NON-INTEGRAL TOROIDAL SURGERY ON HYPERBOLIC KNOTS IN S^{3}

CAMERON MCA. GORDON, YING-QING WU, AND XINGRU ZHANG

Abstract. We show that on any hyperbolic knot in S^{3} there is at most one nonintegral Dehn surgery which yields a manifold containing an incompressible torus.

Let K be a knot in the 3-sphere S^{3} and M = M_{K} the complement of an open regular neighborhood of K in S^{3}. As usual, the set of slopes on the torus ∂M (i.e. the set of isotopy classes of essential simple loops on ∂M) is parameterized by

{m/n : m, n ∈ Z, n > 0, (m, n) = 1} ∪ {1/0},

so that 1/0 is the meridian slope and 0/1 is the longitude slope. A slope m/n is called non-integral if n ≥ 2. The manifold obtained by Dehn surgery on S^{3} along the knot K (equivalently, Dehn filling on M along the torus ∂M) with slope m/n, is denoted by M(m/n). Now suppose that K ⊂ S^{3} is a hyperbolic knot, i.e. the interior of M has a complete hyperbolic metric of finite volume. A basic question in Dehn surgery theory is: when can a surgery on K produce a non-hyperbolic 3-manifold? A special case of this question is: when can a surgery on K produce a toroidal 3-manifold, i.e. a 3-manifold which contains an (embedded) incompressible torus? In [GL1], Gordon and Luecke showed that if m/n is a non-integral slope and M(m/n) contains an incompressible torus, then n = 2. In this paper, we show that there is at most one such surgery slope.

Theorem 1. For a hyperbolic knot in S^{3}, there is at most one surgery with nonintegral slope producing a manifold containing an incompressible torus.

There are examples of hyperbolic knots in S^{3} which admit toroidal surgeries with non-integral slopes [EM]. The best known example is the (−2, 3, 7)-pretzel knot, on which the surgery with slope 37/2 gives a toroidal 3-manifold. It is a conjecture ([Go1], [K, Problem 1.77]) that for a hyperbolic knot in S^{3}, there is at most one surgery with non-integral slope producing a non-hyperbolic 3-manifold, and further, if there is such a surgery, it must be a toroidal surgery and the knot must belong to the collection of examples given in [EM]. Theorem 1 provides some supporting evidence for this conjecture.

1991 Mathematics Subject Classification. Mathematics Subject Classification: 57N10, 57M25.

Gordon was Partially supported by an NSF grant. Research at MSRI supported in part by NSF grant #DMS 9022140.

We now go on to the proof of Theorem 1. Our argument is based on applications of results and combinatorial techniques developed in [CGLS,GL1,Go2]. Recall that the distance between two slopes m_{1}/n_{1} and m_{2}/n_{2} is defined as the number Δ = Δ(m_{1}/n_{1}, m_{2}/n_{2}) = |m_{1}n_{2} − n_{1}m_{2}|, which is equal to the minimal geometric intersection number between simple loops representing the two slopes on ∂M. Suppose that there are two slopes m_{1}/2 and m_{2}/2 such that both M(m_{1}/2) and M(m_{2}/2) contain incompressible tori. It follows from [Go2, Theorem 1.1] that there are four hyperbolic manifolds with toroidal fillings at distance more than 5, but for homological reasons, only one of these is the complement of a knot in S^{3}, namely the figure 8 knot complement, and by [Th] every nonintegral surgery on this manifold is hyperbolic. Hence we have Δ(m_{1}/2, m_{2}/2) = |2m_{1} − 2m_{2}| = 2|m_{1} − m_{2}| ≤ 5. Note that both m_{1} and m_{2} are odd integers. So |m_{1} − m_{2}| is even and thus must be equal to 2. Hence the distance between the two slopes is exactly 4. Our task here is to show that this is impossible. Note, however, that 4 can be realized as the distance between integral toroidal surgery slopes for a hyperbolic knot in S^{3}. For instance the slopes 16 and 20 are both toroidal surgery slopes for the (−2, 3, 7)-pretzel knot. The reason that distance 4 is impossible in our situation is mainly due, as we will see, to the fact that the first homology of M(m_{i}/2) with Z_{2} coefficients is trivial for i = 1, 2.

By [GL1] and [GL2] (see [GL1, Theorem 1.2]), for i = 1, 2, there is an incompressible torus T̂_{i} in M(m_{i}/2) such that M ∩T̂_{i} = T_{i} is an incompressible, ∂-incompressible, twice punctured torus properly embedded in M with each component of ∂T_{i} having slope m_{i}/2 in ∂M. Note that T_{i} separates M since M(m_{i}/2) has finite first homology. By an isotopy of T_{i}, we may assume that T_{1} and T_{2} intersect transversely, and T_{1} ∩T_{2} has the minimal number of components. So T_{1} ∩ T_{2} is a set of finitely many circle components and arc components properly embedded in T_{i}, i = 1, 2. Furthermore, no circle component of T_{1} ∩ T_{2} bounds a disk in T_{i} and no arc component of T_{1} ∩ T_{2} is boundary parallel in T_{i}, i = 1, 2, since T_{i} is incompressible and ∂-incompressible.

We shall use the indices i and j to denote 1 or 2, with the convention that, when they are used together, {i, j} = {1, 2} as a set.

Let V_{i} denote the solid torus that is attached to M in forming M(m_{i}/2). The torus T̂_{i} intersects V_{i} in two disks B_{i(1)} and B_{i(2)}, which cut V_{i} into two 2-handles, which we denote by H_{i(1)} and H_{i(2)}. Correspondingly, ∂T_{i} = ∂B_{i(1)} ∪ ∂B_{i(2)} cuts ∂M into two annuli A_{i(1)} and A_{i(2)}, where A_{i(k)} ⊂ ∂H_{i(k)}. Each ∂T_{j} ∩ A_{i(k)} consists of exactly 8 essential arcs on A_{i(k)}.

The torus T̂_{i} separates M(m_{i}/2) into two submanifolds which we denote by X̂_{i(1)} and X̂_{i(2)}, with H_{i(k)} ⊂ X̂_{i(k)}. Correspondingly T_{i} separates M into two pieces, denoted by X_{i(1)} and X_{i(2)}. Thus X_{i(k)} = M ∩ X̂_{i(k)}. Note that X̂_{i(k)} is obtained by attaching the 2-handle H_{i(k)} to X_{i(k)} along the annulus A_{i(k)}, k = 1, 2. Also note that F_{i(k)} = ∂X_{i(k)} = T_{i} ∪ A_{i(k)} is a closed surface of genus two, k = 1, 2.

Lemma 2. For i = 1, 2, k = 1, 2, we have H_{1}(X̂_{i(k)}, T̂_{i}; Z_{2}) = 0.

Proof. Since H_{1}(M(m_{i}/2); Z_{2}) = 0, and T̂_{i} is connected, we have

0 = H_{1}(M(m_{i}/2), T̂_{i}; Z_{2}) = H_{1}(X̂_{i(1)}, T̂_{i}; Z_{2}) ⊕ H_{1}(X̂_{i(2)}, T̂_{i}; Z_{2}),

hence each H_{1}(X̂_{i(k)}, T̂_{i}; Z_{2}) = 0.

Now, as in [CGLS, 2.5] and [Go2], we construct two graphs Γ_{1} and Γ_{2} in T̂_{1} and T̂_{2} respectively by taking the arc components of T_{1} ∩ T_{2} as edges and T̂_{i} − int(T_{i}) = B_{i(1)} ∪ B_{i(2)} as (fat) vertices. So Γ_{i} is a graph on a torus with two vertices. The exterior of the graph Γ_{i} in T̂_{i} is the set of faces of Γ_{i}. Each face of the graph Γ_{i} is a surface properly embedded in X_{j(k)} for some k = 1 or 2. Since each component of ∂T_{1} intersects each component of ∂T_{2} in exactly 4 points, each of the two vertices of Γ_{i} has valency (i.e. the number of edge endpoints at the vertex) 8 and there are exactly 8 edges in Γ_{i}. If e is an edge with an endpoint at a vertex B_{i(k)} of Γ_{i} (k = 1 or 2), then that endpoint is in ∂B_{i(k)} ∩ ∂B_{j(l)} for some vertex B_{j(l)} (l = 1 or 2) of Γ_{j}, and the endpoint of e at the vertex B_{i(k)} is given the label l. So the labels of edge endpoints around each of the vertices B_{i(k)}, i = 1, 2, k = 1, 2, are as shown in Figure 1.

Figure 1

A cycle in Γ_{i} is a subgraph homeomorphic to a circle (where the fat vertices of Γ_{i} are considered as points). The number of edges in a cycle is its length. A length one cycle is also called a loop. A loop of a graph is said to be trivial if it bounds a disk face of the graph. So Γ_{i} has no trivial loops. We shall also consider two parallel loops as forming a cycle of length two. (Two edges of Γ_{i} are said to be parallel in Γ_{i} if they, together with two arcs in ∂T_{i}, bound a disk in T_{i}.) A cycle in Γ_{i} consisting of two parallel adjacent loops is called an S-cycle. This definition is a specialization to our current situation of the usual definition of an S-cycle given in [W, GL1]. The disk face of Γ_{i} bounded by an S-cycle in Γ_{i} is called the S-disk of the S-cycle.

Figure 2

Let \overline{Γ}_{i} be the reduced graph of Γ_{i}, i.e. the graph obtained from Γ_{i} by amalgamating each complete set of mutually parallel edges of Γ_{i} to a single edge. Then up to homeomorphism of T̂_{i}, \overline{Γ}_{i} is a subgraph of the graph illustrated in Figure 2 (for a proof of this, see [Go2, Lemma 5.2]). It follows in particular that in Γ_{i} the number of loops at the vertex B_{i(1)} is equal to the number of loops at the vertex B_{i(2)}, and thus the number of loops of Γ_{i} is even. From now on we will take Figure 2 as a fixed model graph of which our \overline{Γ}_{i} is a subgraph. Note that each non-loop edge in \overline{Γ}_{i} represents at most two edges of Γ_{i}, otherwise there would be a pair of edges parallel in both Γ_{i}, contradicting Lemma 3(1).

If Γ_{i} contains some loops, then they cut T_{i} into disks together with two annuli, which will be called the loop complement annuli.

Lemma 3. (1) No pair of edges can be parallel in both Γ_{1} and Γ_{2}.

(2) Any closed curve α on T_{i} intersects Γ_{i} in an even number of points. In particular, any loop complement annulus contains an even number of (non-loop) edges.

(3) (The parity rule) A component e of T_{1} ∩ T_{2} is a loop on Γ_{1} if and only if it is a non-loop on Γ_{2}. An edge of Γ_{i} has the same label at its two endpoints if and only if it is not a loop.

Proof. (1) Otherwise, the manifold M would be cabled by [Go2, Lemma 2.1], contradicting the assumption that M is hyperbolic.

(2) We have seen above that T̂_{j} is a separating torus, hence α intersects T̂_{j} in an even number of points. Since α ∩ Γ_{i} = α ∩ T̂_{j}, the result follows. For the statement about a loop complement annulus A, just take α to be the central curve of A.

(3) We fix an orientation on T_{i} and let each component of ∂T_{i} have the induced orientation. Then the two components of ∂T_{i} are not homologous in ∂M since T_{i} separates M. (The corresponding vertices of Γ_{i} are anti-parallel in the terminology of [CGLS, p. 278]). Since M, T_{1} and T_{2} are all orientable, the parity rule given in [CGLS, Page 279] holds and has the above special form in our present situation.

Lemma 4. Γ_{i} cannot have two disk faces D_{1}, D_{2}, each with an even number of edges, such that ∂D_{1} and ∂D_{2} are nonparallel, nonseparating curves on F_{j(k)} for some k = 1, 2.

Proof. Otherwise, since F_{j(k)} has genus 2, the disks D_{i} cuts X_{j(k)} into a manifold with sphere boundary, which must be a 3-ball B because X_{j(k)}, as a submanifold of S^{3} with a single boundary component, is irreducible. Thus we have

X̂_{j(k)} = T̂_{j} ∪ H_{j(k)} ∪ D_{1} ∪ D_{2} ∪ B.

Now H_{1}(T̂_{j} ∪ H_{j(k)}, T̂_{j}; Z_{2}) = Z_{2} is generated by the core of H_{j(k)}. Since ∂D_{i} has an even number of edges, it runs over H_{j(k)} an even number of times, so it represents zero in H_{1}(T̂_{j} ∪ H_{j(k)}, T̂_{j}; Z_{2}). Therefore,

H_{1}(X̂_{j(k)}, T̂_{j(k)}; Z_{2}) = H_{1}(T̂_{j} ∪ H_{j(k)}, T̂_{j(k)}; Z_{2})/ ⟨∂D_{1}, ∂D_{2}⟩ = Z_{2}/ ⟨0, 0⟩ = Z_{2}.

This contradicts Lemma 2, completing the proof.

Lemma 5. Let ∂_{i} be a boundary component of T_{i}, i = 1, 2. If the four points of ∂_{1} ∩ ∂_{2} appear in the order x_{1}, x_{2}, x_{3}, x_{4} on ∂_{1}, then they also appear in the same order on ∂_{2}, in some direction. In particular, if two of the points x_{p}, x_{q} are adjacent on ∂_{1} among the four points, then they are also adjacent on ∂_{2}.

Proof. We may choose coordinates on ∂M so that ∂_{1} is the 1/0 curve and ∂_{2} is the 1/4 curve. The lemma is obvious by drawing such curves on a torus.

We will often apply this lemma to the endpoints of a pair of edges e_{1}, e_{2}. If each e_{i} has an endpoint x_{i} at the vertex B_{1(l)}, with label k, then at B_{2(k)} e_{1} and e_{2} both have label l. The above says that x_{1}, x_{2} are adjacent on ∂B_{1(l)} among all edge endpoints labeled k if and only if they are adjacent on ∂B_{2(k)} among all edge endpoints labeled l.

By the parity rule, one of Γ_{1} and Γ_{2}, say Γ_{1}, contains at least 4 loops. So Γ_{1} has either 8, 6 or 4 loops.

Lemma 6. Γ_{i} cannot have 8 loops.

Proof. Suppose Γ_{1}, say, has 8 loops. Then the four loops based at the vertex B_{1(1)} in Γ_{1} form three S-cycles, bounding three S-disks D_{1}, D_{2} and D_{3}. The disks D_{1} and D_{3} are on the same side of T_{2}, say X_{2(1)}. The four corresponding edges of Γ_{2} connect the two vertices B_{2(1)} and B_{2(2)}, and by Lemma 3(1) they are mutually nonparallel in Γ_{2}. Thus ∂D_{1} and ∂D_{3} are nonparallel curves on the genus 2 surface F_{2(1)} = ∂X_{2(1)}. Since each ∂D_{i} is an S-cycle, it runs over H_{2(1)} twice in the same direction, so it is a nonseparating curve on F_{2(1)}. This contradicts Lemma 4.

Lemma 7. Γ_{i} cannot have 6 loops.

Proof. Suppose Γ_{1}, say, has 6 loops. By Lemma 3(2) the number of non-loop edges in each loop complement annulus A_{k} is even. Thus the two non-loop edges must be on the same A_{k}. Up to isomorphism, the graph Γ_{1} is one of the two graphs shown in Figure 3(a)–(b). However, Figure 3(a) is not possible because the longitude of the torus intersects the graph in an odd number of points, which contradicts Lemma 3(2).

Now consider Γ_{2}. For the same reason as above, the number of non-loop edges on each loop complement annulus A_{k} is even, so we have two possibilities, shown in Figure 3(c)–(d). Again, Figure 3(c) can be ruled out by looking at the intersection number between Γ_{2} and a longitude of the torus.

Consider the two edges e_{1}, e_{2} in Figure 3(d). The two endpoints x_{1}, x_{2} of e_{1}, e_{2} on ∂B_{2(2)} are adjacent among all points of ∂B_{1(1)} ∩ ∂B_{2(2)}, so by Lemma 5 they are also adjacent on ∂B_{1(1)}. From Figure 3(b) we see that the other two endpoints y_{1}, y_{2} of e_{1}, e_{2} are also adjacent on ∂B_{1(1)} among all points of ∂B_{1(1)} ∩ ∂B_{2(1)}. However, on Figure 3(d) the two endpoints of e_{1}, e_{2} on ∂B_{2(1)} are not adjacent among all edge endpoints labeled 1, which contradicts Lemma 5.

Figure 3

We may now assume that Γ_{1} has exactly four loops. By the parity rule Γ_{2} also has exactly four loops. By Lemma 3(2) each loop complement annulus A_{k} contains an even number of edges, so there are four possible configurations for the graph Γ_{i}, according to whether each A_{k} contains exactly two edges, and if there are two, whether they are parallel. See Figure 4(a)–(d). However, Figure 4(a) is impossible for each of Γ_{1} and Γ_{2} because the longitude of the torus intersects the graph in an odd number of points.

Figure 4

Lemma 8. Figure 4(b) is impossible for each of Γ_{1} and Γ_{2}.

Proof. Suppose that one of Γ_{1} and Γ_{2}, say Γ_{1}, is as shown in Figure 4(b). Let D_{1} be the S-disk bounded by the S-cycle {e_{1}, e_{2}}, and D_{2} the disk bounded by the cycle {e_{3}, e_{4}}, as shown in Figure 5.

Figure 5

Then D_{1} and D_{2} are contained in the same side of T_{2}, say in X_{2(1)}. In Γ_{2}, the two edges {e_{1}, e_{2}} also form a cycle which we denote by σ_{1}. There are two possibilities for σ_{1} = {e_{1}, e_{2}}, depending on whether or not e_{1} and e_{2} lie on the same loop complement annulus. See Figure 6.

Figure 6

It is clear that ∂D_{1} and ∂D_{2} are nonparallel curves on F_{2(1)}. Also, as the boundary of an S-disk, ∂D_{1} is always nonseparating. Now ∂D_{2} consists of one loop at each vertex of Γ_{2}, together with two arcs on the boundary of the handle H_{2(1)}. In the case of Figure 6(a), there is an arc with its endpoints on ∂D_{1}, intersecting ∂D_{2} at a single point, and hence ∂D_{2} is also nonseparating. This contradicts Lemma 4.

In the case of Figure 6(b), the two endpoints of e_{1} and e_{2} on ∂B_{2(1)} are labeled 1, say, and are adjacent among all points of ∂B_{1(1)} ∩ ∂B_{2(1)} on ∂B_{2(1)}. But from Figure 5 we can see that they are not adjacent among all edge endpoints labeled 1 on ∂B_{1(1)}, which contradicts Lemma 5.

Lemma 9. Γ_{i} cannot be as in Figure 4(c).

Proof. Suppose Γ_{1}, say, is as shown in Figure 4(c). Consider the disks D_{1}, D_{2} shown in Figure 7. They are on the same side of T_{2}, say in X_{2(1)}. The curve ∂D_{1} is an S-cycle on Γ_{1}, so it is automatically nonseparating on F_{2(1)}. The curve ∂D_{2} is also nonseparating, because it runs over H_{2(1)} three times. Clearly they are nonparallel, so by the proof of Lemma 4 we see that

X̂_{2(1)} = T̂_{2} ∪ H_{2(1)} ∪ D_{1} ∪ D_{2} ∪ B,

where B is a 3-ball. Similarly

X̂_{2(2)} = T̂_{2} ∪ H_{2(2)} ∪ D_{3} ∪ D_{4} ∪ B.

Hence

H_{1}(M(m_{2}/2); Z_{2}) = H_{1}(T̂_{2} ∪ V_{2}; Z_{2})/ ⟨∂D_{1}, ∂D_{2}, ∂D_{3}, ∂D_{4}⟩ = (Z_{2})^{4}/ ⟨∂D_{1}, … ∂D_{4}⟩ .

Each of ∂D_{1} and ∂D_{3} runs over H_{2(k)} an even number of times, so if we denote by σ_{1}, σ_{3} the corresponding cycles on Γ_{2} with each fat vertex shrunk to a point, then homologically we have ∂D_{i} = σ_{i} in H_{1}(T̂_{2} ∪ V_{2}; Z_{2}), for i = 1, 3.

Figure 7

Consider Γ_{2}. We have shown so far that Γ_{2} must be the graph in either Figure 4(c) or 4(d). In either case we have σ_{1} + σ_{3} = 0 in H_{1}(T̂_{2}; Z_{2}), so homologically we have ∂D_{1} = ∂D_{3}. Therefore,

0 = H_{1}(M(m_{2}/2); Z_{2}) = (Z_{2})^{4}/ ⟨∂D_{1}, ∂D_{2}, ∂D_{4}⟩ ≠ 0.

This contradiction completes the proof of the lemma.

Lemma 10. Γ_{1} and Γ_{2} cannot both be as in Figure 4(d).

Proof. In this case the graph Γ_{i} is contained in an annulus Q_{i} in T̂_{i}, so we can redraw the graph on Q_{i}, i = 1, 2, as in Figure 8. Orient the non-loop edges in both Γ_{i} so that they go from B_{i(1)} to B_{i(2)}. This determines the orientations of the loop edges: they must go from the endpoints labeled 1 to those labeled 2. Use e_{i}^{−}, e_{i}^{+} to denote the tail and head of e_{i}, respectively.

Label the edges of Γ_{1} as in Figure 8(a). There is a correspondence between the edges of Γ_{1} and Γ_{2}. Up to isomorphism we may assume that e_{1} on Γ_{2} is as shown in Figure 8(b). There are only two non-loop edges on Γ_{2} labeled 2. They correspond to loops in Γ_{1} based at B_{1(2)}, so the other one must be e_{2}. Now consider the four points of ∂B_{1(2)} ∩ ∂B_{2(1)}. On ∂B_{1(2)} they appear in the order e_{1}^{−}, e_{2}^{−}, e_{7}^{+}, e_{3}^{+}, so by Lemma 5 they appear in the same order on ∂B_{2(1)}. This determines the edges e_{7} and e_{3}. Similarly one can determine the correspondence between the other edges. The labeling of the edges are shown in Figure 8.

Figure 8

Let D_{1}, D_{2} be the two disk faces of Γ_{1} shown in Figure 8(a). Note that they are on different sides of T_{2}. Since D_{1} is an S-disk, and D_{2} has an odd number of edges, both ∂D_{1} and ∂D_{2} are nonseparating curves on F_{2(1)} and F_{2(2)}, respectively. Let P be the twice punctured annulus obtained from Q_{2} by removing the interiors of the fat vertices. Consider the complex Y = ∂M ∪ P ∪ D_{1} ∪ D_{2}. Denote by N(S) the regular neighborhood of a set S in M. Notice that N(∂M ∪ P) has boundary the torus ∂M and a surface F of genus three. We have shown that ∂D_{1}, ∂D_{2} are nonseparating. Since they lie on different sides of P, they are also nonparallel. Thus after adding the two 2-handles N(D_{1}) and N(D_{2}) to N(∂M ∪ P), the manifold N(Y) has boundary consisting of two tori.

We would like to calculate the homology of N(Y). Consider the complex P ∪ ∂M shown in Figure 9(a). We have determined the graph Γ_{2} on P. Up to homeomorphism there is a unique way of connecting the endpoints of e_{1}, e_{2} by arcs on A_{2(1)} to form the loop ∂D_{1}. Similarly for ∂D_{2}. See Figure 9(a)–(b). This also determines the boundary slope m_{1}/2 on ∂M, as shown in Figure 9(c). Now pick a basis {u, v, x_{1}, x_{2}} for H_{1}(∂M ∪ P) = Z^{4} as in Figure 9(d), where u is the loop ∂B_{2(1)}, v is the inner boundary circle of Q_{2}, and x_{j} runs over the annulus A_{2(j)}.

Figure 9

Consider the curves ∂D_{1}, ∂D_{2} shown in Figure 9(a)–(b). Calculating their homology classes in H_{1}(∂M ∪ P), we have

∂D_{1} = x_{1} + v + x_{1} − u = 2x_{1} + v − u,
∂D_{2} = −x_{2} + v + x_{2} + u + v − x_{2} − u = 2v − x_{2}.

The boundary slope m_{2}/2 is represented by u. From Figure 9(c) we can see that the slope m_{1}/2 is a (1, −4) curve, represented by u − 4(x_{1} + x_{2}). The meridian slope r_{0} = 1/0 of ∂M is characterized by the property that it has geometric intersection number 2 with each of m_{1}/2 and m_{2}/2, so it must be a (1, −2) curve, represented by u − 2(x_{1} + x_{2}). Thus in homology we have

r_{0} = u − 2(x_{1} + x_{2}).

Denote by W the manifold obtained from N(Y) by Dehn filling on ∂M along the meridian slope. That is, W = N(Y)(1/0) = N(Y)(r_{0}). Then W is obtained from ∂M ∪ P by adding two 2-handles along ∂D_{1}, ∂D_{2}, then adding a solid torus along the slope r_{0}. So H_{1}(W) has the presentation

⟨u, v, x_{1}, x_{2} : 2x_{1} − u + v = 2v − x_{2} = u − 2(x_{1} + x_{2}) = 0⟩
= ⟨x_{1}, x_{2} : 3x_{2} = 0⟩ = Z ⊕ Z_{3}.

We have seen that ∂N(Y) consists of two tori, hence W = N(Y)(1/0) has boundary a single torus. On the other hand, W is a submanifold of M(r_{0}) = S^{3} with boundary a torus, hence H_{1}(W) = Z. This contradiction completes the proof of the lemma, hence the proof of Theorem 1.

References

[CGLS] M. Culler, C. Gordon, J. Luecke and P. Shalen, Dehn surgery on knots, Ann. Math. 125 (1987) 237-300.

[EM] M. Eudave-Munoz, Non-hyperbolic manifolds obtained by Dehn surgery on hyperbolic knots, Proceedings of Georgia Topology Conference, 1996, Part 1, pp. 35–61.

[Go1] C. Gordon, Dehn filling: a survey, Proc. Mini Semester in Knot Theory, Warsaw 1995, (to appear).

[Go2] ——, Boundary slopes of punctured tori in 3-manifolds, Trans. Amer. Math. Soc. (to appear).

[GL1] C. Gordon and J. Luecke, Dehn surgeries on knots creating essential tori, I, Comm. in Analy. and Geo., 3 (1995) 597-644.

[GL2] ——, Dehn surgeries on knots creating essential tori, II, Comm. in Analy. and Geo., (to appear).

[K] R. Kirby, Problems in low-dimensional topology, Proceedings of Georgia Topology Conference, Part 2, 1996, pp. 35–473.

[Th] W. Thurston, The Geometry and Topology of 3-manifolds, Princeton University Press 1978.

[W] Y-Q. Wu, The reducibility of surgered 3-manifolds, Topology and its Applications 43 (1992) 213-218.

Cameron McA. Gordon, Department of Mathematics, University of Texas at Austin, Austin, TX 78712

Current address: MSRI, 1000 Centennial Drive, Berkeley, CA 94720-5070

  E-mail address: gordon@math.utexas.edu

Ying-Qing Wu, Department of Mathematics, University of Iowa, Iowa City, IA 52242 Current address: MSRI, 1000 Centennial Drive, Berkeley, CA 94720-5070

  E-mail address: wu@math.uiowa.edu

Xingru Zhang, Department of Mathematics, Oklahoma State University, Stillwater, OK 74078

  E-mail address: xingru@math.okstate.edu