Heegaard splitting Bridge splitting Result

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Knots with arbitrary high distance bridge decompositions K.Ichihara

Introduction Heegaard splitting Bridge splitting Result Corollary

Meridional destabilizing number Definition of Hempel distance Outline of Proof

Strategy Claim Braids

Knots with arbitrary high distance bridge decompositions

Kazuhiro Ichihara

Nihon University

College of Humanities and Sciences

Joint works with

Toshio Saito (Joetsu University of Education) Low-dimensional Topology Seminar

Osaka University, October 16, 2012

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Table of contents

Introduction

Heegaard splitting Bridge splitting Result

Corollary

Meridional destabilizing number

Deﬁnition of Hempel distance

Outline of Proof Strategy Claim Braids

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Heegaard splitting

Heegaard splitting ( Heegaard (1898) )

A decomposition of a closed orientable 3-manifold into two handlebodies.

Moise (1952)

Every closed orientable 3-manifold has a Heegaard splitting.

Facts

Certain properties of Heegaard splittings reﬂect topological characteristics of 3-manifolds:

I

any Heegaard splitting of a reducible 3-manifold is reducible (Haken (1968))

I

any Heegaard splitting of a non-Haken 3-manifold

is reducible or strongly irreducible

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Hempel distance

Motivated by such works, in 2001,

Hempel introduced an invariant of a Heegaard splitting, called the distance, or commonly called the Hempel distance.

(Precise deﬁnition will be given later)

This measures certain complexity of Heegaard splittings.

A lot of studies have been done about the distance of Heegaard splittings...

Among them, on the existence of high distance splittings, there are several known results.

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High distance splittings for closed 3-manifolds

Hempel (2001)

There exist Heegaard splittings of closed 3-manifolds with distance at least n for arbitrarily large n.

(adapting an idea of Kobayashi (1988))

Evans (2006)

An inﬁnite sequence of closed 3-manifolds

{

M

_n}

satisﬁes the distance of a Heegaard splitting of M

n

is at least n.

(by using purely combinatorial techniques)

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High distance splittings for knot exteriors

Minsky-Moriah-Schleimer (2007)

There exist knots in

S³

with the exteriors admitting

Heegaard splittings of arbitrarily high distance, in any genus.

(based on Hempel’s idea)

Campisi-Rathbun (2012)

For any closed 3-manifold M with genus g

₀

Heegaard splitting and any integers g

≥

g

₀

+ 1,

there is a knot in

M

with its exterior admitting

a genus g Heegaard splitting of arbitrarily high distance.

(generalizing the method of Minsky-Moriah-Schleimer)

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Bridge splitting

A natural generalization of Heegaard splitting for a link is given by the bridge splitting (or bridge decomposition).

(g, b)-bridge splitting

A decomposition of (M, L) into

two pairs of a genus g handlebody and b trivial arcs for a link L in a closed 3-manifold M .

For bridge splittings, the notion of distance is naturally deﬁned as a generalization of the case of Heegaard splittings for closed manifolds.

(Precise deﬁnition will be given later)

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On high distance bridge splittings

Saito (2004)

In any closed 3-manifold with a Heegaard splitting of genus one, there is a knot with a

(1,1)-bridge splitting

of arbitrary high distance.

Blair-Tomova-Yoshizawa (preprint) For given integers b, c, g, and n,

there exists a c-component link L in a 3-manifold M so that (M, L) admits a (g, b)-bridge splitting of distance at least n.

(based on Evans’s idea)

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Main Theorem

Theorem (I.-Saito)

For any given closed 3-manifold M with a Heegaard surface

of genus g and any given positive integers b & n, there exists

a knot K in M which admits a (g, b)-bridge splitting of

distance greater than n with respect to the Heegaard surface

except for (g, b) = (0, 1), (0, 2).

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Table of contents 9:45

Introduction

Heegaard splitting Bridge splitting Result

Corollary

Meridional destabilizing number

Deﬁnition of Hempel distance

Outline of Proof Strategy Claim Braids

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Tunnel number

Tunnel number t(K) (Clark (1980))

The minimum number of the properly embedded arcs τ in the knot exterior E(K) such that

E(K)

−

intN (τ ) is a handlebody for a knot K in S

³

.

This gives an interesting subject to study in Knot Theory, and there are many works to study it.

NOTE :

t(K) + 1 = minimal genus of Heegaard splittings of E(K)

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Meridional destabilizing number

Consider t(K) and bridge splitting of K simultaneously.

For example, knots of t(K) = 1 are classiﬁed into 3 classes:

·

the two-bridge knots, i.e., knots admitting (0, 2)-splittings,

·

the (1, 1)-knots, i.e., knots admitting (1, 1)-splittings, and

·

the other knots, i.e., knots admitting (2, 0)-splittings.

A generalization of this classiﬁcation is:

The meridional destabilizing number (Saito (2011)) The maximal number of m such that

(M, K) admits a (t(K ) + 1

−

m, m)-bridge position for a knot K in a closed 3-manifold M.

Note :

the meridional destabilizing number is at most t(K) + 1.

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Corollary

Main Theorem and Tomova (2007) implies:

Corollary

For

any

integers t

≥

1 and m

≥

0 with m

≤

t + 1, there exists a knot K in S

³

of tunnel number t & of meridional destabilizing number m.

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Introduction

Heegaard splitting Bridge splitting Result

Corollary

Meridional destabilizing number

Deﬁnition of Hempel distance

Outline of Proof Strategy Claim Braids

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Curve complex

To deﬁne the (Hempel) distance,

we ﬁrst prepare the terminology about the curve complex, originally introduced by Harvey (1981).

Let

F

be a compact orientable surface

possibly with non-empty boundary.

the curve complex C (F )

The simplicial complex whose k-simplexes are the isotopy classes of k + 1 collections of mutually non-isotopic essential loops on F which can be realized disjointly.

Remark

“essential” means non-trivial and not boundary-parallel.

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Distance on the curve complex

For a pair of vertices [x] and [y] in

C

(F ),

the distance d([x], [y]) between [x] and [y] is deﬁned as the minimal number of edges in a path from [x] to [y].

The well-deﬁnedness is due to:

Masur-Minsky (1999)

The curve complex is connected if F is not sporadic,

i.e., ∂F has at least 5 (resp.2) components if g = 0 (resp.1).

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Distance of bridge splitting

For a bridge splitting ((V

₁

, t

₁

), (V

₂

, t

₂

)) of (M, K), set E(K) := M

−

intN(K),

W

_i

:= V

_i∩

E(K) , S

₀

:= ∂V

_i∩

E(K) for i = 1, 2.

For each i = 1, 2, the disk complex

D

(W

_i

) is the maximal subcomplex of

C

(S

₀

) spanned by

the vertices correspond to the curves bounding disks in W

i

.

Hempel distance d( D (W

₁

), D (W

₂

))

min

{

d([x], [y])

|

[x]

∈ D

(W

₁

), [y]

∈ D

(W

₂

)

}

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Introduction

Heegaard splitting Bridge splitting Result

Corollary

Meridional destabilizing number

Deﬁnition of Hempel distance

Outline of Proof

Strategy Claim Braids

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Settings

Let

M

be a closed orientable 3-manifold and

F

a Heegaard surface of M of genus g.

That is, F decomposes M into two handlebodies V

1

and V

2

. Choose arbitrary integers n, b

≥

1 .

Technical assumptions

We assume that b

≥

3 if g = 0 . Also Saito presented a knot with

a (1, 1)-bridge splitting of arbitrary high distance, and so, we assume that b

≥

2 if g = 1 .

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Strategy

Take a 2b-tuple of points

{x₁, x₂,· · · , x_2b₋₁, x_2b}

on F and, ∂-parallel arcs

αi,1, αi,2,· · · , αi,b⊂

V

i

for i = 1, 2,

with ∂α

i,j

=

{

x

2j−1

, x

2j}

for j = 1,

· · ·

, b.

⇒

they compose a

b-component trivial linkL

in M . Take a small regular neighborhood

N(F)

of F in M, which is topologically F

×

[0, 1],

and regard it as lying in

V2

with

F =F × {0}.

We isotope the link L so that L gives

a 2b-string vertical arcs

σ₀ =t⁰₁∪ · · · ∪t⁰_2b

in N (F ).

Our strategy

We will replace σ

0

with new braided arcs to make a knot K in M which admits a b bridge presentation with respect to F of distance greater than n.

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Claim

Set

V

₂⁰

:= V

2−

intN (F ), α

⁰_2,j

:= α

2,j∩

V

₂⁰

. W

₁

:= V

₁−

intN (α

_1,1∪ · · · ∪

α

_1,b

), W

₂⁰

:= V

₂⁰−

intN (α

⁰_2,1∪ · · · ∪

α

⁰_2,b

).

F

0

:= ∂W

1∩

(F

× {0}),

F

1

:= ∂W

₂⁰∩

(F

× {1}).

(both are 2b punctured surfaces of genus g) Since σ

₀

is a set of vertical arcs, we can naturally identify F

0⊂

F

× {0}

and F

1 ⊂

F

× {1}

in N (F )

∼

= F

×

[0, 1].

Thus we regard

D

(W

₂⁰

)

⊂ C

(F

0

) .

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Claim Claim There exists

a

pseudo-Anosov

homeomorphism φ

_N

: F

₀ →

F

₀

such that

d(D(W1),(φN)^m(D(W₂⁰)))→ ∞

as m

→ ∞, and

the caped oﬀ homeomorphism φ

c_N

: F

→

F

is isotopic to the identity map of F, and maps x

j+1

to x

j

and maps x

1

to x

2b

. Here

(φ_N)^m(D(W₂⁰))

denotes

the maximal subcomplex in

C(F0

) spanned by the vertices correspond to the image of curves bounding disks in W

₂⁰

by (φ

_N

)

^m

for m

∈N

. The existence of such a homeomorphism is essentially given in Ichihara-Motegi (2005),

together with a generalization of Campisi-Rathbun.

(based on the idea of Minsky-Moriah-Schleimer)

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From map to braid

Φ : F

×

[0, 1]

→

F

×

[0, 1]

: an isotopy from φ

c_N

to the identity of F

i.e., Φ(x, 0) = ( φ

cN

(x), 0) and Φ(x, 1) = (x, 1).

Deﬁne

t_j

: [0, 1]

→

F

×

[0, 1] as t

_j

(

·

) = Φ(x

_j

,

·

).

This t

_j

gives a monotone arc connecting

( φ

c_N

(x

_j

), 0) = (x

_j₋₁

, 0) and (x

_j

, 1).

Then t

₁

,

· · ·

, t

_2b

give

braided arcs

σ

= t

1∪ · · · ∪

t

2b

in N (F )

∼

= F

×

[0, 1]

⊂

V

2

.

Note that σ = Φ(σ

₀

) holds.

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Replacement

Now we replace the trivial braid σ

0

by the

m-th powerσ^m

of σ as a braid on the surface F with m = 2bm

₀

+ 1 for an integer m

₀

.

After replacing, we have a knot

K_m

in M

which admits a b-bridge presentation with respect to F . It then suﬃces to show that

the distance of the bridge splitting of K

_m

is equal to d(

D

(W

1

), (φ

N

)

^m

(

D

(W

₂⁰

))), which is shown in the next.

Claim

D

(W

₂

) = (φ

_N

)

^m

(

D

(W

₂⁰

)) in

C

(F

₀

).

Here

W2:= (M−intN(Km))∩V2=V2−intN(σ^m∪α⁰_2,1∪ · · · ∪α_2,b⁰ ).

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