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数理解析研究所講究録
第2004巻 2016年 108-114

On annulus twists

Tetsuya Abe
Osaka City University Advanced Mathematical InstituteABSTRACT. We survey some results 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 onS^{3} In other words, every closed connected orientable
3‐manifold is described by a framed link in S^{3} Our interest is insome link inuniqueness 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 framedlinks \mathcal{L} and \mathcal{L}\sqcup \mathcal{O} give the same 3‐manifold. A modified questionis 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 NO20

[16] (see also [8, 9, 17,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sproblem in Kirby problem list [10]:109

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 generalizedannulus 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 thati, \dot{j}\in \mathbb{N}.

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

Here X_{K}(n) denotes the smooth 4‐manifold obtained from the4‐ball B^{4} by attaching a 2‐handle along K with framing n , andthe symbol \approx stands for a diffeomorphism. Due to Akbulut[5,pair of distinct knots K_{n} and K_{n}' such that6],
X_{K_{n}}(n)\approx XKń(n) for each n\in \mathbb{Z} which is a partial answer toProblem 1. In [4], Jong, Omae, Takeuch, and the author solved
Problem 1 for the case n=0, \pm 4 In [3], Jong, Luecke, Osoinach,there existsa,.

and the author also solved Problem 1 affirmatively.

2. OSOINACHS RESULTIn this section, we recall Osoinachs result in[19].

Let K_{n} bethe knots in Figure 1, which is isotopic to the knots in the page731 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,

whereS_{n}^{3}(K)denotes the 3‐manifold obtained by n ‐surgery of a knot K in S^{3}

110

FIGURE 1. The definition of the knots K_{n}.

Note thatwecancheck that

K_{n} and K_{-n} are isotopic, andTakioka proved that the knots K_{n}(n\geq 0) are mutually distinct

by calculating the Gamma polynomial which is a specializationof 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 infollowing.

[19] is theFIGURE 2. The definitions of V 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.
Remark 2.3. 0soinach [19] considered the diffeomorphism $\varphi$_{n}^{-1}

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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 CONCORDANCEWe recall a terminology in knot concordance. Two knots K andK

areconcordant if they cobound a properly embedded annulus

in S^{3}\mathrm{x}I.

In this paper, we do NOT consider orientations of agiven knot.
Dehn surgery on knots and knot concordance are closely related.

A motivating question is the following.

112

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}(KIs the converse,true 9

The following conjecture is due to Akbulut 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)arerelated by a cork twist.

Forthe 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

not concordant(see [2]).

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Acknowledgments. The author was supported by JSPS KAK‐ENHI Grant Number 16\mathrm{K}17597.

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114[24] Problem list Banff 2016,`.

Synchronisation of smooth and topological 4‐manifolds
Banff International Research Station, February 2016.

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