rx and r2, denoted by A(ri, r2), is the minimal geometric intersection number between r\ and r2. As dM is a torus, A(/"i, r2) may be calculated as the absolute value of the algebraic intersection number of the homology classes carried by r\ and r2 . Given a slope r on dM, a well-defined closed 3-manifold M{r) can be constructed by attaching a solid torus to M via a homeomorphism which identifies a meridian curve of the solid torus to a representative curve for r. M{r) is called the r-filling of M. Given a knot K in a closed, connected, orientable 3-manifold W, with tubular neighbourhood N{K), exterior M = W int{N{K)), and slope r on dM, M{r) is also referred to as the r-surgery of W along K. A fundamental result of Wallace [Wa] and Lickorish [Li] states that each closed, orientable 3-manifold results from surgery on some link in the 3-sphere. Thus a natural approach to 3-manifold topology is to analyze to what extent various aspects of the topology of a manifold Af, as above, are inherited by the manifolds Af (r). For instance, one could try to determine when a closed essential surface in Af becomes inessential in some Af(r) or when an irreReceived by the editors September 14, 1993. 1991 Mathematics Subject Classification. Primary 57M25, 57R65. The first author was partially supported by grants NSERC OGP 0009446 and FCAR EQ 3518. The second author was supported by a postdoctoral fellowship from the Centre de Recherches Mathématiques. A version of part of the material of this paper was presented by the first author at the joint AMS-SMM meeting in Merida, Mexico, in December 1993. © 1994 American Mathematical Society 0273-0979/94 $1.00+ $.25 per page 197 198 S. BOYER AND X. ZHANG ducible Af could produce a reducible M(r). An excellent survey of this topic may be found in [Gol]. Another example of some importance arises when Af is a hyperbolic manifold, i.e., int(Af) admits a complete hyperbolic metric of finite volume. Thurston [Th] has shown that in this situation, all but finitely many of the manifolds Af (r) are hyperbolic. The nonhyperbolic slopes include those whose fillings are: (a) manifolds with finite or cyclic fundamental groups, (b) manifolds which are reducible, (c) manifolds which are Seifert-fibred spaces, (d) manifolds which admit an incompressible torus. Thurston's geometrication conjecture [Th] predicts that the remaining slopes yield fillings which are hyperbolic. A basic problem then is to describe the set of exceptional slopes on a torally bounded manifold Af, i.e., those slopes producing nongeneric fillings. An appropriate description should include an upper bound on the number of such slopes as well as a qualitative measure of their relative positions determined by bounds on their mutual distances. We call a slope r on <9Af cyclic if M{r) has cyclic fundamental group; and similarly we shall refer to finite slopes, reducible slopes, Seifert slopes, and essential torus slopes. It is a remarkable result, the cyclic surgery theorem [CGLS], that if Af is not a Seifert-fibred space, then all cyclic slopes on dM have mutual distance no larger than 1, and consequently there are at most three such fillings. Gordon and Luecke obtained similar estimates for the set of reducible slopes [GLul] and also examined essential torus slopes [Go2, GLu2]. In another direction, Bleiler and Hodgson [BH] refined results of Gromov and Thurston to obtain restrictions on the slopes on a hyperbolic Af which do not produce manifolds admitting metrics of strictly negative sectional curvature. We consider the problem of determining the set of finite slopes on Af here. Henceforth, we shall use finite/cyclic to mean either infinite cyclic or finite. Standard arguments show that we may take Af to be irreducible, and we shall assume this below. Then according to [Th], Af belongs to one of the following three mutually exclusive categories: (I) Af is a Seifert-fibred space admitting no essential, i.e., incompressible and nond -parallel torus. (II) Af is a hyperbolic manifold. (Ill) Af contains an essential torus. It turns out to be convenient to consider these three cases separately. In case (I) it is well known that one can completely classify the finite/cyclic fillings of Af. 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 noncyclic) surgery slopes. Our contributions deal with the cases (II) and (III). For the former we obtain Theorem A. Let K be a knot in a closed, connected, orientable, 3-manifold W, such that the interior of M = W in\N{K) admits a complete hyperbolic structure of finite volume. {1 ) There are at most six finite/cyclic surgeries on K, and A{rx, r2 ) < 5 for any two finite/cyclic surgery slopes r\ and r2 of K. EXCEPTIONAL SURGERY ON KNOTS 199 (2) If ri is a finite/cyclic surgery slope of K and r2 is a cyclic surgery slope of K, then A(r., r2)<2. The known realizable maximal number of finite/cyclic surgeries on a knot as in Theorem A and their maximal mutual distance is 5 and 3 [We]. This example may also be used to show that Theorem A(2) is sharp. To discuss case (III), we introduce the following notion. A compact, connected, orientable, 3-manifold Af, with boundary a torus, is called a generalized n-iterated torus knot exterior if Af can be decomposed along disjoint, essential tori into a union of n cabled spaces (in the sense of [GLi]) and a Seifert-fibred space which has a Seifert fibration over the 2-disc with exactly two exceptional fibres. Theorem B. Let K be a knot in a closed, connected, orientable, 3-manifold W, such that M =W ir\\N{K) is irreducible and contains an essential torus. (1) If M is not a generalized I-iterated torus knot exterior, then A(ri, r2) < 1 for any two finite/cyclic surgery slopes rj and r2 of K. In particular, there are at most three finite/cyclic surgeries on K. (2) If M is not a generalized 1or 2-iterated torus knot exterior and if K admits a cyclic surgery of odd order, then K does not admit any other finite/cyclic surgery. Finite/cyclic fillings on a generalized 1or 2-iterated torus knot exterior Af can be completely described. This is essentially done in [BH, §2], where it is shown that if Af is not a union of the twisted /-bundle over the Klein bottle and a cabled space, then there are no more than six finite/cyclic fillings of maximal mutual distance 5 (realized on the exterior of the (11, 2)-cable over the (2, 3)-torus knot in S3). In particular, it is proved that an iterated torus knot in S3, admitting a nontrivial finite surgery, must be a cable over a torus knot. A complete list of all finite surgeries on cabled knots over torus knots in S3 is given in §2 of that paper. It is shown in [BZ1, Example 10.6] that Theorem B(l) is sharp. We also note that as the finite/cyclic fillings on generalized 1or 2-iterated torus knot exteriors are readily determined, Theorem B(2) completes the classification of finite/cyclic surgeries on knots in manifolds of odd-order, cyclic fundamental group whose exteriors contain an essential torus. Consider now surgery on knots in the 3-sphere S3. As is usual, slopes for a knot in S3 are parameterized by Q U {¿} through the use of the standard meridian-longitude coordinates [R]. In S3 only the trivial knot admits a Zsurgery [Ga3]. Corollary C. Let K c S3 be a hyperbolic knot. (1) There are at most six finite surgeries on K, and A(ri, r2) < 5 for any two finite surgery slopes rx and r2 of K. (2) If rx is a finite surgery slope of K and r2 is a cyclic surgery slope of K, then A(ri , r2) < 2. In particular, if m/n is a finite surgery slope of K, then \n\ < 2. It is shown in [BH] that the (-2, 3, 7)-pretzel knot admits at least four finite surgeries of maximal mutual distance 2. We prove in [BZ1, Example 10.1] that this knot has no other finite slopes. This example exhibits the known maximal number of finite surgeries on a hyperbolic knot in S3. 200 S. BOYER AND X. ZHANG Corollary D. Let K c S3 be a satellite knot. If K admits a nontrivial finite surgery, then K is a cabled knot over a torus knot. It follows from our previous remarks that Corollary D classifies finite surgeries on satellite knots in S3. In their recent work [BH], Bleiler and Hodgson obtained, using a completely different approach, the number 24 and the distance 23 for finite/cyclic surgery on a knot as in Theorem A and the number 8 and the distance 5 for finite/cyclic surgery on a knot as in Theorem B. It is a classic result [Mi] that a finite group, which is the fundamental group of a 3-manifold, must belong to one of the following types: C-type, cyclic groups; D-type, dihedral-type groups; T-type, tetrahedral-type groups; O-type, octahedral-type groups; I-type, icosahedral-type groups; and Q-type, quaternionic-type groups. It is shown in [BZ1] that more precise information on finite/cyclic surgeries of a given type may be obtained. For knots in S3 these yield the following result. Proposition E. Let K c S3 be a hyperbolic knot. (1) (i) Any D-type slope of K must be an integral slope. (ii) There is at most one D-type finite surgery on K. (iii) If there is a D-type surgery on K, then there is no even-order cyclic surgery on K. (iv) If there is a D-type surgery a on K, then there is at most one nontrivial cyclic surgery on K ; and if there is one, ß, say, then a and ß are consecutive integers. (2) There are at most two T-type slopes on K ; and if two, one is integral, the other has denominator 2, and their distance is 3. (3) (i) Any O-type slope of K must be an integral slope. (ii) There are at most two O-type slopes on K ; and if two, their distance is 4. (iii) If there is an O-type surgery on K, then there is no even-order cyclic surgery on K. (iv) If there is an O-type surgery a on K, then there is at most one nontrivial cyclic surgery on K ; and if there is one, ß, say, then a and ß are consecutive integers. (v) If there is an O-type surgery slope and a D-type surgery slope on K, then they are consecutive even integers. Proposition E may be complemented by various examples of hyperbolic knots in S3 which admit O-type, or I-type, or D-type surgery [BH]. In [BZ1] we produce a few more examples. Notably, [BZ1, Example 10.2] provides a hyperbolic knot in S3 which admits a D-type surgery and an odd-order nontrivial cyclic surgery, and [BZ1, Example 10.4] provides a hyperbolic knot in S3 which admits a T-type surgery and a nontrivial cyclic surgery. Based on the results obtained in [BZ1] and known examples, we raise the following problem. Finite/cyclic surgery problem. (I) Let K be a knot in a connected, closed, orientable 3-manifold W suchthat Af = W -vo\{N(K)) is a hyperbolic manifold. EXCEPTIONAL SURGERY ON KNOTS 201 Show that there are at most five finite/cyclic surgeries on K and that the distance between any two finite/cyclic surgery slopes is at most 3. (II) Let K c S3 be a hyperbolic knot. Show that: (1) there are at most four finite surgeries on K ; (2) the nontrivial finite surgery slopes on K form a set of consecutive integers; (3) the distance between any two finite surgery slopes of K is at most 2; (4) there is at most one finite surgery on K with an even integral slope. Evidence supporting a positive solution to this problem may be found in [BZ1]. For instance, if the minimal norm [CGLS, Chapter 1] amongst all nontrivial elements of Hi{dM) is greater than 24 (16 for knots whose exteriors have no 2-torsion in their homology), then the methods of [BZ1] show that for a knot K as in Theorem A there are at most four finite/cyclic surgeries on K of maximal mutual distance no more than 2. These methods may also be used to solve the finite/cyclic surgery problem for various families of knots, such as those with 2-bridges [Ta]. Proofs of the results listed above may be found in [BZ1]. The techniques used there are based on the work on M. Culler, C. M. Gordon, J. Luecke, and P. Shalen [CS, CGLS]. In particular, we derive results on the norm defined in Chapter 1 of [CGLS] which, when combined with Chapter 2 of that paper, yields Theorem A. Theorem B is proven by considering the torus decomposition of Af and then applying Theorem A and results from [Gal, Ga2, Sch, CGLS] to determine what the pieces of this decomposition are, under the added assumption that there are two finite/cyclic slopes such that either (i) they are of distance greater than 1 apart, or (ii) one of them is a cyclic slope of odd order. Our study of exceptional fillings is continued in [BZ2]. Theorem F. Let M be a compact, connected, orientable, irreducible 3-manifold with dM a torus. Assume that M is not an atoroidal Seifert-fibred manifold. Fix slopes rx and r2 on dM and suppose that Af(ri) is a reducible manifold. (1) If M{r2) has a cyclic fundamental group, then A(ri , r2) < 1. (2) If M is hyperbolic and r2 is a finite slope, then A{ri, r2) < 5 unless Af(ri) = RP3#RP3 and 7ii{M{r2)) isa D-type group or a Q-type group. {3) If M contains an essential torus and r2 is a finite slope, then A(n , r2) < 1 unless M is a cable on the twisted I-bundle over the Klein bottle or a cable on a hyperbolic manifold for which the inequality of part (2) does not hold. We also obtain a new proof of the following result of Gordon and Luecke. Theorem G ([GLul]). Let M be a compact, connected, orientable, irreducible 3-manifold with dM a torus. If M{r¡) is a reducible manifold, for i = 1,2, then A(ri, r2) < 1. Combining the last two results with the cyclic surgery theorem, we obtain Corollary H. Let M be a compact, connected, orientable, irreducible 3-manifold with dM a torus. Suppose that M is not an atoroidal Seifert-fibred space. If for i = 1,2, M{r¡) is either a reducible manifold or a manifold with cyclic fundamental group, then A{ri, r2) < 1. Consequently there are at most three cyclic/reducible fillings on M. 202 S. BOYER AND X. ZHANG Suppose now that ri is a slope on <9Af suchthat M(r{) has the fundamental group of a Seifert-fibred manifold W and that r2 is a finite/cyclic slope. Theorems A and B provide upper bounds for A(/"i, r2) when W admits a Seifert-fibration with base orbifold the 2-sphere having no more than two exceptional fibres or three such fibres if their indices form a platonic triple. Our next theorem deals with most of the remaining cases. Theorem I. Let M be a compact, connected, orientable, 3-manifold with dM a torus. Suppose further that M is neither Seifert-fibred nor a cable on a Seifertfibred manifold. Let r¡ be a slope on dM suchthat M(r\) has the fundamental group of a Seifert-fibred space which admits no Seifert fibration having base orbifold the 2-sphere with exactly three exceptional fibres. Then (1) A(ri, r2) <\ if M{r2) has a cyclic fundamental group; (2) A(/"i, r2) < 5 if M{r2) has a finite fundamental group unless Af(ri) is either RP3#RP3 or a union of two copies of the twisted I-bundle over the Klein bottle, and 7ii{M{r2)) isa D-type group or a Q-type group. Applying this result to knots in S3, we obtain Corollary J. Let M be the exterior of a knot K in S3 and r a slope on dM suchthat M{r) has the fundamental group of a Seifert-fibred space. (1) If M{r) has the fundamental group of a Seifert-fibred space which is Haken, then r is an integral slope. (2) If K is a satellite knot which is not cabled exactly once, then r is an integral slope. Finally we return to the question of quantifying nonhyperbolic slopes on a hyperbolic manifold Af. Call a slope on dM big Seifert, or bS for short, if the associated filling yields a Seifert-fibred manifold whose base orbifold is not a 2-sphere with fewer than four cone points. Using the results above as well as results from [GLi] and [Go2], we may prove the following. Corollary K. Let M be a compact, connected, orientable, hyperbolic 3-manifold with dM a torus. Let r and s be two slopes that are contained in the set of all reducible/cyclic/finite/bS slopes. If neither M{r) not M{s) is RP3#RP3 ora union of two copies of the twisted I-bundle over the Klein bottle, then A{r, s) < 5. Hence there are at most eight such slopes. The reader will find further discussion and results on exceptional slopes in [BZ2].