KazuhiroIchihara OnexceptionalsurgeriesonMontesinosknots

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Exceptional surgeries on Montesinos

knots K.Ichihara

Introduction Dehn surgery Exceptional surgery Montesinos knot Problem Known facts Toroidal Seifert surgery

Known facts Result Cyclic/Finite surgery

Cyclic/Finite surgery Results On alternating knots

alternating knot Results Remains

On exceptional surgeries on Montesinos knots

Kazuhiro Ichihara

Nihon University

College of Humanities and Sciences joint works with

In Dae Jong

(OCAMI)

Shigeru Mizushima

(Tokyo Institute of Technology)

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Preprints

This talk is based on

• K. Ichihara and I.D. Jong

Cyclic and finite surgeries on Montesinos knots Algebr. Geom. Topol. 9 (2009) 731–742.

Preprint version, arXiv:0807.0905

• K. Ichihara and I.D. Jong

Toroidal Seifert fibered surgeries on Montesinos knots Preprint, arXiv:1003.3517

To apear in Comm. Anal. Geom.

• K. Ichihara, I.D. Jong and S. Mizushima

Seifert fibered surgeries on alternating Montesinos knots in preparation.

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1. Introduction

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Dehn surgery on a knot

• K : a knot in the 3-sphere S

³

• E(K): the exterior of K (:= S

³

−(open nbd. of K)) Dehn surgery on K

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

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Dehn surgery on a knot

• K : a knot in the 3-sphere S

³

• E(K): the exterior of K (:= S

³

−(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 .

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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.

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knots K.Ichihara

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.

Exceptional surgery

Dehn surgery on a hyperbolic knot

yielding a non-hyperbolic manifold is called exceptional surgery.

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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.

Exceptional surgery

Dehn surgery on a hyperbolic knot

yielding a non-hyperbolic manifold is called exceptional surgery.

An exceptional surgery is either:

• Reducible surgery

(yielding a manifold. containing an essentialS²)

• Toroidal surgery

(yielding a manifold. containing an essential T²)

• Seifert surgery

(yielding a Seifert fibered manifold.)

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

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knots K.Ichihara

Montesinos knot

Montesinos knot M (R

₁

, . . . , R

l

) in S

³

A knot admitting a diagram obtained by putting rational tangles R

₁

, . . . , R

^l

together in a circle.

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

length of the knot

= minimal number of

rational tangles M (

¹₂

,

¹₃

, −

²₃

)

P (a

1

, · · · , a

ⁿ

) = M (

a¹

, · · · ,

a¹

) : (a

1

, · · · , a

ⁿ

)-pretzel knot.

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knots K.Ichihara

Problem

Classify all the exceptional surgeries on hyperbolic Montesinos knots.

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knots K.Ichihara

Problem

Classify all the exceptional surgeries on 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 knots

have been completely classified by Moser (1971).

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knots K.Ichihara

Known facts : Length other than 3

K : hyperbolic Montesinos knot with length l

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knots K.Ichihara

Known facts : Length other than 3

K : hyperbolic Montesinos knot with length l

• l ≤ 2 ⇒ K is a two-bridge knot.

Exceptional surgeries for them are completely classified

[Brittenham-Wu (1995)].

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knots K.Ichihara

Known facts : Length other than 3

K : hyperbolic Montesinos knot with length l

• l ≤ 2 ⇒ K is a two-bridge knot.

Exceptional surgeries for them are completely classified [Brittenham-Wu (1995)].

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

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Known facts : Length other than 3

K : hyperbolic Montesinos knot with length l

• l ≤ 2 ⇒ K is a two-bridge knot.

Exceptional surgeries for them are completely classified [Brittenham-Wu (1995)].

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

Remains

Exceptional surgeries on M (R

₁

, R

₂

, R

₃

) (i.e. l = 3)

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knots K.Ichihara

Known facts : Reducible / Toroidal surgery

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

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Known facts : Reducible / Toroidal surgery

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

• Toroidal surgeries on Montesinos knots

are completely classified [Wu (2006)].

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knots K.Ichihara

Known facts : Reducible / Toroidal surgery

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

• Toroidal surgeries on Montesinos knots

are completely classified [Wu (2006)].

Remains

Seifert surgeries on M(R

₁

, R

₂

, R

₃

)

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2. Toroidal Seifert surgery

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knots K.Ichihara

Known facts : Toroidal Seifert surgery

Recall: Each exceptional surgery is either:

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

• Toroidal,

• Seifert.

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Known facts : Toroidal Seifert surgery

Recall: Each exceptional surgery is either:

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

• Toroidal,

• Seifert.

Remark [Eudave-Mu˜noz (2002)]

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

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Known facts : Toroidal Seifert surgery

Recall: Each exceptional surgery is either:

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

• Toroidal,

• Seifert.

Remark [Eudave-Mu˜noz (2002)]

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

Theorem [Motegi (2003)]

A hyperbolic knot K with |Sym

^∗

(K)| > 2

has no toroidal Seifert surgery.

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Known facts : Toroidal Seifert surgery

Recall: Each exceptional surgery is either:

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

• Toroidal,

• Seifert.

Remark [Eudave-Mu˜noz (2002)]

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

Theorem [Motegi (2003)]

A hyperbolic knot K with |Sym

^∗

(K)| > 2

has no toroidal Seifert surgery.

In particular, other than the trefoil knot,

no two-bridge knots admit toroidal Seifert surgeries.

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knots K.Ichihara

Results : Toroidal Seifert surgery

Theorem [I.-Jong (2010)]

Montesinos knots admit no toroidal Seifert surgeries other than the trefoil knot.

Corollary

No hyperbolic Montesinos knot admit toroidal Seifert surgery.

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Results : Toroidal Seifert surgery

Theorem [I.-Jong (2010)]

Montesinos knots admit no toroidal Seifert surgeries other than the trefoil knot.

Corollary

No hyperbolic Montesinos knot admit toroidal Seifert surgery.

Remains

Atoroidal Seifert surgeries on M (R

₁

, R

₂

, R

₃

)

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knots K.Ichihara

Results : Toroidal Seifert surgery

Theorem [I.-Jong (2010)]

Montesinos knots admit no toroidal Seifert surgeries other than the trefoil knot.

Corollary

No hyperbolic Montesinos knot admit toroidal Seifert surgery.

Remains

Atoroidal Seifert surgeries on M (R

₁

, R

₂

, R

₃

) Theorem [Wu (2009, 2010)]

If K = M (R

₁

, R

₂

, R

₃

) admits an atoroidal Seifert surgery, then K = P (q

₁

, q

₂

, q

₃

) or P (q

₁

, q

₂

, q

₃

, −1)

with (|q

1

|, |q

2

|, |q

3

|) = (2, ∗, ∗), (3, 3, ∗), (3, 4, 5).

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3. Cyclic/Finite surgery

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knots K.Ichihara

cyclic / finite surgery

Problem

On (hyperbolic) knots in S

³

,

determine all Dehn surgeries giving 3-manifolds with cyclic or finite fundamental groups.

We call such surgeries

cyclic surgeries / finite surgeries respectively.

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knots K.Ichihara

cyclic / finite surgery

Problem

On (hyperbolic) knots in S

³

,

determine all Dehn surgeries giving 3-manifolds with cyclic or finite fundamental groups.

We call such surgeries

cyclic surgeries / finite surgeries respectively.

Remark

· Such manifolds are all Seifert fibered.

· On non-hyperbolic knots, such surgeries are classified.

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knots K.Ichihara

Theorem We give a complete classification of

cyclic / finite surgeries on Montesinos knots.

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knots K.Ichihara

Theorem We give a complete classification of

cyclic / finite surgeries on Montesinos knots.

Theorem [I.-Jong (2009)]

K : hyperbolic Montesinos knot

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

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knots K.Ichihara

Theorem We give a complete classification of

cyclic / finite surgeries on Montesinos knots.

Theorem [I.-Jong (2009)]

K : hyperbolic Montesinos knot

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

• If π

₁

(K(r)) is cyclic,

then K = P (−2, 3, 7) and r = 18 or 19.

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knots K.Ichihara

Theorem We give a complete classification of

cyclic / finite surgeries on Montesinos knots.

Theorem [I.-Jong (2009)]

K : hyperbolic Montesinos knot

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

• If π

₁

(K(r)) is cyclic,

then K = P (−2, 3, 7) and r = 18 or 19.

• If π

₁

(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.

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4. On alternating knots

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Alternating knots

alternating knot

An alternating diagram = the crossings alternate

under, over, under, over, as you travel along the knot.

A knot is alternating if it admits an alternating diagram.

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knots K.Ichihara

Alternating knots

alternating knot

An alternating diagram = the crossings alternate

under, 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 if

its reduced Montesinos diagram is alternating.

In particular, M(R

₁

, . . . , R

l

) is alternating

if R

₁

, . . . , R

l

have the same sign.

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knots K.Ichihara

Results : atoroidal Seifert surgery

Theorem [I.-Jong-Mizushima]

If K = M (R

₁

, R

₂

, R

₃

) with R

₁

, R

₂

, R

₃

> 0

(i.e. K is alternating) admits an atoroidal Seifert surgery,

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

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knots K.Ichihara

Results : atoroidal Seifert surgery

Theorem [I.-Jong-Mizushima]

If K = M (R

₁

, R

₂

, R

₃

) with R

₁

, R

₂

, R

₃

> 0

(i.e. K is alternating) admits an atoroidal Seifert surgery, then K = P (a, b, c) with odd integers 3 ≤ a<b<c.

Theorem [Wu (2009, 2010)]

If K = M (R

₁

, R

₂

, R

₃

) admits an atoroidal Seifert surgery, then K = P (q

₁

, q

₂

, q

₃

) or P (q

₁

, q

₂

, q

₃

, −1)

with (|q

₁

|, |q

₂

|, |q

₃

|) = (2, ∗, ∗), (3, 3, ∗), (3, 4, 5).

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knots K.Ichihara

Results : atoroidal Seifert surgery

Theorem [I.-Jong-Mizushima]

If K = M (R

₁

, R

₂

, R

₃

) with R

₁

, R

₂

, R

₃

> 0

(i.e. K is alternating) admits an atoroidal Seifert surgery, then K = P (a, b, c) with odd integers 3 ≤ a<b<c.

Theorem [Wu (2009, 2010)]

If K = M (R

₁

, R

₂

, R

₃

) admits an atoroidal Seifert surgery, then K = P (q

₁

, q

₂

, q

₃

) or P (q

₁

, q

₂

, q

₃

, −1)

with (|q

₁

|, |q

₂

|, |q

₃

|) = (2, ∗, ∗), (3, 3, ∗), (3, 4, 5).

Corollary

An alternating hyperbolic Montesinos knot with length 3

admits no Seifert surgery.

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5. Remains

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Remains

Atoroidal Seifert surgeries (not cyclic/finite) on P (q

₁

, q

₂

, q

₃

) or P (q

₁

, q

₂

, q

₃

, −1)

non-alternating and (|q

₁

|, |q

₂

|, |q

₃

|) = (2, ∗, ∗), (3, 3, ∗), (3, 4, 5).

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knots K.Ichihara

Remains

Atoroidal Seifert surgeries (not cyclic/finite) on P (q

₁

, q

₂

, q

₃

) or P (q

₁

, q

₂

, q

₃

, −1)

non-alternating and (|q

₁

|, |q

₂

|, |q

₃

|) = (2, ∗, ∗), (3, 3, ∗), (3, 4, 5).

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

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

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Remains

Atoroidal Seifert surgeries (not cyclic/finite) on P (q

₁

, q

₂

, q

₃

) or P (q

₁

, q

₂

, q

₃

, −1)

non-alternating and (|q

₁

|, |q

₂

|, |q

₃