On annulus twists (Intelligence of Low-dimensional Topology)

On annulus twists

Tetsuya

Abe

Osaka

_City

_University

Advanced Mathematical

Institute

ABSTRACT. We _survey someresults about annulus twists which are related

to _{Dehn surgery} on knots, knot concordance, and 4‐manifold _theory.

1. INTRODUCTION

Lickorish

_[15]

and Wallace

_[22]

_proved

that _every closed con‐

nected orientable 3‐manifold can be obtained

by

Dehn _surgery on

some link in

S^{3}

Inother

words,

_every closed connected orientable

3‐manifold is described

_by

a framed link in

S^{3}

Our interest is in

uniqueness

of framed link

_descriptions

of a

given

3‐manifold. \mathrm{A}

natural

_question

is the

_following.

Question

1. If two framed links \mathcal{L} and \mathcal{L}'

_give

the same 3‐

manifold,

then are \mathcal{L} and \mathcal{L}

isotopic

as framed links?

It is well‐known that the answer of

Question

1 is NO.

Indeed,

for a

given

framed link \mathcal{L} and a

1/n

‐framed unknot \mathcal{O}_, two framed

links \mathcal{L} and \mathcal{L}\sqcup \mathcal{O}

_give

the same 3‐manifold. A modified

question

is the

_following.

Question

2. If two framed knots \mathcal{K} and \mathcal{K}

_give

the same 3‐

manifold,

then are \mathcal{K} and \mathcal{K}

isotopic

as framed knots?

The answer of

Question

2 is

again

NO

[16]

(see

also

[8,

_9,

_17,

20 The

_{remaining questions}

are the

following.

(1)

Under what

_conditions,

are framed knot

descriptions

of a 3‐

manifold

_unique?

(2)

To what

_extent,

are framed knot

descriptions

of a 3‐manifold

far from

_unique?

For the

_question

_(1),

for

_example,

see

[12,

13,

14,

18].

We con‐

centrate on the

question

(2).

More

precisely,

we consider Clark’s

problem

in

_Kirby

_problem

list

_[10]:

Problem

_3.6(D).

Fix an

integer

n. Is there a

homology 3‐sphere

(or

any

3‐manifold)

which can be obtained

by

n_‐surgery on an

infinite number of distinct knots?

In

_[19],

Osoinach solved Problem

_3.6(D)

for the case n=0

by

constructing

knots

_using

the method of

_twisting

_along

an an‐

nulus,

which we call an annulus twist. After

Teragaito‘s

work

[21] (see

also

_[7,

11

_{Jong, Luecke, Osoinach,}

and the author

[3]

solved Problem

_3.6(D)

_{affirmatively,}

where

_they

_generalized

annulus twists.

In

_[4],

a4‐dimensional extension of Problem

_3.6(D)

was _pro‐

posed

as follows:

Problem 1. Let n be an

integer.

Find

infinitely

_many

mutually

distinct knots

_{K_{1}, K_{2},}

\cdots such that

X_{K_{i}}(n)\approx X_{K_{j}}(n)

for each

i,

\dot{j}\in \mathbb{N}.

Here

_{X_{K}(n)}

denotes the smooth 4‐manifold obtained from the

4‐ball

B^{4}

_{by attaching}

a 2‐handle

along

K with

framing

n_, and

the

_symbol

\approx stands for a

diffeomorphism.

Due to Akbulut

[5,

6],

there exists a

pair

of distinct knots

K_{n}

and

K_{n}'

such that

X_{K_{n}}(n)\approx

XKń(n)

for each n\in \mathbb{Z}, which is a

partial

answer to

Problem 1. In

_[4],

_{Jong, Omae, Takeuch,}

and the author solved

Problem 1 for the case

n=0,

\pm 4. In

[3],

Jong,

Luecke,

Osoinach,

and the author also solved Problem 1

_{affirmatively.}

2. OSOINACH’S RESULT

In this

_section,

we recall Osoinach‘s result in

[19].

Let

K_{n}

be

the knots in

_Figure

_1,

which is

_isotopic

to the knots in the _page

731 in

_[19].

One of the main results in

_[19]

is the

_following.

Theorem 2.1

_{(Osoinach [19]).}

We have the

_following.

(1)

The

_3‐manifold

obtained

_by

_{0 ‐surgery}

_of

_{K_{0}}

is toroidal.

(2)

The _sequence

_{\{K_{n}\}}

contains

_infinitely

_many distinct

_hyper‐

bolic knots.

(3)

S_{0}^{3}(K_{0})\approx S_{0}^{3}(K_{1})\approx S_{0}^{3}(K_{2})\approx S_{0}^{3}(K_{3})\approx\cdots

, where

S_{n}^{3}(K)

FIGURE 1. The definition ofthe knots _{K_{n}.}

Note that we can check that

K_{n}

and

K_{-n}

are

isotopic,

and

Takioka

_proved

that the knots

_{K_{n}(n\geq 0)}

are

mutually

distinct

by calculating

the Gamma

_polynomial

which is a

specialization

of the HOMFLYPT

_polynomial.

Let V be the solid torus

_standardly

embedded in

S^{3}

and V‘

the

3‐manifold as in

Figure

2. The main observation in

[19]

is the

following.

FIGURE 2. The definitions ofV and V‘.

Lemma 2.2

_(cf.

Theorem 2.1 in

_[19]).

There exists a

(natural)

diffeomorphism

$\varphi$_{n}:V\rightarrow V

such that

_{$\varphi$_{n}|_{\partial V'}=id.}

Figure

2

_explains

a

proof

of

(3)

in Theorem 2.1. Note

that,

by

Lemma

_2.2,

the

_picture

on the bottom‐left is

diffeomorphic

to

S_{0}^{3}

(K0).

FIGURE 3. A_proof of

₍₃₎

in Theorem 2.1.

3. DEHN SURGERY AND KNOT CONCORDANCE

We recall a

terminology

in knot concordance. Two knots K and

K‘

are concordant if

they

cobound a

properly

embedded annulus

in

S^{3}\mathrm{x}I

. In this paper, we do NOT consider orientations of a

given

knot.

Dehn _surgeryon knots and knot concordance are

closely

related.

Question

3.1

_{(A.Levine [24]).}

_If

K is concordant to K', then

for

all _n,

_{S_{n}^{3}(K)}

is

_homology

cobordant to

S_{n}^{3}(K

Is the converse

true9

The

_following

_conjecture

is due toAkbulut and

_Kirby

_(see

Prob‐

lem 1.19 in the

_{Kirby’s problem}

list

_[10]).

Conjecture.

If 0‐framed

_surgeries

on two knots

give

the same

3‐manifold,

then the knots are concordant.

Tagami

and the author

_[2]

_proved

that

_{Akbulut‐Kirby’s}

_conjec‐

ture is false if the slice‐riuUon

_conjecture

is true.

_{Subsequently,}

Yasui

_[23]

_proved

that

_{Akbulut‐Kirby’s}

_conjecture

is false

_by

con‐

structing

knots K and K'

_satisfying

(1) X_{K}(0)

and

_{X_{K'}(0)}

are exotic

(i.e.

homeomorphic

but non‐

diffeomorphic).

(2)

K and K' are not concordant.

Note that

_{X_{K}(0)}

and

_{X_{K'}(0)}

are related

by

a cork twist. For

the

_details,

see

[23].

The

remaining conjecture

is the

following.

Conjecture.

Let K and K‘

be knots. If

_{X_{K}(0)}

and

_{X_{K'}(0)}

are

diffeomorphic,

then K and K' are concordant.

K_{\mathrm{O}} K_{1}

FIGURE 4. The definition of _{K_{0}} and _{K_{1}.}

Remark: Let

_{K_{0}}

and

_{K_{1}}

be the knots in

_Figure

4.

_By

the re‐

sult in

_[4],

the 4‐manifolds

_{X_{K_{0}}(0)}

and

_{X_{K_{1}}(0)}

are

diffeomorphic.

Furthermore,

if the slice‐riuUon

_conjecture

is

_true,

_{K_{0}}

and

_{K_{1}}

are

Acknowledgments.

The author was

supported by

JSPS KAK‐

ENHI Grant Number 16\mathrm{K}17597.

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Banff International Research_{Station, February}2016.

Osaka

_{City University}

Advanced Mathematical Institute

3‐3‐138

_{Sugimoto, Sumiyoshi‐ku}

Osaka 558‐8585 JAPAN

\mathrm{E}‐mail address:

_{[email protected]‐cu.ac.jp}