On exceptional surgeries on Montesinos knots

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Montesinos knots K.Ichihara

Introduction Dehn surgery Exceptional surgery Montesinos knot

Known facts Length other than 3 Reducible/Toroidal surgery

Results Cyclic / Finite surgery On alternating knots Toroidal Seifert surgery

Proof Claim 1 Claim 2 Claim 3 Claim 4

On exceptional surgeries on Montesinos knots

市原一裕

(Kazuhiro Ichihara)

奈良教育大学(Nara University of Education)

joint work with In Dae Jong

(Osaka City University) and Shigeru Mizushima (Tokyo Institute of Technology)

Intelligence of Low Dimensional Topology, Nov.14,2009

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1. Introduction

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Dehn surgery on knot

γ m

f

γ = [ f (m) ] , (isotopy class)

: surgery slope, identiﬁed with r ∈ Q ∪ { 1/0 } K(r): the manifold obtained by surgery on K

along the slope corresponding to r

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Exceptional surgery

Dehn surgeries on a hyperbolic knot giving non-hyperbolic mfds [Thurston] They are only ﬁnitely many.

Each exceptional surgery is either:

•

Reducible (essential 2-sphere)

•

Toroidal (essential torus)

•

Seifert ﬁbered (foliation by circles) as a consequence of

the Geometrization Conjecture

established by G.Perelman (2002-03).

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Montesinos knot

M (

¹₂

,

¹₃

, −

²₃

) ⇒

length of the knot

= minimal number of rational tangles

In particular, a Montesinos knot K is called a (a

₁

, · · · , a

_n

)-pretzel knot, P (a

₁

, · · · , a

_n

) if the rational tangles are

_a¹

1

, · · · ,

_a¹

n

.

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2. Known facts

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Length other than 3

Montesinos knot of length ≤ 2 is two-bridge, on which exceptional surgeries

are completely classiﬁed [Brittenham-Wu ’95].

Montesinos knot of length ≥ 4

has NO exceptional surgery [Wu ’96].

Remark:

Non-hyperbolic Montesinos knots are T (2, n) (two-bridge) and,

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

([Oertel ’84], [Bonahon-Siebenmann])

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Reducible/Toroidal surgery

There are NO reducible surgeries

on Montesinos knots [Wu ’96].

Toroidal surgeries on Montesinos knots

are completely classiﬁed [Wu ’06].

¶ ³

In the following:

K: hyperbolic Montesinos knot of length 3 and consider Seifert ﬁbered surgery

µ ´

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3. Results

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Cyclic / Finite surgery

i.e. π

1

(K(r)) is cyclic / ﬁnite.

Theorem 1 [I.-Jong ’09]

(i) If π

1

(K(r)) is cyclic,

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

K. Ichihara and I.D. Jong

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

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Seifert ﬁbered surgery on alternating knots

K: hyperbolic Montesinos knot of length 3

Theorem 2 [I.-Jong-Mizushima]

If K is alternating,

then K(r) is never Seifert ﬁbered.

Remark:

A Montesinos knot is alternating if and only if

its reduced Montesinos diagram is alternating.

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The new result of Wu

Theorem [Wu ’09]

If M (

^p_q¹

1

,

^p_q²

2

,

^p_q³

3

) with q

₁

≤ q

₂

≤ q

₃

admits an atoroidal Seifert ﬁbered surgery, then

q₁ = 2, (q₁, q₂) = (3,3), or (q₁, q₂, q₃) = (3,4,5).

[arXiv:0910.4882] Ying-Qing Wu

Immersed surfaces and Seifert ﬁbered surgery on Montesinos knots

On the other hand, we have already proved:

q

₁

, q

₂

, q

₃

are all odd and mutually distinct.

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Toroidal Seifert surgery

Recall: Each exceptional surgery is either:

•

Reducible (essential 2-sphere)

•

Toroidal (essential torus)

•

Seifert ﬁbered (foliation by circles) Remark: They are NOT exclusive.

i.e., there are non-empty intersection.

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Known results on toroidal Seifert surgery

Example: [Eudave-Mu˜ noz ’02]

∃ knots with toroidal Seifert ﬁbered surgery.

Theorem [Motegi ’03]

Knots with | Sym

^∗

(K ) | > 2 has

no toroidal Seifert ﬁbered surgery.

In particular, other than trefoil ,

no two-bridge knots admit such surgeries.

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Results on Toroidal Seifert surgery

Theorem 3 [I.-Jong]

Non-pretzel Montesinos knots admit

no toroidal Seifert ﬁbered surgeries.

Theorem 4 [I.-Jong-Mizushima]

Alternating knots, other than

trefoil and connected sums of T(2,n)’s ,

admit no toroidal Seifert ﬁbered surgeries.

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4. Proof

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Proof of [Thm 3.]

K: non-pretzel Montesinos knot

Suppose : K(r) is toroidal Seifert ﬁbered.

We may assume: K is hyperbolic & length=3.

Claim 1.

K must be a ﬁbered knot and r = 0.

This follows from:

Proposition 1 [I.-Motegi-Song ’08]

and fact that K is small [Oertel ’84].

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Proof of [Thm 3.]

Claim 1 + Classiﬁcation of toroidal surgery on Montesinos knots [Wu ’06]:

⇓ Claim 2.

K must be M ( −

¹₂

,

¹₃

,

¹

6±¹₂

).

Now we show:

Claim 3.

K 6=M(−¹₂,¹₃, ¹

6+¹₂)

Claim 4.

K 6=M(−¹₂,¹₃, ¹

6−¹₂)

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Proof of [Clm 3.]

Suppose: K = M ( −

¹₂

,

¹₃

,

₆₊¹1 2

)

& K(0) is Seifert ﬁbered.

⇒ Monodromy map for K(0) is periodic .

⇒ ∆

_K

(t) has a root of unity as zeros .

⇒⇐ ∆

K

(t) = − (t

²

+ t

⁻²

) + 3

for K = M ( −

¹₂

,

¹₃

,

₆₊¹1 2

).

¤

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Proof of [Clm 4.]

Suppose:

K = M ( −

¹₂

,

¹₃

,

₆₋¹1 2

)

& K(0) is Seifert ﬁbered.

Kirby moves

= ⇒

framed link presentationof K(0)

−1/3

0 0

−1/3

0

−1 −1

1 1

−1/3

=

⇒ Monodromy of K(0) is

τ

₅³

◦ τ

₃

◦ τ

₁⁻¹

◦ τ

₄

◦ τ

₂⁻¹

(τ_i: Dehn twist along i-th Lickorish generator on F₂)

.

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Proof of [Clm 4.]

⇒ K (0) → S

²

× S

¹

, double-branched cover along ˆ β with β = σ

³₅

σ

₃

σ

⁻₁¹

σ

₄

σ

₂⁻¹

(σ_i: i-th generator of braid groupB₆(S²))

.

⇒ exp(β) = 3 6≡ 0, 5 mod 10.

⇒ β 6∼ = ∆

^k

for

^∀

k

i.e., β is not conjugate to a periodic braid .

by

Proposition 2.3, Chap.11, Murasugi-Kurpita ’99

exp(β) mod 2(m − 1) is

an invariant for braids in B

_m

( S

²