Dehn Surgery on Knots - International Mathematical Union

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In [D], Dehn considered the following method for constructing 3-manifolds: remove a solid torus neighborhood N(K) of some knot X in the 3-sphere S3 and sew it ...
Dehn Surgery on Knots Cameron McA. Gordon Department of Mathematics, The University of Texas at Austin Austin, TX 78712, USA

1. Introduction In [D], Dehn considered the following method for constructing 3-manifolds: remove a solid torus neighborhood N(K) of some knot X in the 3-sphere S 3 and sew it back differently. In particular, he showed that, taking X to be the trefoil, one could obtain infinitely many non-simply-connected homology spheres o

in this way. Let Mx = S3 —N(K). Then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus ôMK that bounds a meridional disk in the re-attached solid torus. We denote the resulting closed oriented 3-manifold by MK(V), and say that it is obtained by r-Dehn surgery on X. More generally, one can consider the manifolds ML(*) obtained by r-Dehn surgery on a fc-component link L = Xi U • • • U Kk in S3, where r = (r\,..., ;>). It turns out that every closed oriented 3-manifold can be constructed in this way [Wal, Lie]. Thus a good understanding of Dehn surgery might lead to progress on general questions about the structure of 3-manifolds. Starting with the case of knots, it is natural to extend the context a little and consider the manifolds M(r) obtained by attaching a solid torus V to an arbitrary compact, oriented, irreducible (every 2-sphere bounds a 3-ball) 3-manifold M with dM an incompressible torus, where r is the isotopy class (slope) on dM of the boundary of a meridional disk of V. We say that M(r) is the result of r-Dehn filling on M. An observed feature of this construction is that generically, the topology of M persists in M(r). We shall illustrate this slogan by stating some results that give restrictions on the "exceptional" slopes r for which M(r) represents some sort of degeneration of M. Specifically, we shall consider the following questions: (1) When is an essential surface destroyed by Dehn filling? (2) When is an essential surface created by Dehn filling? (3) When is M(r) "small"? The results we shall state show that the present state of knowledge on these questions is quite good. Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990

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First notice that, roughly speaking, anything can happen under a single Dehn filling. For if we are interested in when M(r) has a certain property, then we can o

simply start with a closed 3-manifold Q with that property, and let M = Q—N(K) for some suitable knot X in Q. Then there is a Dehn filling on M that yields Q. This puts a limitation on the kind of results one expects to obtain. However, the theorems we shall state will usually assert that (for any M) the set of exceptional slopes (for the particular kind of degeneration in question) is small. In fact, if A(r,s) denotes the minimal geometric intersection number of two slopes r,s on dM, then the theorems will often be of the form that give an upper bound on A(r,s) for any pair of exceptional slopes r and s. We conclude this introduction with some conventions that we shall adopt throughout the paper. All 3-manifolds and surfaces will be assumed to be compact and orientable, and M will always denote (as above) an irreducible 3-manifold with dM an incompressible torus. A knot X will always be assumed to be a (non-trivial) knot in S3 unless otherwise stated, and we reserve the notation MK o

(= S3 — N(K)) for this case. Slopes on 3Mx will be parametrized by (Qu {00} in the usual way, using a meridian-longitude basis {p, X} for Hi(dMKÌ- Thus r 2) and push it into intfS1 xD2). Then it turns out that there are infinitely many Dehn surgeries on X that yield a solid torus again. In particular, there are infinitely many Dehn surgeries on X under which ^(S1 x D2) compresses. To put this in the setting of our question, we o

simply take a knot Xo in a closed 3-manifold Q (such that MQ = Q — N(KQ) is irreducible and ôMQ is incompressible), and identify N(KQ) with S1 x D2 by some homeomorphism, so that X becomes a knot in Q. (We say that X is a (p, q)-cable

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of Xo.) Then the torus S = dN(KQ) is essential in M = Q — N(K), but compresses in M(r) for infinitely many slopes r. Analogous examples with surfaces S of higher genus can be constructed by starting with a curve C in the boundary of a handlebody X of genus > 2 such o

that dX — C is incompressible in X, pushing C into X to get X, say, taking a o

suitable embedding of X in some closed 3-manifold Q, and letting M = Q — N(K) and S = SX An example of a different kind is the knot X in S1 x D2 illustrated in Fig. 1.

Fig.l

Let W = S1 x D2 — N(K). Then, parametrizing slopes on dN(K) using the meridian and longitude that come from the embedding of X in S3 defined by Fig. 1, we have W(oo) s W(1S) = W(19) ^ S 1

xD2,

so d(S{ x D2) compresses in W(r) for 3 distinct slopes r. (In fact X is the unique non-cable curve in S1 x D2 (which is not a core and does not lie in a 3-ball) with this property [Ga2, Bl, Sc].) Note that ^1(18,19) = ,4(18,oo) = zl(19,oo) = 1. Similar examples in handlebodies of genus > 2 have been constructed by Berge (unpublished). Again, as above, these can be used to construct essential closed surfaces S in irreducible 3-manifolds M with torus boundary such that S compresses under 3 distinct Dehn fillings on M. With these examples in mind we have the following result of Wu. Theorem 2.1 [Wu2]. Let S be a closed essential surface in M which compresses in M(r) and M(s). Then either A(r,s) = 1, or the core X of the solid torus V in M(s) = MUV can be isotoped onto S and S compresses in M(r)for infinitely many r.

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In particular, if S is a torus then either A(r,s) = 1 or X is a cable. In this case the theorem was originally proved in [CGLS]. Let us now consider when an essential surface with boundary in M is destroyed by Dehn filling. By this we mean the following. Let F be an essential surface in M with non-empty boundary, so each boundary component of F has slope r, say. (We say that r is a boundary-slope.) Then F gives rise to a closed surface F in M(r) = M U V by capping off the boundary components of F with meridian disks of V, and we are interested in the question: is F essential in M(r)? It is clear that the answer is not always "yes." For instance, we can start with a closed surface S in S3, and take a knot X which punctures S in such a way o

^

that F = S — N(K) is essential in MK- But then F = S is necessarily inessential in 3 MK(OO) = S . (It is not hard to explicitly construct such examples.) However, it turns out that the failure of F to be essential can be accounted for either by the presence of a closed essential surface in M (in which case Theorem 2.1 applies), or by a bad choice of F as a representative of the boundary-slope r. Theorem 2.2 [CGLS]. Suppose that M contains no closed essential surface, and let r be a boundary-slope on dM. Then there exists an essential surface F in M with boundary-slope r such that F is essential in M(r). One can also show that if the surface F is not a 2-sphere, then M(r) is irreducible, and if F is a 2-sphere, then M(r) is either S1 x S2 or a connected sum of two lens spaces.

3. When Is an Essential Surface Created by Dehn Filling? Suppose that M(r) = MUV contains an essential surface S. It is straightforward to show that if S is moved (by an isotopy if M(r) is irreducible, or in general, by a sequence of disk-swappings) so as to minimize the number of components of o

S nV, then either S lies in M or F = S — V is an essential surface in M with boundary-slope r. The title of this section refers to the second possibility. In this context we have the following basic result of Hatcher. Theorem 3.1 [Ha]. M has onlyfinitelymany boundary-slopes. Corollary 3.2. If M does not contain a closed essential surface then M(r) contains a closed essential surface for onlyfinitelymany r. Since essential spheres and tori play a special role in 3-manifold theory, it is of interest to consider when they are created by Dehn filling. In the case of essential spheres, the cabling construction described in Sect. 2 provides some examples. For if X is a (p, g)-cable of a knot Xo in a 3-manifold o

Q, then M = Q — N(K) contains an essential annulus with boundary-slope r where A(r,s) = 1, s being the meridian of X. This annulus becomes a 2-sphere in M(r), and in fact it splits M(r) as a connected sum Mo(ro)#L(q,p), where o

Mo = Q — N(KQ) and ro is some slope on dMo. If we now choose Q to contain an essential sphere (and Xo to be any knot in Q such that Mo is irreducible), then

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M is irreducible, whilst M(r) and M(s) = Q contain essential spheres, (It is not obvious, but in fact follows from [GLul], that, here, Mo(ro) cannot be S3.) Another example where M(r) and M(s) contain essential spheres with A(r,s) = 1, which does not come from cabling, is given in [GLi]. The next theorem says that this is the worst that can happen. Theorem 3.3 [GLu3]. If M(r) and M(s) each contain an essential sphere then A(r,s) = l. Turning to essential tori, a key example here is the exterior MK of the figure eight knot X (see Fig. 2). MK does not contain an essential torus, but M#(4), which is homeomorphic to MK(—4) since X is amphicheiral, does. Note that A(4,-4) = 8.

Fig. 2

Theorem 3.4 [Go]. If M does not contain an essential torus, but M(r) and M(s) do, then A(r,s) < 8. In fact one can do a little better. The figure eight knot exterior belongs to the family of manifolds obtained by Dehn surgery on one component of o

the Whitehead link L (see Fig. 3). More precisely, let W = S3 — N(L) and let W(p/q) be the manifold with torus boundary obtained by p/q-Dehn filling on one boundary component of W (where we use the obvious meridian-longitude parametrization). Then W(l) is homeomorphic to the exterior of the figure eight knot. It also turns out that for M = W(-5), W(-5/2), and W(2), M is irreducible and contains no essential torus, whilst there are r,s such that M(r) and M(s) contain essential tori with A(r,s) = 8,7, and 6 respectively. One can show that these examples are the only ones with A(r,s) > 5. Addendum 3.5 [Go]. In the setting of Theorem 3.4, if A(r,s) = 8, then M is homeomorphic to W(l) or W(—5); if A(r,s) = 1, then M is homeomorphic to W(—5/2); if A(r,s) = 6, then M is homeomorphic to W(2).

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Fig. 3

4. When Is M{r) Small? Examples of what we might mean by "small" are : When is m(M(r)) trivial? finite? cyclic? When is M(r) homeomorphic to S 3 ? a lens space? S 1 x 5 2 ? a Seifert fibred space? This is a convenient point at which to recall that by [T], M is either hyperbolic, or Seifert fibred, or contains an essential torus. (For instance, if M = Mx then the last two possibilities correspond respectively to X being a torus knot or a satellite knot.) As it is straightforward to describe M(r) when M is Seifert fibred [He], and as the case when M contains an essential torus is to a large extent covered by Theorem 2.1, the hyperbolic case is the most important to understand. Here there is the following result of Thurston. Theorem 4.1 [T]. If M is hyperbolic then M(r) is hyperbolic for all but finitely many r. A hyperbolic manifold is one whose interior has a complete Riemannian metric of constant negative (sectional) curvature. If we are willing to sacrifice "constant" in the conclusion of Theorem 4.1, then we have the following theorem, due to Gromov and Thurston, and improved in [BH] using a result of Adams. Theorem 4.2. / / M is hyperbolic then M(r) has a Riemannian metric of negative curvature for all but at most 24 values of r. It is conjectured that a 3-manifold with negative curvature is in fact hyperbolic. Even in the absence of a proof of this conjecture it is known that such a manifold must have infinite fundamental group, cannot contain an essential sphere or torus, cannot be Seifert fibred, etc. Let us turn to the more specific question: When is ni(M(r)) cyclic? As a first set of examples, let X be a torus knot. Then MK(r) is a lens space for infinitely many r [Mo]. Note that in this case MK is Seifert fibred. A set of examples of a different flavor comes from the knot X in S 1 x D2 depicted in Fig. 1 and discussed in Sect. 2. Embedding S 1 xD2 in S3 by applying fc meridional twists to the embedding shown in Fig. 1, we obtain an infinite family of

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knots X/f, —oo < fc < oo, (which are in fact hyperbolic), The two non-tri vial Dehn surgeries on X that yield SixD2 determine two Dehn surgeries on Kk, with slopes r* and s*, say, that yield lens spaces, where A(rk,Sk) = A(i\,co) = zl(s/c,oo) = 1. For instance Xo is the (—2,3,7) pretzel knot shown in Fig, 4 (this example was discovered earlier by Fintushel and Stern [FS]), for which

M* 0 (18)^L(18,5),

M* 0 (19)^L(19,7),

M*0(oo)^S3.

Fig. 4

The following Cyclic Surgery Theorem shows that these examples are extremal. Theorem 4.3 [CGLS]. Suppose that M is not Seifert fibred. If ni(M(r)) and m(M(s)) are cyclic then A(r,s) = 1. Less is known about when n\(M(r)) is finite. An interesting example here is the manifold W(—5) defined in Sect. 3. It is shown in [We] that there exist r,s with A(r,s) = 3 such that ni(W(—5)(r)) and n\(W(—5)(s)) are finite. Question 4.4. If M is hyperbolic and n\(M(r)) and ni(M(s)) are finite, is A(r,s) < 3? The case where M = MK for a satellite knot X (so M contains an essential torus) is discussed in [BH], where it is shown that A(r,s) < 5, and that this bound is attained. Regarding the question of when ni (M(r)) is trivial, the following conjecture is still open. Conjecture 4.5. ni (M(r)) is trivial for at most one slope r.

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5. Dehn Surgery on Knots in S3 Continuing the theme of the previous section, let us ask: when can we have two Dehn fillings on M that give small manifolds, one of which is S31 Clearly there is a Dehn filling on M that gives S3 if and only if M is homeomorphic to MK for some knot X in S3, with the slope that determines the filling corresponding to the meridian of X. So our question becomes: given a knot X in S3, when does non-trivial Dehn surgery on X yield a small manifold? First we have the result that S 3 can never be obtained by non-trivial Dehn surgery. Theorem 5.1 [GLu2]. MK(r) £ S3 ifr ± oo. An immediate corollary of this theorem is that knots are determined by their complements. Corollary 5.2. If Ki and X2 are knots in S3 such that S3 — Xi is homeomorphic to S3 —K2, then there is a homeomorphism of pairs (S3,K\) = (S3,K2). The following so-called Property P Conjecture, however, is still open, although it is known to be true for many classes of knots. Conjecture 5.3. %i(MK(r)) =fc 1 if r ^ co. Of course, Conjectures 4.5, 5.3, and Theorem 5.1 are closely related, and are equivalent if the Poincaré Conjecture is true. After S3 (the unique 3-manifold of Heegaard genus zero), we might consider when Dehn surgery on a knot X yields a manifold of Heegaard genus one, i.e., S1 x S2 or a lens space. For homological reasons, the only Dehn surgery that could possibly give S1 x S2 is O-Dehn surgery. However, a result of Gabai implies that this never happens. Theorem 5.4 [Gal]. M#(0) is irreducible. Regarding lens spaces, Berge [B2] gives an explicit construction which yields several infinite classes of knots with a Dehn surgery yielding a lens space. (These include the infinite family described in Sect. 4 with two such Dehn surgeries.) It appears to be not entirely outside the bounds of possibility that every such knot can be constructed in this way. Question 5.5. Does every knot K such that Mx(r) is a lens space for some r appear in Berge's list? An affirmative answer to this question would imply, in particular, the truth of the following conjecture. Conjecture 5.6. MK(r) 9* RP3 (= L(2,1)), L(3,1), or L(4,1).

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Finally, we mention that for satellite knots, the question of when 7ri(Mjc(r)) is cyclic is completely solved. Theorem 5.7 [Wan, Wul, BL]. Let X be a satellite knot. Then 7ii(Mx(r)) is cyclic if and only ifX is the (2pq ± l,2)-cable of a (p,q)-torus knot and r = 4pq + 1, in which case MK(r) = L(4pq ± i,4q2). We now turn to the creation of essential spheres and tori by Dehn surgery on a knot X. As far as spheres are concerned, recall the discussion of cables in Sect. 3. In our present setting of knots in S3, if X is the (p, q)-cable of Xo then the boundary-slope of the essential annulus in MK is pq, and MK(pq)^MK,(p/q)#L(q,p). (It is convenient here to allow Xo to be unknotted, in which case X is the (p, g)-torus knot and Mx(/;g) = L(p,q)#L(q,p). Also, if Xo is non-trivial then ni(MKo(p/q)) j= 1 by [CGLS, Corollary 2].) The following Cabling Conjecture of Gonzalez-Acuna and Short asserts that these are the only examples where essential spheres arise. Conjecture 5.8 [G-AS]. MK(V) contains an essential sphere if and only ifK (p, q)-cable and r = pq.

is a

Although this conjecture is still open, several partial results exist. For instance, there is Theorem 5.4 above. Also, it is known to be true if X is a satellite knot [Sc], an alternating knot [MT], a strongly invertible knot [E-M], and others [G-AS]. It is also known that if M^(r) contains an essential sphere then r is an integer [GLul] and MK(V) has a lens space summand [GLu2]. (Incidentally, it is an immediate consequence of this last result that no non-prime homology sphere can be obtained by Dehn surgery on a knot in S3. There remains the interesting question of whether there are prime homology spheres that cannot be so obtained.) Regarding essential tori, some examples are known, but the general picture is not yet clear. However, the following conjecture seems reasonable. (Recall that MK contains an essential torus if and only if X is a satellite knot.) Conjecture 5.9. If K is not a satellite knot and Mx(r) contains an essential torus then A (r, oo) < 2. (There are examples with A(r,oo) = 2.)

6. The General Knot Complement Problem Theorem 5.1 above asserts that if M(r) = M(s) = S3, then r = s. More generally, one can ask: when is M(r) = M(s)l One has to be a little careful with orientations here, for M may have an orientation-reversing automorphism. For instance, if X is an amphicheiral knot (such as the figure eight knot), then MK(T) = MK(—r). However, if we orient M, then this determines an orientation of M(r), and we

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can ask: when is M(r) = M(s) by an orientation-preserving homeomorphism? No counterexamples are known to the following conjecture. Conjecture 6.1. 1/ M(r) = M(s) by an orientation-preserving homeomorphism then r = s. This would imply the truth of the following Oriented Knot Complement Conjecture. Conjecture 6.2. IfKi and K2 are knots in a closed, oriented 3-manifold Q such that Q—Ki and Q—K2 are homeomorphic by an orientation-preserving homeomorphism, then there exists an orientation-preserving homeomorphism h : Ô —> Q such that h(K{) = K2. Mathieu [Ma] has given examples where Q—Ki and Q—K2 are homeomorphic (by an orientation-reversing homeomorphism), but there is no homeomorphism of pairs (Q,Ki)^(Q,K2).

7. Intersections of Surfaces The proofs of the theorems stated above use several techniques from modern 3dimensional topology. See [Sh] for instance, for a nice account of some of the ideas that went into the proof of the Cyclic Surgery Theorem (Theorem 4.3). However, here we shall focus on just one technique, namely, the analysis of intersections of properly embedded surfaces. The point is that many of the particular forms of what we have called the "degeneration" of the topology of M in M(r) imply the existence of a (useful) surface F in M with non-empty boundary such that each boundary component has slope r on dM. For instance, as was pointed out in Sect. 3, if M(r) contains an essential surface S which cannot be moved into M, then M contains an essential surface F with boundary-slope r such that F = S. Hence if M(ri) and M(r2) contain such surfaces Si,S2 respectively, then we get corresponding essential surfaces Fi,F2 in M with boundary-slopes ri,r2. Putting Fi and F2 in mutual general position, Fi n F2 will consist of circles and arcs properly embedded in Fi and F2. Note that the pattern of intersections on the boundary, that is, the triple (dM;dFi,dF2) is completely standard: all the components of dFt are parallel, i = 1,2, and each component of dFi meets each component of dF2 in A(ri,r2) points. A final important point is that we may assume that no arc of F\ n F2 is isotopie in F,- (keeping its endpoints fixed) into dFi, i= 1,2. This easily follows from the fact that Fi and F2 are essential. The idea is to then analyze the pattern of arcs of Fi n F2 as they lie on Fi and F2. We record the intersections on the boundary by numbering the boundary components of Ft in order as they appear on dM, i = 1,2, and labelling the points of dFi n dF2 on Fi with the number of the corresponding boundary component ^

o

of F2 (and vice versa). It is convenient to regard the disks Fi—Fi as "fat" vertices, and the arcs of F\ nF2 as edges, so that we get graphs Fi, F2 in Fi, F2 respectively, whose vertices and edge-endpoints are labelled as just described. Typically, one now proceeds to show that if A(ri,r2) is greater than some Ao, then the pair of graphs Fi,F2 must contain certain configurations of faces which have topological

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implications for M (or M(ri) or M(r2)) inconsistent with the hypotheses. This type of argument was first used by Litherland [Lit]. Theorems 3.3 and 3.4 are proved in this way, the surfaces Sj being spheres and tori respectively. The proof of Theorem 2.1 is similar in spirit, only here the surfaces S,- are disks, whose boundaries lie on the closed surface S, realizing the compression of S in M(ri), i = 1,2. (This set-up is also an ingredient of the proof of Theorem 4.3.) In Theorem 5.1, the hypothesis is that M(r\) = M(r2) ^ S3, which of course does not contain any essential surface. Nevertheless, using an idea of Gabai [Gal], 2-spheres Sj in M(r\), i = 1,2, can be chosen so that the corresponding punctured spheres F\ in M still satisfy the key condition that no arc of Fi n F2 can be isotoped into the boundary in either Fi or F2. This allows one to apply the same philosophy.

8. Conclusion As mentioned in the Introduction, the importance of Dehn surgery lies in the fact that every closed oriented 3-manifold can be obtained by Dehn surgery on some link in S3, Thus one might regard a Dehn surgery presentation of a 3-manifold as being analogous to a Heegaard splitting, with the study of Dehn surgery on knots corresponding to the study of Heegaard splittings of genus one. However, I think this analogy is misleading, because, unlike Heegaard splittings of genus one, Dehn surgery on knots already seems to contain much of the complexity of the general case. This, together with the progress that has been made in the case of a single component, makes it hopeful that Dehn surgery might be a useful way to approach general problems about 3-manifolds. At any rate, the next step should be to find appropriate generalizations of the results discussed above to the case where solid tori are attached along the boundary components of an irreducible 3-manifold M whose boundary is a disjoint union of an arbitrary number of tori.

References Berge, J.: The knots in D2 x S1 which have nontrivial Dehn surgeries that yield D2 x S1. Topology and its Applications 38 (1991) 1-19 [B2] Berge, J. : Obtaining lens spaces by surgery on knots. To appear in Proceedings of International Conference on Knots 90. de Gruyter [BH] Bleiler, S.A., Hodgson, CG. : Spherical space forms and Dehn surgery. To appear [BL] Bleiler, S.A, Litherland, R.A. : Lens spaces and Dehn surgery. Proc. Amer. Math. Soc. 107 (1989) 1127-1131 [CGLS] Culler, M., Gordon, C.McA., Luecke, J., Shalen, P.B.: Dehn surgery on knots. Ann. Math. (2) 125 (1987) 237-300 [D] Dehn, M.: Über die Topologie des dreidimensionalen Raumes. Math. Ann. 69 (1910) 137-168 [E-M] Eudave-Munoz, M. : Band sum of links which yield composite links. To appear [FS] Fintushel, R., Stern, R. : Constructing lens spaces by surgery on knots. Math. Z. 175 (1980) 33-51 [Gal] Gabai, D.: Foliations and the topology of 3-manifolds. III. J. Diff. Geom. 26 (1987) 479-536 [Ga2] Gabai, D.: Surgery on knots in solid tori. Topology 28 (1989) 1-6 [Bl]

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[G-AS] Gonzalez-Acuna, F., Short, H. : Knot surgery and primeness. Math. Proc. Camb. Phil. Soc. 99 (1986) 89-102 [Go] Gordon, C.McA. : Boundary slopes of punctured tori in 3-manifolds. To appear [GLi] Gordon, C.McA., Litherland, R.A.: Incompressible planar surfaces in 3-manifolds. Topology and its Applications 18 (1984) 121-144 [GLul] Gordon, C.McA., Luecke, J.: Only integral Dehn surgeries can yield reducible manifolds. Math. Proc. Camb. Phil. Soc. 102 (1987) 97-101 [GLu2] Gordon, C.McA., Luecke, J.: Knots are determined by their complements. J. Amer. Math. Soc. 2 (1989) 371-415 [GLu3] Gordon, C.McA., Luecke, J. : Reducible manifolds and Dehn surgery. To appear [Ha] Hatcher, A.E.: On the boundary curves of incompressible surfaces. Pacific J. Math. 99 (1982) 373-377 [He] Heil, W.: Elementary surgery on Seifert fiber spaces. Yokohama Math. J. 22 (1974) 135-139 [Lie] Lickorish, W.B.R.: A representation of orientable combinatorial 3-manifolds. Ann. Math. 76 (1962) 531-540 [Lit] Litherland, R.A. : Surgery on knots in solid tori, II. J. London Math. Soc. (2) 22 (1980) 559-569 [Ma] Mathieu, Y. : Sur des noeuds qui ne sont pas déterminés par leur complément et problèmes de chirurgie dans les variétés de dimension 3. Thèse, L'Université de Provence, 1990 [MT| Menasco, WW, Thistlethwaite, M.B. : Surfaces with boundary in alternating knot exteriors. To appear [Mo] Moser, L.: Elementary surgery along a torus knot. Pacific J. Math. 38 (1971) 734-745 [Sc] Scharlemann, M. : Producing reducible 3-manifolds by surgery on a knot. Topology 29 (1990) 481-500 [Sh] Shalen, P.B. : Representations of 3-manifold groups and applications in topology. Proceedings ICM Berkeley, 1986, pp. 607-614 [T\ Thurston, W; Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. 6 (1982) 357-381 [Wal] Wallace, A.H.: Modifications and cobounding manifolds. Can. J. Math. 12 (1960) 503-528 [Wan] Wang, S.: Cyclic Surgery on knots. Proc. Amer. Math. Soc. 107 (1989) 1091-1094 [We] Weeks, J.R.: Hyberbolic structures on three-manifolds. Ph.D. thesis, Princeton University, 1985 [Wul] Wu, Y-Q.: Cyclic surgery and satellite knots. Topology and its Applications 36 (1990) 205-208 [Wu2] Wu, Y-Q.: Incompressibility of surfaces in surgered 3-manifolds. To appear in Topology