unknotting tunnel

(1)

(1, 1)-knot

_の

unknotting tunnel

東京農工大学合田洋

(

林忠一郎氏（日本女子大学）との共同研究

)

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M : a closed orientable 3-manifold, K (⊂ M ) : a knot

Definition. A knot K is called a (1, 1)-knot if the pair (M, K ) can be decomposed into (H ₁ , K ₁ ) ∪ (H ₂ ∪ K ₂ ) where H _i is a solid torus and K _i is a boundary parallel arc properly embedded in H _i .

H ²

1 K ¹ K ²

H (1,1)-decomposition

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Definition. Let τ be an arc with τ ∩K = ∂τ . τ is called an unknotting tunnel of K if M − IntN (K ∪ τ ) ia s genus 2 handlebody.

K

τ

V V

1 2

unknotting tunnel

A knot which has an unknotting tunnel is called a tunnel number

one knot.

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Proposition. (1, 1)-knots are tunnel number one.

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Proposition. (1, 1)-knots are tunnel number one.

H ²

1 K ¹ K ²

H

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Proposition. (1, 1)-knots are tunnel number one.

K

¹

K

²

K

1

K

2

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Proposition. (1, 1)-knots are tunnel number one.

K

¹

K

²

K

1

K

2

I ^I ~ I ^I ~

K

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Proposition. (1, 1)-knots are tunnel number one.

K

¹

K

²

K

1

K

2

I ^I ~ I ^I ~

τ

₁

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Proposition. (1, 1)-knots are tunnel number one.

K K

1

2

K

1

K

2

I ^I ~ I ^I ~

τ

₁

τ

₁

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Definition. The unknotting tunnel obtained from a (1, 1)-decomposition is called a (1, 1)-tunnel. It has a dual tunnel.

K

¹

^K

²

τ

₁

τ

₂

In what follows, we consider the case of M = S ³ .

Theorem.[Boileau-Rost-Zieschang (88),

森元

(89)] A torus knot has

at most 3 types unknotting tunnels. Two of them are (1, 1)-tunnels.

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Torus knot T (5, 8)

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Torus knot T (5, 8)

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Torus knot T (5, 8)

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τ

0 1 2

Torus knot T (5, 8)

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Theorem.[

小林

(99)] A two birdge knot has at most 4 unknotting tunnels (up to homeo.). All of them are (1, 1)-tunnels.

Theorem.[

森元−作間

(91)] A satellite knot has at most 4 unknotting tunnels (up to homeo.). All of them are (1, 1)-tunnels.

Problem. Is there a tunnel number one knot which does not have a (1, 1)-tunnel ?

Answer. Yes.

森元−作間−横田

knot (The twisted torus knot, (96)).

MSY(7, 17, 10m − 4) (m ∈ N ) is not (1, 1).

Theorem.[Scharlemann

−

Thompson -

合

(00)] Any unknotting tun-

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( p,q ) - torus knot

r - half twists unknotting tunnel

p,q r ( ; ) MSY

From the observation of the torus knots and the previous theorem,

Question. Let K be a (1, 1)-knot. If K has an unknotting tunnel τ

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§ Results of Rubinstein-Scharlemann and

小林−佐伯

Let M = H ₁ ∪ _H H ₂ = V ₁ ∪ _V V ₂ be two Heegaard splittings of M . Roughly speaking, Rubinstein-Scharlemann proved that H and V may intersect in a non-empty collection of simple closed curves which are essential in both H and V .

小林−佐伯

studied the case of 3-manifolds with links. Applying

their results to our case, we may assume that:

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Let (M, K ) = (H ₁ , K ₁ ) ∪ _H (H ₂ , K ₂ ) : a (1, 1)-decomposition

= (V ₁ , K ) ∪ _V (V ₂ , ∅) : a genus 2 Heegaard split.

Then, H and V may intersect in a non-empty collection of simple closed curves which are K -essential in both H and V .

Let ` be the minimum number of such simple closed curves, and τ be the unknotting tunnel associated with (V ₁ , K ) ∪ _V (V ₂ , ∅).

Theorem.[

林−合

] If ` 6= 2, one of the following conditions holds:

(1) τ is a (1, 1)-tunnel;

(2) τ may be isotoped into H fixing the knot K ;

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Examples by Song : MSY(5,7,2).

τ

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A criterion for deciding ‘not (1, 1)-tunnel’

Let θ be a θ-graph obtained from K ∪ τ . We denote by θ ₁ and θ ₂ the consituent knots of θ.

K

τ

θ θ

θ

1

2

Proposition.

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Examples by Song : MSY(5,7,2).

τ

(22)

Examples by Song : MSY(5,7,2).

τ

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ǫ

Both θ ₁ and θ ₂ are not the trivial knots. Hence, τ ₂ is not a (1, 1)-

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Question.[Scharlemann] If we slide τ enough, both θ ₁ and θ ₂ become non-trivial knots although τ is a (1, 1)-tunnel ?

Answer. Yes. So, we need some conditions.

Suggestion.[Song] How about a thin position for τ ?

このあと，進展なし，，，

論文，ホームページに置きっぱ，，，

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In the recent 10 years, some invariants for the (unknotting) tunnels were defined: ‘distance’ (Johnson), ‘depth’ (Cho-McCullough),

‘t-distance’ (

古宇田

).

Theorem.[Scharlemann-Tomova, Johnson (06)]

dist(τ ) ≥ 6 = ⇒ τ is the unique unknotting tunnel.

Theorem.[Johnson] For any n, there is an unknotting tunnel τ such that dist(τ ) > n.

（印象）えーっつ！ほとんどユニーク！？

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Problem. dist(τ ) ≤ 5

の

unknotting tunnel

ってどのようなものか？

分類可能か？

このうちユニークなものは？ユニークでない場合，そのうち一つが

(1, 1)-tunnel

ならばどんなものか？

Fact. (1) depth(τ ) = 1 ⇐⇒ τ is a (1, 1)-tunnel.

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石原

found an algorithm for calculating the depth of unknotting tunnels.

Theorem.[

石原

] The depth of the unknotting tunnel τ ₂ for MSY(5,7,2)

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MSY(5,7,2)

と

τ ₂

の組はある操作によって一般化出来るが，これらについても同様のことが言えることを石原は示した．

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謝辞

本講演をするにあたり古宇田悠哉さん，石原海さんに様々な事を教えて頂きました．感謝いたします．

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ご静聴ありがとうございました．

unknotting tunnel

(1, 1)-knot

unknotting tunnel

(

)

M : a closed orientable 3-manifold, K (⊂ M ) : a knot

Definition. A knot K is called a (1, 1)-knot if the pair (M, K ) can be decomposed into (H 1 , K 1 ) ∪ (H 2 ∪ K 2 ) where H i is a solid torus and K i is a boundary parallel arc properly embedded in H i .

H 2

1

K 1 K 2

H (1,1)-decomposition

Definition. Let τ be an arc with τ ∩K = ∂τ . τ is called an unknotting tunnel of K if M − IntN (K ∪ τ ) ia s genus 2 handlebody.

K

τ

V V

unknotting tunnel

A knot which has an unknotting tunnel is called a tunnel number

one knot.

Proposition. (1, 1)-knots are tunnel number one.

Proposition. (1, 1)-knots are tunnel number one.

H 2

1

K 1 K 2

H

Proposition. (1, 1)-knots are tunnel number one.

K

K

K

K

Proposition. (1, 1)-knots are tunnel number one.

K

K

K

K

I I ~ I I ~

K

Proposition. (1, 1)-knots are tunnel number one.

K

K

K

K

I I ~ I I ~

τ

Proposition. (1, 1)-knots are tunnel number one.

K K

K

K

I I ~ I I ~

τ

τ

Definition. The unknotting tunnel obtained from a (1, 1)-decomposition is called a (1, 1)-tunnel. It has a dual tunnel.

K

K

τ

τ

In what follows, we consider the case of M = S 3 .

Theorem.[Boileau-Rost-Zieschang (88),

(89)] A torus knot has

at most 3 types unknotting tunnels. Two of them are (1, 1)-tunnels.

Torus knot T (5, 8)

Torus knot T (5, 8)

Torus knot T (5, 8)

τ

τ

τ

0

1 2

Torus knot T (5, 8)

Theorem.[

(99)] A two birdge knot has at most 4 unknotting tunnels (up to homeo.). All of them are (1, 1)-tunnels.

Theorem.[

(91)] A satellite knot has at most 4 unknotting tunnels (up to homeo.). All of them are (1, 1)-tunnels.

Problem. Is there a tunnel number one knot which does not have a (1, 1)-tunnel ?

Answer. Yes.

knot (The twisted torus knot, (96)).

MSY(7, 17, 10m − 4) (m ∈ N ) is not (1, 1).

Theorem.[Scharlemann

Thompson -

(00)] Any unknotting tun-

( p,q ) - torus knot

Definition. A knot K is called a (1, 1)-knot if the pair (M, K ) can be decomposed into (H ₁ , K ₁ ) ∪ (H ₂ ∪ K ₂ ) where H _i is a solid torus and K _i is a boundary parallel arc properly embedded in H _i .

H ²

K ¹ K ²

H ²

K ¹ K ²

I ^I ~ I ^I ~

I ^I ~ I ^I ~

I ^I ~ I ^I ~

^K

In what follows, we consider the case of M = S ³ .

Let M = H ₁ ∪ _H H ₂ = V ₁ ∪ _V V ₂ be two Heegaard splittings of M . Roughly speaking, Rubinstein-Scharlemann proved that H and V may intersect in a non-empty collection of simple closed curves which are essential in both H and V .

Let (M, K ) = (H ₁ , K ₁ ) ∪ _H (H ₂ , K ₂ ) : a (1, 1)-decomposition

= (V ₁ , K ) ∪ _V (V ₂ , ∅) : a genus 2 Heegaard split.

Let ` be the minimum number of such simple closed curves, and τ be the unknotting tunnel associated with (V ₁ , K ) ∪ _V (V ₂ , ∅).

Let θ be a θ-graph obtained from K ∪ τ . We denote by θ ₁ and θ ₂ the consituent knots of θ.

Both θ ₁ and θ ₂ are not the trivial knots. Hence, τ ₂ is not a (1, 1)-

Question.[Scharlemann] If we slide τ enough, both θ ₁ and θ ₂ become non-trivial knots although τ is a (1, 1)-tunnel ?

] The depth of the unknotting tunnel τ ₂ for MSY(5,7,2)

τ ₂