Seifert Fibered Slopes and Boundary Slopes on Small Hyperbolic Knots

奈良教育大学学術リポジトリNEAR

Seifert Fibered Slopes and Boundary Slopes on Small Hyperbolic Knots

著者 ICHIHARA Kazuhiro, MOTEGI Kimihiko, SONG Hyun‑Jong

journal or

publication title

奈良教育大学紀要. 自然科学

volume 57

number 2

page range 21‑25

year 2008‑10‑31

URL http://hdl.handle.net/10105/713

1. Introduction

Let K be a knot in S

with the exterior E(K); the complement of an open tubular neighborhood of K in S

. On the boundary ∂ E(K) of E(K), let γ be a slope, i.e.

the isotopy class of an essential unoriented simple closed curve. Then, by K(γ ), we denote the 3-manifold obtained by Dehn surgery on K along γ; that is, K( γ ) is constructed by attaching a solid torus to E(K) so that a representative of γ bounds a disk in the attached solid torus.

We call a slope γ a boundary slope if γ is repre- sented by a boundary of an essential (i.e. incompress- ible and boundary-incompressible) surface in E(K).

We say that a knot K is small if there is no closed incompressible surface except for a boundary parallel torus in the exterior E(K).

For small knots, we have the following as an implicit corollary of Theorem 2.0.3 in the paper

, see also the paper

.

Theorem. Let K be a small hyperbolic knot in S

and γ a boundary slope of K. Then the result K(γ ) of γ -surgery on K can be neither a lens space nor a

Seifert fiber space with finite fundamental group.

Along the same line with the proof of this theo- rem, we have:

Proposition 1. Let K be a small hyperbolic knot in S

and γ a boundary slope of K. If K(γ ) is a Seifert fiber space with infinite fundamental group, then K is fibered and γ is the slope of the fiber, i.e. the preferred longitude of K.

The purpose in this note is to show an existence of such hyperbolic knots.

Theorem 2. There are infinitely many small, fibered hyperbolic knots K in S

with longitudinal, boundary slope γ for each of which K( γ ) is a Seifert fiber space.

This result is motivated by also works of Teragaito in the papers

Acknowledgments: The first author is supported in part by Grant-in-Aid for Young Scientists (B) (No.

18740038), and the second author is supported in part by Grant-in-Aid for Scientific Research (No.17540097), Bull. Nara Univ. Educ., Vol. 57, No.2 (Nat. ) , 2008

Seifert Fibered Slopes and Boundary Slopes on Small Hyperbolic Knots

Kazuhiro ICHIHARA, Kimihiko MOTEGI ^* and Hyun-Jong SONG

(Department of Mathematics Education, Nara University of Education, Nara 630 −8528, Japan)

(Received May 7, 2008)

Abstract

Let K be a small hyperbolic knot in the 3-sphere S

and γ a boundary slope of K. Then it is known by the cyclic surgery theorem that the result K( γ ) of γ-surgery on K can be neither a lens space nor a Seifert fiber space with finite fundamental group. In this note, we show an existence of a small hyperbolic knot K in S

such that K(γ ) is a Seifert fiber space with infinite fundamental group for a boundary slope of K. (AMS Subject Classification: 57M25.)

*Nihon University,

Pukyong National University Key Words : Dehn surgery, hyperbolic knot, Seifert

fiber space, boundary slope

The Ministry of Education, Culture, Sports, Science and Technology, Japan.

2. Proofs

Proof of Proposition 1. It follows immediately from Theorem 2.0.3 in the paper

and Corollary 8.19 in the paper

. Since K is a small knot and K( γ ) is irreducible by VI.13.Example in the book

, referring to the possi- bilities of the conclusion of Theorem 2.0.3 in the paper

, we see that K( γ ) is a Haken manifold or E(K) fibers over S

with fiber a planar surface having bound- ary slope γ. In the former case, VI.13.Example in the book

shows that H

(K(γ )) is infinite and K(γ ) is a sur- face bundle over S

and hence, γ is a longitudinal slope. Then Corollary 8.19 in the paper

shows that K is a fibered knot in S

with fiber slope γ. In the latter case, K(γ ) contains a non-separating 2-sphere, in par- ticular it is reducible, a contradiction. This completes a

proof of Proposition 1. [Q.E.D.]

Proof of Theorem 2. We use the construction given in the paper

with a slight modification. Let k∪t

∪t

∪t

be the four component link given by Figure 1; each component is a trivial knot in S

.

Figure 1: Surgery description of K

Let K

be a knot obtained from k by performing

− , − and −(2n+1)＋ -surgeries on t

, t

and t

, respectively.

Lemma 3. The knot K

is lying in S

and K

(0) is a manifold obtained from S

by performing 1, − ,

− and − (2n+1) ＋ -surgery on k, t

, t

and t

,

respectively.

Proof. After − -surgery on t

(i.e., n＋1-twist along t

) and − -surgery on t

(i.e., n-twist along t

), k

∪t

become a 2-bridge link in (new) S

with their link- ing number (n＋1) −n＝1 and new framings 2n＋2 and on k and t

, respectively (Figure 2).

Then after −(2n＋2)-twist along t

, we obtain a knot K

in S

with the framing (2n ＋2) −1

× (2n ＋2) ＝0.

[Q.E.D.]

We shall say that a Seifert fiber space is of type S

(n

, n

, n

) if it has a Seifert fibration over S

with three exceptional fibers of indices n

, n

and n

(n

≥ 2).

Lemma 4. The resulting 3-manifold K

(0) (n ≠0, −1,

−2) is a small Seifert fiber space of type S

( | ²ⁿ ＋1 | ^,

| ²ⁿ ＋3 | ^, | ^{(2n ＋1)(2n ＋3)} | ^).

Proof. To prove this we take the quotient by the strong inversion of S

as shown in Figure 3. Then we obtain a branch knot c ′ which is the image of the axis C. The Montesinos trick developed in the paper

(also see the paper

) shows that r

-surgery on t

in the upstairs corresponds to r

untangle surgery in the downstairs, i.e., a replacement of 1/0-untangle by r

- untangle. Please remark that we are adopting Bleiler's convention in the paper

on the parametrization of rational tangles. These untangle surgeries convert the knot c ′ into a link c (Figure 3).

We follow the sequence of isotopies in Figures 3 and 4. Finally we obtain a Montesinos link M( ,

− , ). Since K

(0) is a double branched

cover of S

branched over the Montesinos link M( ,

− , ), K

(0) is a Seifert fiber space of type

S

( | 2n＋1 | , | 2n＋3 | , | (2n＋1)(2n＋3) | ^{). [Q.E.D.]}

Since K

(0) is a Seifert fiber space, it follows from Proposition 1 that K

is a fibered knot in S

.

It is easy to see that K

is a trivial knot if n＝0, −1 or −2. Except for these values, we have:

2n+2 (2n+1)(2n+3) n+1

2n+1

n+1 2n+3 2n+2

(2n+1)(2n+3) n+1

2n+1

n+1 2n+3

1 2n+2

1 n

1 n+1

1 2n+2 1

n

1 n+1 1

2n+2 1

n 1 n+1

Kazuhiro Ichihara

Kimihiko Motegi

Hyun-Jong Song 22

Figure 2: Modification of the surgery description

Lemma 5. The knot K

is hyperbolic if n ≠0, −1,

−2.

Proof. Note that the 2-bridge link given in Figure 2 is a

(2, p)-torus link if and only if n＝0, −1, −2. In fact,

the 2-bridge link is a Hopf link when n ＝0, −1, −2.

Hence by the result in the paper

it is a hyperbolic link if n≠0, −1, −2. Since | 2n＋2 | > 1 if n≠−1, it follows from Theorem 1 in the paper

(see also Theorem 1.2 in the paper

) that K

is a hyperbolic knot.

See also Corollary A.2 in the paper

, Theorem 1.2 in the paper

and Theorem 1.1 in the paper

. [Q.E.D.]

Let us show that the knot K

is a small knot.

Lemma 6. Each hyperbolic knot K

(n≠0, −1, −2) is a small knot.

Proof. As we observed in the proof of Lemma 3, the knot K

is obtained from a component k of a two-bridge link L([−2n−1, 2n＋3])＝k ∪t

by twisting along the other component t

, see Figure 2.

Note that L([ −2n −1, 2n ＋3]) is isotopic to L([ − 2n −3, 2n ＋1]), and isotopic to the mirror image of L([2n＋1, −2n−3]) by their symmetries.

Now assume for a contradiction that K

(n≠0, −1,

−2) is not small. Then there is a closed essential sur- face F in E(K

). Since there is no closed essential sur- face in a two-bridge link exterior by Theorem 1 and its Remarks (1) in the paper

F intersects the dual t

of t

in E(K

). We may isotope F so that F ∩ N(t

) consists of meridian disks of N(t

) and the number of such disks is minimal. Put S = F∩E(K

∪t

), which is a properly embedded essential surface in E(K

∪t

)＝E(k ∪t

).

Then a component of ∂ S has a slope ( ≠ ) on

∂ N(t

). If S is meridionally compressible, i.e., there is a

disk D⊂S

such that D∩S＝∂ D is essential in S and D meets K

transversely in one point, then apply merid- ional surgeries until it is meridionally incompressible.

It is easy to check that meridional surgeries preserve essentiality of surfaces. Thus we obtain an essential, meridionally incompressible surface S in the 2-bridge link exterior E(k ∪ t

) satisfying property ( ＊ ):

(i) ∂ S ∩∂ N(k) ＝0 / ^{or ∂ S ∩∂ N(k) has slope , and}

(ii) ∂ S∩∂ N(t

) has slope .

Now we apply a classification of such surfaces in a 2-bridge link exterior due to the result in the paper

by Goda, Hayashi and Song, which is based on the paper

. For adaptation of their setting, if 2n＋1 > 0 (i.e., −2n

−1 < 0), then we once take the mirror image L([2n＋1,

1 2n+2

1 0

1 0 1 2n+2

Figure 3:

Figure 4: Continued from Figure 3

−2n−3]) of L([ −2n−1, 2n+ 3]) and then to obtain correct boundary slopes, we change the signs of bound- ary slopes given in the paper

. Then we have the following tables for pairs of boundary slopes on ∂ N(k) and ∂ N(t

) for some integers α, β and m; ( α , β ) ≠ (0, 0). In the table, we use the (longitude, meridian)-coor- dinates with multiplication; The usual boundary slope can be obtained by dividing each coordinate by their greatest common divisor.

Table 1: Boundary slopes in case of 2n＋1 > 0

Table 2: Boundary slopes in case of 2n＋1 < 0

Referring these tables, simple computation shows that there is no essential surface S with property (＊).

It follows that the hyperbolic knot K

(n≠0, −1, −2) is

a small knot as desired. [Q.E.D.]

Finally we show:

Lemma 7. For K

and K

(m ≠ n), we have the fol- lowing.

(1) If m＋ n＝ − 2, then there is an orientation reversing diffeomorphism of S

sending K

to K

. (2) If m ＋ n ≠−2, then there is no diffeomorphism

of S

sending K

to K

. In particular, K

and K

are not isotopic in S

if 1 ≤ m < n.

Proof. First assume that m ＋ n ＝−2. Then 2m ＋3＝

−(2n＋1) and 2m＋1＝−(2n＋3). Thus the 2-bridge link L([−2m −1, 2m＋3])＝L([2n+3, −2n−1]), which is isotopic to L([2n＋1, −2n−3]), is the mirror image of L([ −2n −1, 2n ＋3]) as we mentioned in the proof of Lemma 6. Furthermore, since 2m＋2＝−(2n＋2), K

is the mirror image of K

.

Now we suppose that m＋n≠−2. Recall that K

(0) is a Seifert fiber space of type S

( | ²ⁿ ＋1 | ^, | ²ⁿ ＋3 | ^, | ⁽²ⁿ

＋1)(2n＋3) | ). Assume that we have a diffeomorphism of S

sending K

to K

. Then K

(0) is diffeomorphic to

K

(0). It follows from VI.17.Theorem in the book

that

{ | ^2m ＋1, 2m ＋3 | ^, | ^{(2m ＋1)(2m ＋3)} | ^{} = {} | ²ⁿ ＋1 | ^, | ²ⁿ ＋

3 | ^, | (2n＋1)(2n＋3) | }. Then a simple computation shows

that m＝n. [Q.E.D.]

References

(1) Aït Nouh, M., Matignon, D. and Motegi, K.: Twisted unknots, C. R. Acad. Sci. Paris, Ser. I 337 (2003), pp.321

326.

(2) Aït Nouh, M., Matignon, D. and Motegi, K.: Obtaining graph knots by twisting unknots, Topology Appl. 146

147 (2005), pp.105−121.

(3) Aït Nouh, M., Matignon D. and Motegi, K.: Geometric types of twisted knots, Annales mathématiques Blaise Pascal, 13 (2006), pp.31− 85.

(4) Bleiler, S.A.: Prime tangles and composite knots, Lect.

Notes in Math., Vol. 1144, Springer-Verlag, 1985, pp.1−13.

(5) Bleiler, S.A.: Knots prime on many strings, Trans. Amer.

Math. Soc. 282 (1984), pp.385

401.

(6) Culler, M. , Gordon, C.McA., Luecke, J. and Shalen, P.B.:

Dehn surgery on knots, Ann. Math. 125 (1987), pp.237−

300.

(7) Floyd, W. and Hatcher, A.: The space of incompressible surfaces in a 2-bridge link complement, Trans. Amer. Math.

Soc. 305 (1988), pp.575

599.

(8) Gabai, D.: Foliations and the topology of 3manifolds III, J.

Diff. Geom. 26 (1987), pp.479

536.

(9) Goda, H. , Hayashi, C. and Song, H.-J.: Dehn surgeries on 2- bridge links which yield reducible 3-manifolds, preprint, arXiv:math/0512116.

(10) Gordon, C.McA. and Luecke, J.: Non-integral toroidal Dehn surgeries, Comm. Anal. Geom. 12 (2004), pp.417−485.

(11) Hatcher, A. and Thurston, W.: Incompressible surfaces in 2-bridge knot complements, Invent. Math. 79 (1985), pp.225−246.

(12) Jaco, W.: Lectures on three-manifold topology, Conf. Board of Math. Sci. 43, Amer. Math. Soc. 1980.

(13) Mattman, T.: The Culler-Shalen seminorms of pretzel knots, Ph.D. thesis, McGill University, 2000.

(14) Menasco, W.W.: Closed incompressible surfaces in alter- nating knot and link complements, Topology 23 (1984), pp.37

44.

(15) Miyazaki, K. and Motegi, K.: Seifert fibered manifolds and Kazuhiro Ichihara

Kimihiko Motegi

Hyun-Jong Song

24

slope on ∂ N(k) (resp. ∂ N(t

)) ∂ N(t

) (resp.∂ N(k)) ( β, (2n＋3) β − 2m) ( β, (2n＋1) β＋2m)

( α, β ) ( β, − (2n ＋1) α )

(α, (n＋1)α−n β) (β, −n α＋(n＋1)β)

(α, β) (β, α)

( α, (n＋1) α＋n β ) ( β, n α＋ (n＋3) β ) ( α, (2n ＋1) α ) ( β, (2n ＋3) β )

(α, (2n＋1)α) (β, (2n＋1)α＋2β)

(α, (2n＋3)α) (β, (2n＋1)β)

slope on ∂ N(k) (resp. ∂ N(t

)) ∂ N(t

) (resp. ∂ N(k)) ( β, (2n＋1) β＋2m) ( β, (2n＋3) β−2m)

( α, β ) ( β, − (2n ＋3) α )

(α, (n＋2)α−(n＋1)β) (β, −(n＋1)α＋(n＋2)β)

(α, β) (β, α)

( α, (n＋2) α + (n＋1) β ) ( β , (n＋1) α＋nβ ) ( α, (2n ＋3) α ) ( β, (2n ＋1) β )

(α, (2n＋3)α) (β , (2n＋3)α−2β)

(α, (2n＋1)α) (β , (2n＋3)β)

Dehn surgery III, Comm. Anal. Geom. 7 (1999), pp.551

582.

(16) Montesinos, J.M.: Surgery on links and double branched coverings of S

, Ann. Math. Studies 84 (1975), pp.227

260.

(17) Motegi, K. and Song, H.-Y.: All integral slopes can be Seifert fibered slopes for hyperbolic knots, Algebr. Geom.

Topol. 5 (2005), pp.369−378.

(18) Teragaito, M. ; Roll-spun knots, Math. Proc. Camb. Phil.

Soc. 113 (1993), pp.91

96.

(19) Teragaito, M. ; Corrigenda: “Roll-spun knots”, Math. Proc.

Camb. Phil. Soc. 116 (1994), p.191.

Address

Kazuhiro Ichihara: Nara University of Education Department of Mathematics Education, Takabatake, Nara 630−8528, Japan

e-mail: [email protected] Kimihiko Motegi: Nihon University

Department of Mathematics, 3−25−40 Sakurajosui, Setagaya-ku, Tokyo 156−8550, Japan

e-mail: [email protected] Hyun-Jong Song: Pukyong National University

Division of Mathematical Sciences, 599−1 Daeyondong, Namgu, Pusan 608−737, Korea

e-mail: [email protected]