On exceptional surgeries on Montesinos knotsichihara/Research/... · Algebr. Geom. Topol. 9 (2009)...

Exceptionalsurgeries onMontesinos

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

ToroidalSeifert surgery

Known facts

Result

Cyclic/Finitesurgery

Results

On alternatingknots

alternating knot

Results

Remains

On exceptional surgerieson Montesinos knots

Kazuhiro Ichihara

Nihon UniversityCollege of Humanities and Sciences

joint works with

In Dae Jong(OCAMI)

Shigeru Mizushima(Tokyo Institute of Technology)

V-JAMEX, Colima, Mexico, Sep 29, 20101 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Preprints

This talk is based on

• K. Ichihara and I.D. JongCyclic and finite surgeries on Montesinos knotsAlgebr. Geom. Topol. 9 (2009) 731–742.Preprint version, arXiv:0807.0905

• K. Ichihara and I.D. JongToroidal Seifert fibered surgeries on Montesinos knots

Preprint, arXiv:1003.3517To apear in Comm. Anal. Geom.

• K. Ichihara, I.D. Jong and S. MizushimaSeifert fibered surgeries on alternating Montesinos knots

in preparation.

2 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

1. Introduction

3 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Dehn surgery on a knot

• K : a knot in the 3-sphere S3

• E(K): the exterior of K (:= S3−(open nbd. of K))

Dehn surgery on K

Gluing a solid torus V to E(K) to obtain a closed manifold.

4 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Dehn surgery on a knot

• K : a knot in the 3-sphere S3

• E(K): the exterior of K (:= S3−(open nbd. of K))

Dehn surgery on K

Gluing a solid torus V to E(K) to obtain a closed manifold.

K(r): the manifold obtained by Dehn surgery on K along r.

r ∈ Q ∪ {1/0}: surgery slope, corresponding to [ f(m) ] ,

where f : ∂V → ∂E(K), m: meridian of V .

4 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Exceptional surgery

Theorem [Thurston (1978)]

Dehn surgeries on a hyperbolic knot(i.e., knot with hyperbolic complement)

yielding a non-hyperbolic manifold are only finitely many.

5 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Exceptional surgery

Dehn surgery on a hyperbolic knotyielding a non-hyperbolic manifold is called exceptional surgery.

5 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Exceptional surgery

Dehn surgery on a hyperbolic knotyielding a non-hyperbolic manifold is called exceptional surgery.

An exceptional surgery is either:

• Reducible surgery (yielding a manifold. containing an essential S2)

• Toroidal surgery (yielding a manifold. containing an essential T2)

• Seifert surgery (yielding a Seifert fibered manifold.)

as a consequence of the Geometrization Conjectureestablished by Perelman (2002-03).

5 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Montesinos knot

Montesinos knot M(R1, . . . , Rl) in S3

A knot admitting a diagram obtained by putting rationaltangles R1, . . . , Rl together in a circle.

arcs on a 4-punctured sphere, and 12-tangle

length of the knot= minimal number of

rational tangles M(12 , 1

3 ,−23)

P (a1, · · · , an) = M( 1a1

, · · · , 1an

) : (a1, · · · , an)-pretzel knot.

6 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Problem

Classify all the exceptional surgerieson hyperbolic Montesinos knots.

7 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Problem

Classify all the exceptional surgerieson hyperbolic Montesinos knots.

Remark [Menasco], [Oertel], [Bonahon-Siebenmann]

Non-hyperbolic Montesinos knots are

T (2, n), P (−2, 3, 3)(=T (3, 4)), P (−2, 3, 5)(=T (3, 5)).

T (x, y) : the (x, y)-torus knot.

Remark

Dehn surgeries on torus knotshave been completely classified by Moser (1971).

7 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Known facts : Length other than 3

K : hyperbolic Montesinos knot with length l

8 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

• l ≤ 2 ⇒ K is a two-bridge knot.Exceptional surgeries for them are completely classified

[Brittenham-Wu (1995)].

8 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

• l ≥ 4 ⇒ K admits no exceptional surgery [Wu (1996)].

8 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

• l ≥ 4 ⇒ K admits no exceptional surgery [Wu (1996)].

Remains

Exceptional surgeries on M(R1, R2, R3) (i.e. l = 3)

8 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Known facts : Reducible / Toroidal surgery

• 6 ∃ reducible surgeries on Montesinos knots [Wu (1996)].

9 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

• Toroidal surgeries on Montesinos knotsare completely classified [Wu (2006)].

9 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

• Toroidal surgeries on Montesinos knotsare completely classified [Wu (2006)].

Remains

Seifert surgeries on M(R1, R2, R3)

9 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

2. Toroidal Seifert surgery

10 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Known facts : Toroidal Seifert surgery

Recall: Each exceptional surgery is either:

• Reducible (conjectured:6 ∃ (Cabling Conjecture)),

• Toroidal,

• Seifert.

11 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

• Toroidal,

• Seifert.

Remark [Eudave-Munoz (2002)]

They are not exclusive (i.e., there are non-empty intersection).

11 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

• Toroidal,

• Seifert.

Theorem [Motegi (2003)]

A hyperbolic knot K with |Sym∗(K)| > 2has no toroidal Seifert surgery.

11 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

• Toroidal,

• Seifert.

Theorem [Motegi (2003)]

A hyperbolic knot K with |Sym∗(K)| > 2has no toroidal Seifert surgery.

In particular, other than the trefoil knot,no two-bridge knots admit toroidal Seifert surgeries.

11 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Results : Toroidal Seifert surgery

Theorem [I.-Jong (2010)]

Montesinos knots admit no toroidal Seifert surgeriesother than the trefoil knot.

Corollary

No hyperbolic Montesinos knot admit toroidal Seifert surgery.

12 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Corollary

Remains

Atoroidal Seifert surgeries on M(R1, R2, R3)

12 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Corollary

Remains

Atoroidal Seifert surgeries on M(R1, R2, R3)

Theorem [Wu (2009, 2010)]

If K = M(R1, R2, R3) admits an atoroidal Seifert surgery,then K = P (q1, q2, q3) or P (q1, q2, q3,−1)

with (|q1|, |q2|, |q3|) = (2, ∗, ∗), (3, 3, ∗), (3, 4, 5).

12 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

3. Cyclic/Finite surgery

13 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

cyclic / finite surgery

Problem

On (hyperbolic) knots in S3,determine all Dehn surgeries giving 3-manifoldswith cyclic or finite fundamental groups.

We call such surgeriescyclic surgeries / finite surgeries respectively.

14 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

cyclic / finite surgery

Problem

On (hyperbolic) knots in S3,determine all Dehn surgeries giving 3-manifoldswith cyclic or finite fundamental groups.

We call such surgeriescyclic surgeries / finite surgeries respectively.

Remark

· Such manifolds are all Seifert fibered.· On non-hyperbolic knots, such surgeries are classified.

14 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Theorem

We give a complete classification ofcyclic / finite surgeries on Montesinos knots.

15 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Theorem

K : hyperbolic Montesinos knotK(r): the manifold obtained by surgery on K along r.

15 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Theorem

• If π1(K(r)) is cyclic,then K = P (−2, 3, 7) and r = 18 or 19.

15 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Theorem

• If π1(K(r)) is cyclic,then K = P (−2, 3, 7) and r = 18 or 19.

• If π1(K(r)) is acyclic finite,then K = P (−2, 3, 7) and r = 17, or

K = P (−2, 3, 9) and r = 22 or 23.

15 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

4. On alternating knots

16 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Alternating knots

alternating knot

An alternating diagram = the crossings alternateunder, over, under, over, as you travel along the knot.

A knot is alternating if it admits an alternating diagram.

17 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Alternating knots

alternating knot

An alternating diagram = the crossings alternateunder, over, under, over, as you travel along the knot.

A knot is alternating if it admits an alternating diagram.

Remark [Lickorish-Thistlethwaite]

A Montesinos knot is alternating if and only ifits reduced Montesinos diagram is alternating.

In particular, M(R1, . . . , Rl) is alternatingif R1, . . . , Rl have the same sign.

17 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Results : atoroidal Seifert surgery

Theorem [I.-Jong-Mizushima]

If K = M(R1, R2, R3) with R1, R2, R3 > 0(i.e. K is alternating) admits an atoroidal Seifert surgery,

then K = P (a, b, c) with odd integers 3 ≤ a<b<c.

18 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Theorem [Wu (2009, 2010)]

with (|q1|, |q2|, |q3|) = (2, ∗, ∗), (3, 3, ∗), (3, 4, 5).

18 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Theorem [Wu (2009, 2010)]

with (|q1|, |q2|, |q3|) = (2, ∗, ∗), (3, 3, ∗), (3, 4, 5).

Corollary

An alternating hyperbolic Montesinos knot with length 3admits no Seifert surgery.

18 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

5. Remains

19 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

Atoroidal Seifert surgeries (not cyclic/finite) onP (q1, q2, q3) or P (q1, q2, q3,−1)

non-alternating and (|q1|, |q2|, |q3|) = (2, ∗, ∗), (3, 3, ∗), (3, 4, 5).

20 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

In progress : Exceptional surgeries on P (−2, n, n)

(joint work with I.D. Jong and Y. Kabaya)

20 / 20

K.Ichihara

Introduction

Dehn surgery

Exceptionalsurgery

Montesinos knot

Problem

Known facts

Result

Results

On alternatingknots

alternating knot

Results

Remains

In progress : Exceptional surgeries on P (−2, n, n)

(joint work with I.D. Jong and Y. Kabaya)

Thanks!

20 / 20

On exceptional surgeries on Montesinos knotsichihara/Research/... · Algebr. Geom. Topol. 9 (2009)...

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