On a Heegaard surface homeomorphism obtained by bridge position of a knot

On a Heegaard surface homeomorphism

obtained by

bridge position of a knot

Shin ⁰ ya Okazaki (Osaka City University)

(

)

December 20 , 2010

Contents

1 . Background

2 . Main theorem

1 . Background Fact

Every closed connected orientable 3-manifold M is obtained by the 0-surgery of the 3-sphere S ³ along a link .

We introduce two kinds of genera g _bridge ( M ) and

g _braid ( M ) .

Definition (0-surgery of S ³ along L ) L : a link in S ³ .

χ ( L, 0)

We call χ ( L, 0) the 3-manifold obtained by the

0-surgery of S ³ along L.

bridge( L ): the bridge index of L.

braid( L ): the braid index of L.

Definition (bridge genus and braid genus)

g _bridge ( M ) = min { bridge( L ) | χ ( L, 0) = M } . g _braid ( M ) = min { braid( L ) | χ ( L, 0) = M } .

We have

g ( M ) ≤ g ( M ) .

Definition (Heegaard splitting)

M : a closed connected orientable 3-manifold .

M = ∪ _f

H H ⁰

We call the triple ( H, H ⁰ ; f ) a Heegaard splitting

of M. The Heegaard genus g _H ( M ) of M is the

minimal genus of ∂H.

Theorem 1 [O . ]

g _H ( M ) ≤ g _bridge ( M ) ≤ g _braid ( M ) .

The invariants g _H , g _bridge and g _braid are linearly

independent .

Outline of proof of Theorem 1

S ³ = ∪

B ₁ B ₂

S ³ = ∪ _f

H H ⁰

Example 1

K : the trivial knot ⇒ χ ( K, 0) = S ¹ × S ² . Since g _H ( M ) = 0 ⇔ M = S ³ ,

we have g _H ( S ¹ × S ² ) ≥ 1 .

braid( K ) = 1 ⇒ g _braid ( S ¹ × S ² ) ≤ 1 . Therefore , we have

g _H ( S ¹ × S ² ) = g _bridge ( S ¹ × S ² )

= g _braid ( S ¹ × S ² ) = 1 .

Example 2

L : the Hopf link ⇒ χ ( L, 0) = S ³ . We have g _H ( S ³ ) = 0 .

braid( L ) = 2 ⇒ g _braid ( S ³ ) ≤ 2 .

Since g _bridge ( M ) = g _braid ( M ) = 1

⇔ M = S ¹ × S ² ,

we have g _bridge ( S ³ ) ≥ 2 . Therefore , we have

g _H ( S ³ ) = 0 < g _bridge ( S ³ ) = g _braid ( S ³ ) = 2 .

2 . Main theorem B ³ : a 3-ball .

h _i := D ² × I ( i = 1 , 2 , . . . , n ) .

H : a genus n handlebody in S ³ s . t . B ³ ∪

ⁿ _{i =1} h _i . F := ∂H.

H :

h ₁

h _n h ₂

K : a knot which satisfies condition ( ∗ ) . K ⊂ int H,

h _i ∩ K = { 0 } × I,

B ³ ∩ K is an n -string trivial tangle .















( ∗ )

K ₀ : the knot satisfying condition ( ∗ ) as following .

H :

h ₁

h _n h ₂

Objective of this talk

1 . We show that for any K which satisfies con- dition ( ∗ ) , K is represented by the product of Suzuki generators from K ₀ .

2 . We show that a Heegaard surface homeomor-

phism of χ ( K, 0) in outline of proof of Theorem

1 is represented by the product of Suzuki gen-

erators and two homeomorphisms .

M ( F ) : the mapping class group of F.

ρ, τ ₁ , θ ₁₂ , µ ₁ : the homeomorphisms in M ( F ) . ρ

−→ ρ

h ₁ h _n

h _n h ₂ h _{n − 1} h ₁ τ ₁

h ₁ τ ₁

−→

θ ₁₂

θ ₁₂

−→

h ₁ h ₂ h ₁ h ₂

µ ₁

h ₁ µ ₁

−→

S. Suzuki, On Homeomorphisms of a 3-dimensional handlebody, Canad. J. Math., Vol. 29, (1977), 111–124.

Theorem 2 [Suzuki]

M ( F ) is generated by ρ, τ ₁ , θ ₁₂ , µ ₁ .

ω ₁ , ρ ₁₂ , ξ ₁₂ ⁰ : the homeomorphisms in M ( F ) . ω ₁

ω ₁

−→

h ₁ h ₁

ρ ₁₂

ρ ₁₂

−→

h ₁ h ₂ h ₁ h ₂

ξ ₁₂ ⁰

ξ ₁₂ ⁰

−→

h ₁ h ₂ h ₁ h ₂

ρ, τ ₁ , θ ₁₂ , ω ₁ , ρ ₁₂ , ξ ₁₂ ⁰ can be extended to homeo-

morphisms of H onto itself . But µ ₁ cannot .

M ( H ) : the mapping class group of H.

ρ, ˜ ω ˜ ₁ , ρ ˜ ₁₂ , ξ ˜ ₁₂ ⁰ : the homeomorphisms in M ( H ) . ρ ˜

ρ ˜

−→

h ₁ h _n

h _n h ₂ h _{n − 1} h ₁ ω ˜ ₁

ω ˜ ₁

−→

h ₁ h ₁

ρ ˜ ₁₂

ρ ˜ ₁₂

−→

h ₁ h ₂ h ₁ h ₂

ξ ˜ ₁₂ ⁰

ξ ˜ ₁₂ ⁰

−→

h ₁ h ₂ h ₁ h ₂

Lemma 1

For any n -bridge knot K satisfying ( ∗ ) , there ex- ists a power product ˜ φ of ˜ ρ, ω ˜ ₁ , ρ ˜ ₁₂ , ξ ˜ ₁₂ ⁰ ∈ M ( H ) s . t . K = ˜ φ ( K ₀ ) .

Lemma 2

For any n -braid knot K satisfying ( ∗ ) , there ex-

ists a power product ˜ φ of ˜ ρ, ρ ˜ ₁₂ , ω ˜ ₁ ξ ˜ ₁₂ ⁰ ω ˜ ₁ ^{− 1} ∈ M ( H )

s . t . K = ˜ φ ( K ₀ ) .

Example 3

K : the trefoil satisfying ( ∗ ) .

ω ˜ ₁ ^{− 1} ξ ˜ ⁰⁻ ₁₂ ¹

−→ −→

Therefore , K = ξ ⁰⁻ ₁₂ ¹ ω ₁ ^{− 1} ( K ₀ )

ψ : an orientation reversing homeomorphism of F onto itself .

−→ ψ

c ₁ ψ ( c ₁ ) c ₂

c _n

ψ ( c _n ) ψ ( c ₂ )

κ : the homeomorphism in M ( F ) as following .

−→ κ

h ₁ h ₁

h _n h ₂ h _n h ₂

Theorem 3 [O . ]

K : an n -bridge knot satisfying ( ∗ ) . φ ˜ : a power product of ˜ ρ, ω ˜ ₁ , ρ ˜ ₁₂ , ξ ˜ ₁₂ ⁰ s . t . K = ˜ φ ( K ₀ ) .

φ = ˜ φ | F .

Then χ ( K, 0) has the Heegaard splitting

( H, H ⁰ ; φκφ ^{− 1} ψ ) .

Theorem 4 [O . ]

K : an n -braid knot satisfying ( ∗ ) .

φ ˜ : a power product of ˜ ρ, ρ ˜ ₁₂ , ω ˜ ₁ ξ ˜ ₁₂ ⁰ ω ˜ ₁ ^{− 1} s . t . K = ˜ φ ( K ₀ ) .

φ = ˜ φ | F .

Then χ ( K, 0) has the Heegaard splitting

( H, H ⁰ ; φκφ ^{− 1} ψ ) .

Example 4

K : the trefoil satisfying ( ∗ ) .

ω ˜ ₁ ^{− 1} ξ ˜ ⁰⁻ ₁₂ ¹

−→ −→

Therefore , χ ( K, 0) has the Heegaard splitting

( H, H ⁰ ; ξ ⁰⁻ ₁₂ ¹ ω ₁ ^{− 1} κω ₁ ξ ⁰ ₁₂ ψ ) .

Corollary 1

M : a closed connected orientable 3-manifold . g _bridge ( M ) = min { g ( H ) | M = H ∪ _φκφ

ψ H ⁰ } . Here , φ is a power product of ρ, ω ₁ , ρ ₁₂ , ξ ₁₂ ⁰ .

g _braid ( M ) = min { g ( H ) | M = H ∪ _φ

κφ

ψ H ⁰ } .

Here , φ ⁰ is a power product of ρ, ρ ₁₂ , ω ₁ ξ ₁₂ ⁰ ω ₁ ^{− 1} .