Toroidal Seifert fibered surgeries on Montesinos knots

.

... .

.

.

Toroidal Seifert fibered surgeries on Montesinos knots

鄭 仁大 (In Dae Jong)

Osaka City University Advanced Mathematical Institute (OCAMI)

Joint work with Kazuhiro Ichihara (Nihon University)

E-KOOK Seminar 2010 2010/8/27 10:40–11:10

Osaka City University, Media center

Dehn surgery on a knot

K : a knot in S³

E(K): the exterior of K (i.e., S³\N^◦(K))

.Dehn surgery : Gluing a solid torus to E(K)

. .

... .

.

.

γ = [f(m) ] : surgery slope, identified with r∈Q∪ {1/0}. K(r): the manifold obtained by Dehn surgery on K along γ =r. .Theorem [Lickorish, Wallace]

. .

... .

.

.

Every pair of closed orientable 3-manifolds are related by a finite sequence of Dehn surgeries.

Dehn surgery on a knot

K : a knot in S³

E(K): the exterior of K (i.e., S³\N^◦(K))

.Dehn surgery : Gluing a solid torus to E(K)

. .

... .

.

.

γ m

f

γ = [f(m) ] : surgery slope, identified with r∈Q∪ {1/0}. K(r): the manifold obtained by Dehn surgery on K along γ =r.

.Theorem [Lickorish, Wallace]

. .

... .

.

.

Every pair of closed orientable 3-manifolds are related by a finite sequence of Dehn surgeries.

Dehn surgery on a knot

K : a knot in S³

E(K): the exterior of K (i.e., S³\N^◦(K))

.Dehn surgery : Gluing a solid torus to E(K)

. .

... .

.

.

γ m

f

γ = [f(m) ] : surgery slope, identified with r∈Q∪ {1/0}. K(r): the manifold obtained by Dehn surgery on K along γ =r.

.Theorem [Lickorish, Wallace] ^.

Exceptional surgery

.Exceptional surgery

. .

... .

. .Dehn surgery on ahyperbolic knot yielding a non-hyperbolic mfd.

.Theorem [Thurston]

. .

... .

.

.

Exceptional surgeries are only finitely manyfor each hyperbolic knot.

Each exceptional surgery is either:

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

Exceptional surgery

.Exceptional surgery

. .

... .

. .Dehn surgery on ahyperbolic knot yielding a non-hyperbolic mfd.

.Theorem [Thurston]

. .

... .

.

.

Exceptional surgeries are only finitely manyfor each hyperbolic knot.

Each exceptional surgery is either:

Reducible surgery (yielding a mfd. containing an essentialS²)

Toroidalsurgery (yielding a mfd. containing an essentialT²)

Seifert surgery (yielding a Seifert fibered mfd.)

Exceptional surgery

.Exceptional surgery

. .

... .

. .Dehn surgery on ahyperbolic knot yielding a non-hyperbolic mfd.

.Theorem [Thurston]

. .

... .

.

.

Exceptional surgeries are only finitely manyfor each hyperbolic knot.

Each exceptional surgery is either:

Reducible surgery (yielding a mfd. containing an essentialS²)

Toroidalsurgery (yielding a mfd. containing an essentialT²)

Seifert surgery (yielding a Seifert fibered mfd.)

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

Montesinos knot

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

.

. .

A knot admitting a diagram obtained by putting rational

tangles R₁, . . . , R_l together in a circle.

M(¹₂,¹₃,−²₃)

lengthof the knot = minimal number of rational tangles.

Known facts

Recall: Each exceptional surgery is either:

Reducible, ← Cabling conjecture says “cannot occur”.

Toroidal, Seifert.

.Remark [Eudave-Mu˜noz]

. .

... .

. .They arenot exclusive. (i.e., there are non-empty intersection.)

.Theorem [Motegi]

. .

... .

.

.

A knot with|Sym^∗(K)|>2 admitsno toroidal Seifert surgery. In particular, other thanthe trefoil knot, no two-bridge knots admit toroidal Seifert surgeries.

Known facts

Recall: Each exceptional surgery is either:

Reducible, ← Cabling conjecture says “cannot occur”.

Toroidal, Seifert.

.Remark [Eudave-Mu˜noz]

. .

... .

. .They arenot exclusive. (i.e., there are non-empty intersection.) .Theorem [Motegi]

.

. .

A knot with|Sym^∗(K)|>2 admitsno toroidal Seifert surgery.

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

Known facts

Recall: Each exceptional surgery is either:

Reducible, ← Cabling conjecture says “cannot occur”.

Toroidal, Seifert.

.Remark [Eudave-Mu˜noz]

. .

... .

. .They arenot exclusive. (i.e., there are non-empty intersection.) .Theorem [Motegi]

. .

... .

.

.

A knot with|Sym^∗(K)|>2 admitsno toroidal Seifert surgery.

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

Result

.Theorem [Ichihara-J.]

. .

... .

.

.

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

.Corollary

. .

... .

. .A hyperbolic Montesinos knot admits no toroidal Seifert surgery.

Proof of Theorem: K is hyperbolic & l = 3.

.Theorem [Ichihara-J.]

. .

... .

.

.

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

K: a Montesinos knot of length l

s.t.K(r) is toroidal Seifert fibered for some r∈Q.

.Claim 1

. .

... .

. .K is ofl = 3 and hyperbolic.

Proof.

l≤2 ⇒K is a two-bridge knot. ⇒ K = 3₁ &r = 0 [Motegi]. l≥4 ⇒ K admits no exceptional surgery [Wu].

K: non-hyperbolic ⇒K =T(3,4), or T(3,5)

[Oertel, Bonahon-Siebenmann]. They admit no toroidal Seifert surgeries [Moser]. ¤

Proof of Theorem: K is hyperbolic & l = 3.

.Theorem [Ichihara-J.]

. .

... .

.

.

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

K: a Montesinos knot of length l

s.t.K(r) is toroidal Seifert fibered for some r∈Q. .Claim 1

. .

... .

. .K is ofl = 3 and hyperbolic.

Proof.

l≤2 ⇒K is a two-bridge knot.

⇒ K = 3₁ &r = 0 [Motegi]. l≥4 ⇒ K admits no exceptional surgery [Wu].

K: non-hyperbolic ⇒K =T(3,4), or T(3,5)

[Oertel, Bonahon-Siebenmann]. They admit no toroidal Seifert surgeries [Moser]. ¤

Proof of Theorem: K is hyperbolic & l = 3.

.Theorem [Ichihara-J.]

. .

... .

.

.

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

K: a Montesinos knot of length l

s.t.K(r) is toroidal Seifert fibered for some r∈Q. .Claim 1

. .

... .

. .K is ofl = 3 and hyperbolic.

Proof.

l≤2 ⇒K is a two-bridge knot. ⇒ K = 3₁ &r = 0 [Motegi].

l≥4 ⇒ K admits no exceptional surgery [Wu]. K: non-hyperbolic ⇒K =T(3,4), or T(3,5)

[Oertel, Bonahon-Siebenmann]. They admit no toroidal Seifert surgeries [Moser]. ¤

Proof of Theorem: K is hyperbolic & l = 3.

.Theorem [Ichihara-J.]

. .

... .

.

.

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

K: a Montesinos knot of length l

s.t.K(r) is toroidal Seifert fibered for some r∈Q. .Claim 1

. .

... .

. .K is ofl = 3 and hyperbolic.

Proof.

l≤2 ⇒K is a two-bridge knot. ⇒ K = 3₁ &r = 0 [Motegi].

l≥4 ⇒ K admits no exceptional surgery [Wu].

K: non-hyperbolic ⇒K =T(3,4), or T(3,5)

[Oertel, Bonahon-Siebenmann]. They admit no toroidal Seifert surgeries [Moser]. ¤

Proof of Theorem: K is hyperbolic & l = 3.

.Theorem [Ichihara-J.]

. .

... .

.

.

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

K: a Montesinos knot of length l

s.t.K(r) is toroidal Seifert fibered for some r∈Q. .Claim 1

. .

... .

. .K is ofl = 3 and hyperbolic.

Proof.

l≤2 ⇒K is a two-bridge knot. ⇒ K = 3₁ &r = 0 [Motegi].

l≥4 ⇒ K admits no exceptional surgery [Wu].

K: non-hyperbolic ⇒K =T(3,4), or T(3,5)

[Oertel, Bonahon-Siebenmann].

They admit no toroidal Seifert surgeries [Moser]. ¤

Proof of Theorem: K is fibered and r = 0.

K: a hyperbolic Montesinos knot of length 3

s.t.K(r) is toroidal Seifert fibered for some r∈Q. We will show that suchK does not exist.

.Claim 2

. ..

.. .

. .K must be a fibered knotand r = 0.

Proof.

Since K is small [Oertel], if K(r) is toroidal ⇒

r is a ∂-slope of an essential surface of E(K). Furthermore, ifK(r) is Seifert fibered

⇒ K is fibered and r= 0 [Ichihara-Motegi-Song].

¤

Proof of Theorem: K is fibered and r = 0.

K: a hyperbolic Montesinos knot of length 3

s.t.K(r) is toroidal Seifert fibered for some r∈Q. We will show that suchK does not exist.

.Claim 2

. ..

.. .

. .K must be a fibered knotand r = 0.

Proof.

Since K is small [Oertel], if K(r) is toroidal ⇒

r is a ∂-slope of an essential surface of E(K).

Furthermore, ifK(r) is Seifert fibered

⇒ K is fibered and r= 0 [Ichihara-Motegi-Song].

¤

Proof of Theorem: K is fibered and r = 0.

K: a hyperbolic Montesinos knot of length 3

s.t.K(r) is toroidal Seifert fibered for some r∈Q. We will show that suchK does not exist.

.Claim 2

. ..

.. .

. .K must be a fibered knotand r = 0.

Proof.

Since K is small [Oertel], if K(r) is toroidal ⇒

r is a ∂-slope of an essential surface of E(K).

Furthermore, ifK(r) is Seifert fibered

⇒ K is fibered and r= 0 [Ichihara-Motegi-Song].

¤

Proof of Theorem: Using Wu’s classification

.Claim 3

. .

... .

.

.

K must be one of the following.

(I) P(2m,2n+ 1,−2n−1) for integers m ̸= 0 andn≥1 (II) M(−¹₂,¹₃,_6±¹1

2

)

Proof. Wu classified all toroidal surgeries on Montesinos knots. The 0-surgeries contained in Wu’s list are those on

.^.1. ^{P(2p1−1,2p2−1,2p3−1) for pi ̸= 0,1,}

.^.2. ^{P(2m,2n+ 1,−2n−1) for m ̸= 0 andn≥1, and}

.^.3. ^M(−12,¹₃, ¹

6+_2s¹ ) for s̸= 0.

The first ones arenot fibered since they aregenus 1 knots

and a genus 1 fibered knot is either 31 or 41. Since the Alexander polynomial of a fibered knot ismonic and

∆_M(−¹

2,¹₃, ¹

6+ 12s

) =−s(t²+t⁻²) + 2s+ 1, s=±1. ¤

Proof of Theorem: Using Wu’s classification

.Claim 3

. .

... .

.

.

K must be one of the following.

(I) P(2m,2n+ 1,−2n−1) for integers m ̸= 0 andn≥1 (II) M(−¹₂,¹₃,_6±¹1

2

)

Proof. Wu classified all toroidal surgeries on Montesinos knots.

The 0-surgeries contained in Wu’s list are those on

.^.1. ^{P(2p1−1,2p2−1,2p3−1) for pi ̸= 0,1,}

.^.2. ^{P(2m,2n+ 1,−2n−1) for m ̸= 0 andn≥1, and}

.^.3. ^M(−12,¹₃, ¹

6+_2s¹ ) for s̸= 0.

The first ones arenot fibered since they aregenus 1 knots

and a genus 1 fibered knot is either 31 or 41. Since the Alexander polynomial of a fibered knot ismonic and

∆_M(−¹

2,¹₃, ¹

6+ 12s

) =−s(t²+t⁻²) + 2s+ 1, s=±1. ¤

Proof of Theorem: Using Wu’s classification

.Claim 3

. .

... .

.

.

K must be one of the following.

(I) P(2m,2n+ 1,−2n−1) for integers m ̸= 0 andn≥1 (II) M(−¹₂,¹₃,_6±¹1

2

)

Proof. Wu classified all toroidal surgeries on Montesinos knots.

The 0-surgeries contained in Wu’s list are those on

.^.1. ^{P(2p1−1,2p2−1,2p3−1) for pi ̸= 0,1,}

.^.2. ^{P(2m,2n+ 1,−2n−1) for m ̸= 0 andn≥1, and}

.^.3. ^M(−12,¹₃, ¹

6+_2s¹ ) for s̸= 0.

The first ones arenot fibered since they are genus 1 knots

and a genus 1 fibered knot is either 3₁ or 4₁.

Since the Alexander polynomial of a fibered knot ismonic and

∆_M(−¹

2,¹₃, ¹

6+ 12s

) =−s(t²+t⁻²) + 2s+ 1, s=±1. ¤

Proof of Theorem: Using Wu’s classification

.Claim 3

. .

... .

.

.

K must be one of the following.

(I) P(2m,2n+ 1,−2n−1) for integers m ̸= 0 andn≥1 (II) M(−¹₂,¹₃,_6±¹1

2

)

Proof. Wu classified all toroidal surgeries on Montesinos knots.

The 0-surgeries contained in Wu’s list are those on

.^.1. ^{P(2p1−1,2p2−1,2p3−1) for pi ̸= 0,1,}

.^.2. ^{P(2m,2n+ 1,−2n−1) for m ̸= 0 andn≥1, and}

.^.3. ^M(−12,¹₃, ¹

6+_2s¹ ) for s̸= 0.

The first ones arenot fibered since they are genus 1 knots

and a genus 1 fibered knot is either 3₁ or 4₁.

Proof of Theorem: Remains

.Claim 4

. .

... .

.

.

K =P(2m,2n+ 1,−2n−1) for integers m̸= 0 and n≥1

⇒ K(0) is not Seifert fibered.

.Claim 5

. .

... .

.

.

K =M(−¹₂,¹₃,_6±¹1 2

)⇒ K(0) is not Seifert fibered.

If Claims 4 & 5 are true, then we complete the proof of Theorem.

Today we only prove Claim 4.

(We already introduced the proof of Claim 5 in ILDT 2009.)

Proof of Claim 4

.Claim 4

. .

... .

.

.

K =P(2m,2n+ 1,−2n−1) for integers m̸= 0 and n≥1

⇒ K(0) is not Seifert fibered.

Set K_m,n =P(2m,2n+ 1,−2n−1) andM_m,n =K_m,n(0). .Subclaim 1

. .

... .

. .Mm,1 is not Seifert fibered for m̸= 0.

.Subclaim 2

. .

... .

. .M_±1,n is not Seifert fibered for n≥2.

.Subclaim 3

. .

... .

. .Mm,n is not Seifert fibered for |m| ≥2 and n≥2.

Proof of Claim 4

.Claim 4

. .

... .

.

.

K =P(2m,2n+ 1,−2n−1) for integers m̸= 0 and n≥1

⇒ K(0) is not Seifert fibered.

Set K_m,n =P(2m,2n+ 1,−2n−1) andM_m,n =K_m,n(0).

.Subclaim 1

. .

... .

. .Mm,1 is not Seifert fibered for m̸= 0.

.Subclaim 2

. .

... .

. .M_±1,n is not Seifert fibered forn ≥2.

.Subclaim 3

. .

... .

. .M_m,n is not Seifert fibered for|m| ≥2 andn ≥2.

Proof of Subclaim 1

.Subclaim 1

. .

... .

. .M_m,1 is not Seifert fibered for m̸= 0.

Proof. M_m,1 is the double branched cover of S³ along λ_m.

⇒λ_m is an arborescent link & M_m,1 is not Seifert fibered.

} ^m

}

m

}

m

}

・・・・・・ ・・・

・・・

・・・

}

m =

Cut open M_m,n along the essential torus

Suppose thatM_m,n is Seifert fibered.

Fb:=F ∪D : Klein bottle in M_m,n

∂F =∂D

F

∪

D

V

E(K_m,n) M_m,n=

··· ······

···

······

⇒M_m,n \N^◦(Fb) is also Seifert fibered.

Cut open M_m,n along the essential torus

Suppose thatM_m,n is Seifert fibered.

Fb:=F ∪D : Klein bottle in M_m,n

∂F =∂D

F

∪

D

V

E(K_m,n) M_m,n=

··· ······

···

······

Consider M_m,n \N^◦(Fb) as double branched cover

ι

∂F =∂D

F

∪

D

−→

q

n{ ⁿ{

V Mm,n=

m{

Mm,n/ι=

··· ······

···

······ ···

···

···

(∂F q ) =

∪ (∂D q )

Consider M_m,n \N^◦(Fb) as double branched cover

(^Mm,n/ι)−N^◦(q(Fb)) =^m

{

{

{

=^m

{

{

{

Deforming T_m,n

=

=

=

T_m =

,n m{

{ {

m{

{ {n _n

{ {

m{

··· ···

···

n

{

n

m

{

{

···

··· ···

···

{ {n ^···

···

···

Deforming T_m,n

n : odd

n : even

{ {

n−1

2 × ⁿ₋₁

{

2 ×

n

2×

{

=

n 2×

· · ·

· · ·

· · ·

· · ·

m−1

m−1

{ {n m{

··· ···

···

Deforming T_m,n

n 2×

n+1

2 ×

{

{

2 × n−1

{

2 × ⁿ⁺₂¹×

{

{ { { {

2× (ⁿ₂ −1)× (ⁿ₂ + 1)×

n : odd

n : even

=

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

···m−1 m−1

M_m,n −N^◦(Fb) ∼= E(J_m,n)

We obtain the knotJ_m,n inS²×S¹.

· · · · · ·

· · · · · ·

n : odd n : even

0 0

{{

{{ ^{{

{{n+ 1 {{_n

n + 1

−1 {{ _n

−1 {{

n n n

−2 n+ 2

· · · · · · · · · · · ·

m

−1 m−1

m

−1 m−1

m,n

J

.

To show

.Subclaim 2’

. .

... .

. .E(J_±1,n) is not Seifert fibered for n≥2.

.Subclaim 3’

. .

... .

. .E(Jm,n) is not Seifert fibered for |m| ≥2 and n≥2.

β_m,n :=σ₁· · ·σ_nσ^2m_n+1⁻¹σ⁻_n+2¹ · · ·σ_2n⁻¹ ∈B_2n+1(S²) ⇒ J_m,n =βd_m,n. .Lemma B [Murasugi-Kurpita]

. .

... .

.

.

b∈B_l(S²),b^′ ∈B_l′(S²). Ifbb=bb^′ in S²×S¹, then l=l^′.

exp(b)≡exp(b^′) mod 2(l−1). Remark : exp(β_m,n)≡2m mod 4n.

To show

.Subclaim 2’

. .

... .

. .E(J_±1,n) is not Seifert fibered for n≥2.

.Subclaim 3’

. .

... .

. .E(Jm,n) is not Seifert fibered for |m| ≥2 and n≥2.

β_m,n :=σ₁· · ·σ_nσ^2m_n+1⁻¹σ⁻_n+2¹ · · ·σ_2n⁻¹ ∈B_2n+1(S²) ⇒ J_m,n =βd_m,n. .Lemma B [Murasugi-Kurpita]

. .

. b∈B_l(S²),b^′ ∈B_l′(S²). Ifbb=bb^′ in S²×S¹, then

l=l^′.

exp(b)≡exp(b^′) mod 2(l−1).

Proof of Subclaim 2’

.Subclaim 2’

. .

... .

. .E(J_±1,n) is not Seifert fibered for n≥2.

Proof. To show : E(J_1,n) is not Seifert fibered.

(E(J₋1,n) = the mirror image of E(J1,n). )

Suppose thatE(J_1,n) in S²×S¹ is Seifert fibered.

⇒J_1,n is the closure of (σ₁. . . σ_2n)^k [Kohn],

i.e. β1,n = (σ1. . . σ2n)^k for some k ∈Z.

⇒exp(β_1,n)≡0 or 2n mod 4n. On the other hand, exp(β_1,n) = 2.

By Lemma B, we haven = 1. ¤

Proof of Subclaim 2’

.Subclaim 2’

. .

... .

. .E(J_±1,n) is not Seifert fibered for n≥2.

Proof. To show : E(J_1,n) is not Seifert fibered.

(E(J₋1,n) = the mirror image of E(J1,n). )

Suppose thatE(J_1,n) in S²×S¹ is Seifert fibered.

⇒J_1,n is the closure of (σ₁. . . σ_2n)^k [Kohn],

i.e. β1,n = (σ1. . . σ2n)^k for some k ∈Z.

⇒exp(β_1,n)≡0 or 2n mod 4n.

On the other hand, exp(β_1,n) = 2.

By Lemma B, we haven = 1. ¤