Toroidal Dehn Surgery on Hyperbolic Knots and Hitting Number ∗ Masakazu Teragaito ,1

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Topology and its Applications

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Toroidal Dehn surgery on hyperbolic knots and hitting number ∗ Masakazu Teragaito ,1

Department of Mathematics and Mathematics Education, Hiroshima University, 1-1-1 Kagamiyama, Higashi-hiroshima 739-8524, Japan

article info abstract

Article history: In this paper, we prove that for any positive even integer m,thereexistsahyperbolic DedicatedtoProfessorAkioKawauchifor knot such that its longitudinal Dehn surgery yields a 3-manifold containing a unique his 60th birthday separating, incompressible torus, which meets the core of the attached solid torus in m points minimally. MSC: © 2009 Elsevier B.V. All rights reserved. 57M25

Keywords: Dehn surgery Hyperbolic knot Toroidal manifold Hitting number

1. Introduction

For a hyperbolic knot K in the 3-sphere S3, there are at most ﬁnitely many Dehn surgeries which yield non-hyperbolic 3-manifolds by Thurston’s hyperbolic Dehn surgery theorem [15]. Such surgeries are called exceptional Dehn surgeries.Fur- thermore, any exceptional Dehn surgery is conjectured to yield either a Seifert ﬁbered manifold or a toroidal manifold [6]. Recall that a closed 3-manifold is said to be toroidal if it contains an incompressible torus. When Dehn surgery yields a toroidal manifold, it is called a toroidal Dehn surgery. Let E(K ) = S3 − Int N(K ) be the knot exterior of K .Aslope r is the isotopy class of an essential simple closed curve on ∂ E(K ). As usual [13], r is parameterized as p/q ∈ Q ∪{1/0},ifr runs p times along a meridian and q times along a longitude. We denote by K (r) the resulting 3-manifold by r-Dehn surgery on K . That is, K (r) is the union of E(K ) and a solid torus V along their boundaries such that a meridian of V is mapped to a simple closed curve on ∂ E(K ) with slope r. ∗ The core of the attached solid torus V is denoted by K . ∗ Suppose that K (r) is toroidal. For an incompressible torus T in K (r),let|K ∩ T | denote the minimal geometric inter- ∗ section number between K and an incompressible torus T .Forapair(K , r), ∗ min K ∩ T : T is an incompressible torus in K (r)

gives one way to measure the complexity of the toroidal Dehn surgery. We call this number the hitting number of (K , r) or the toroidal Dehn surgery of K .SinceK is hyperbolic, a hitting number is positive. When K has genus one, 0-surgery is toroidal [3]; indeed, its minimal genus Seifert surface becomes an incompressible torus after 0-surgery, so that the pair (K , 0) has hitting number one. In general, if the hitting number of (K , r) is an odd

* Tel.: +81 824 247079; fax: +81 824 247084. E-mail address: [email protected]. 1 Partially supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientiﬁc Research (C), 19540089.

Fig. 1. The link K0 ∪ C1 ∪ C2.

Fig. 2. The same link K0 ∪ C1 ∪ C2. integer, then K (r) contains a non-separating torus, hence r = 0. Furthermore, K must have genus one by [3, Corollary 8.3], and the hitting number is forced to be one. Thus a natural problem is the realization of even positive integers as hitting numbers. By [7], any toroidal Dehn surgery corresponds to an integer or a half-integer. Also, the knots constructed by Eudave- Munõz [1] are the only hyperbolic knots that admit non-integral toroidal Dehn surgeries [9]. It is shown in [7,8] that any non-integral toroidal Dehn surgery on a hyperbolic knot has hitting number two. Since most familiar examples of integral toroidal Dehn surgery have hitting number two, Luecke asked whether this is always the case in [10, p. 592], but Eudave-Munõz [2] gave an infinite family of strongly-invertible hyperbolic knots, each having a toroidal Dehn surgery with hitting number 4. We note that Eudave-Munõz’s knots are complicated, in particular, have large genus, but 1-surgery on the (−3, 3, 7)-pretzel knot (of genus one) is a toroidal Dehn surgery with hitting number 4, according to [16]. Starting from the (−3, 3, 7)-pretzel knot, we can construct an infinite family of non-invertible hyperbolic knots of genus one, each having a toroidal Dehn surgery with hitting number 4, but this will be treated in a future paper. Along this line, Osoinach [11] gave an infinite family of knots {Kn}, each of whose longitudinal Dehn surgery, in other words, 0-surgery, is a toroidal Dehn surgery with hitting number at least (2n − 72)/35. (In the published paper [12], the family {Kn} is described, but the estimate for hitting number is omitted.) Also, he only showed that the family contains an infinite subset of hyperbolic knots. The purpose of this paper is to confirm that Osoinach’s knots Kn (the construction will be given in Section 2) are hyperbolic unless n = 0, and that the hitting number of (Kn, 0) is exactly 2n. Thus we obtain the complete answer to the realization problem of even integers as hitting numbers.

Theorem 1. For any even positive integer m, there exists a hyperbolic knot in S3 for which longitudinal Dehn surgery is toroidal and its hitting number is equal to m.

2. Osoinach’s construction

First, we review Osoinach’s construction in [11,12]. Let us consider the link K0 ∪ C1 ∪ C2 as illustrated in Fig. 1. Here, K0 is the connected sum of the ﬁgure-eight knot and its mirror image. Fig. 2 shows the same link, but K0 is deformed into a ribbon presentation. 3 For a positive integer n,letKn be the knot in S obtained from K0 by 1/n-Dehn surgery on C1 and (−1/n)-Dehn surgery on C2. Thus Fig. 2 is helpful to see Kn,becauseKn is obtained by adding (−n)-full twists along C1 and n-full twists along C2 to K0. M. Teragaito / Topology and its Applications 157 (2010) 269–273 271

Fig. 3. Aknot in a solid torus.

Fig. 4. (−1)-surgery on yields the ﬁgure-eight knot exterior.

Proposition 2. ([11,12]) Longitudinal Dehn surgery on Kn yields the same toroidal manifold K0(0) that contains a unique (separating) incompressible torus.

Proof. The ﬁrst conclusion follows from Theorem 2.3 of [12]. We note that K0(0) is the union of two copies of the ﬁgure- eight knot exterior E. (The identifying map sends a meridian and longitude on one side to a meridian and longitude of the other side.) Since E is hyperbolic, K0(0) admits a unique incompressible torus T which separates K0(0) into two copies of E. 2

In [11,12], Osoinach shows that the link K0 ∪ C1 ∪ C2 is hyperbolic by considering its tangle decomposition, and con- cludes that the family {Kn} contains an inﬁnite subset consisting of hyperbolic knots by Thurston’s hyperbolic Dehn surgery theorem. We will show that Kn is hyperbolic, unless n = 0, in the next section.

Remark 3. In Fig. 2, the π -rotation around an axis perpendicular to the projection plane shows the amphicheirality of Kn for 1 any n.Also,Kn is a genus two fibered knot. For, since K0 is fibered, K0(0) is a surface bundle over S .ThenKn(0) = K0(0) implies that Kn is fibered by [3]. In addition, the genus of a knot equals to the minimal genus of non-separating closed orientable surface contained in the 0-surgered manifold. That is, the genus depends only on the 0-surgered manifold. Thus any Kn has genus two as well as K0.

3. Hitting number

Lemma 4. Let be the knot in the standard solid torus N in S3 as illustrated in Fig. 3.LetN be the resulting manifold by (−1)-surgery on .ThenN is the ﬁgure-eight knot exterior, and the arc γ as shown there gives an unknotting tunnel for N , that is, N − Int N(γ ) isagenustwohandlebody.

Proof. Let K be the core of the complementary solid torus cl(S3 − N).After(−1)-surgery on , the image of K ,denotedby K again, is a knot in S3 whose exterior is N . See Fig. 4. The conclusion follows from the deformation as shown in Fig. 4. 2 272 M. Teragaito / Topology and its Applications 157 (2010) 269–273

Fig. 5. AsurgerydescriptionofK0(0).

Fig. 6. A modiﬁed surgery description of K0(0).

∗ ∗ Fig. 7. The core Kn is obtained by twisting K0 along A .

By Proposition 2, longitudinal Dehn surgery on Kn is toroidal. Let T be the unique separating incompressible torus ∗ in Kn(0), and let Kn be the core of the attached solid torus.

Lemma 5. The hitting number of (Kn, 0) is 2n.

∗ Proof. It suﬃces to show that the minimal geometric intersection number between the core Kn and T is exactly 2n. Fig.5showsasurgerydescriptionofK0(0) obtained from Fig. 1 by inserting two extra unknotted components with coeﬃcients 1 and −1. There is a once-punctured annulus A bounded by K0 ∪ C1 ∪ C2, each of whose boundary components has slope 0 on N(K0), N(C1) or N(C2).Byperforming0-surgeryonK0, A is capped off with a disk, and becomes an annulus A bounded by C1 and C2 in K0(0).Ifweperform1/n-surgery on C1 and (−1/n)-surgery on C2 in K0(0), K0(0) will be changed to Kn(0). However, the twisting along A , which does not change the ambient manifold, cancels these surgeries on C1 and C2. This is the reason why Kn(0) is the same as K0(0) (see [12]). We modify the surgery description of Fig. 5 keeping A visible into Fig. 6, where an extra unknotted circle with framing 1 is inserted. Then (−1)-twisting along K0 yields the surgery description of K0(0) as in Fig. 7, where the annulus A bounded ∗ ∗ ∗ by C1 and C2 meets K0 in one point. The core Kn is obtained from K0 by n-time-twisting along A in K0(0).Thedirection of the twisting is indicated by an arrow in Fig. 7. M. Teragaito / Topology and its Applications 157 (2010) 269–273 273

Fig. 8. Two incompressible tori.

As shown in Fig. 8, there are two incompressible tori T1 and T2 in K0(0), but they are indeed parallel to the unique essential torus T . Also, each of T1 and T2 bounds the ﬁgure-eight knot exterior by Lemma 4, where the core of A is an unknotting tunnel. ∗ Clearly, Kn meets T1,say,in2n points. Then Corollary 1.2 and Lemma 1.3 of [2] (or [11, Claim 3.1.4]) claim that this ×[ ] ∗ number is minimal. (It is not hard to see that the region between T1 and T2 has a product structure T1 0, 1 , where Kn is decomposed into 2n vertical arcs. We omit the detail.) 2

Lemma 6. For n > 0,Kn is hyperbolic.

Proof. It is well known that no surgery on a torus knot produces a toroidal manifold containing a separating incompressible torus. Hence Kn is not a torus knot by Proposition 2. Suppose that Kn is a satellite knot. Then its knot exterior contains an essential torus F .Let J be the solid torus bounded 3 by F in S , which contains Kn in its interior. Also, let J be the resulting manifold obtained from J by 0-surgery on Kn. In Kn(0), F must be compressible by Lemma 5. This implies that J has a compressible boundary. Hence, J is either a solid torus or the connected sum of a solid torus and a lens space by [14]. Since Kn(0) is irreducible and not a lens space, the latter is not the case. Thus J is a solid torus, and Kn is a 0- or 1-bridge braid in J by [4,14]. In particular, its winding number w is not zero. Then H1( J ) = Z ⊕ Zw (see [5, Lemma 3.3(i)]), a contradiction. 2

Proof of Theorem 1. This immediately follows from Proposition 2, Lemmas 5 and 6. 2

References

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