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数理解析研究所講究録
第2004巻 2016年 30-37

Surface‐links which bound immersed handlebodies

Kengo Kawamura
Osaka City University Advanced Mathematical Institute

Introduction

1This article is a summary for ribbon‐clasp surface‐links defined in our paper
aregeneralization of ribbon surface‐links.

[4], which

Throughout this article, we work in the PL or smooth category. An immersed surface‐link or simply a surface‐link means a closed oriented surface immersed in \mathbb{R}^{4} such thateach multiple point isa transverse double point.
In the PL category, we assume that
locally flat. When it is embedded, we also call it an embedded surface‐link. Two surface‐links are said to be equivalent if they are ambient isotopic.
A surface‐link is trivial if it bounds a disjoint union of handlebodies embedded in \mathbb{R}^{4}.
In particular, a trivial 2‐link means the boundary of a disjoint union of 3‐balls embedded
in \mathb {R}^{4} A surface‐link is ribbon if it bounds a disjoint union of handlebodies immersed in \mathbb{R}^{4}whose multiple point set consists of ribbon singularities. (Note that a ribbon surface‐link
is an embedded surface‐link.) A surface‐link is ribbon‐clasp if it bounds a disjoint unionof handlebodies immersed in \mathbb{R}^{4} whose multiple point set consists of ribbon singularities
and clasp singularities. We give definitions of a ribbon singularity and a clasp singularity
in Section 2. (For an immersion f : M\rightarrow \mathbb{R}^{4} of a compact 3‐manifold M the boundaryof the immersed 3‐manifold f(M) means the image f(\partial M) of the boundary \partial M of M. )In this article, we show two characterizations of a ribbon‐clasp surface‐link. First, wecharacterize it in terms of 1‐handle surgeries and finger moves (Theorem 3.2). Second, we

immersions

are.

,

characterize it in terms of normal forms for immersed surface‐links (Theorem 4.4). We

introduce 1‐handle surgeries and finger moves in Section 3 and normal forms for immersed

surface‐links in Section 4.

2

Ribbon singularities and clasp singularities

In this section, we explain a ribbon singularity and a clasp singularity.
Let M beacompact 3‐manifold with non‐empty boundary and f : M\rightarrow \mathbb{R}^{4} an\mathbb{R}^{4} Let \triangle be a connected component of the multiple point set

immersion of M into.

\{x\in f(M)|\# f^{-1}(x)\geq 2\}\subset \mathbb{R}^{4}.

We say that \triangle is a ribbon singularity if \triangle is a 2‐disk in \mathbb{R}^{4} and the preimage of \triangle is thedisjoint union of embedded 2‐disks \triangle_{1} and \triangle_{2} in M such that \triangle_{1} is properly embedded
Figure 1 shows a local model of a ribbon
inmotion
method.
thepicturesingularity

in M and \triangle_{2} is embedded in the interior of M.

31\downar owf

Figure 1: A local model of a ribbon singularity
We say that \triangle is a clasp singularity if \triangle is a 2‐disk in \mathbb{R}^{4} and the preimage of \triangle is thedisjoint union of embedded 2‐disks \triangle_{1} and $\Delta$_{2} in M such that for each i\in\{1 2 \}, \partial\triangle_{i} isthe union of two arcs $\alpha$_{i} and $\beta$_{i} where $\alpha$_{i} is a properly embedded arc in M and $\beta$_{i} is asimple arc in \partial M which connects endpoints of $\alpha$_{i} Figures 2 and 3 show local models ofa clasp singularity in the motion picture method.
,,.

Figure 2: A local model of a clasp singularity\mathscr{M}\ovalbox{\t smalREJ CT}
Figure 3: Another local model of a clasp singularity

Example 2.1 A Montesinos twin is a surface‐link which is the boundary of a pair ofembedded 3‐disks B_{1} and B_{2} with a single clasp singularity between B_{1} and B_{2} Figure4 shows a Montesinos twin T=S_{1}\cup S_{2} where S_{i}=\partial B_{i}(i\in\{1,2.

,A Montesinos twin has two double points with opposite signs. Note that the equivalenceclass, as a surface‐link, of a Montesinos twin is unique.

32

S_{1}\cup S_{2}Figure 4: A Montesinos twin T=S_{1}\cup S_{2}Definition 2.2 An M ‐trivial 2‐link is a split union of a trivial 2‐link and some (or no)Montesinos twins.

1‐handle surgeries and finger moves

3

Let F be a surface‐link. A chord attached to F means an unoriented simple arc $\gamma$ in \mathbb{R}^{4}F\cap $\gamma$=\partial $\gamma$ which misses the double points of F Two chords attached to F
equivalent if they are ambient isotopic by an isotopy of \mathbb{R}^{4} keeping F setwise fixed.

such thatare.

,

A1‐handle attached to F means an embedded 3‐disk B in \mathb {R}^{4} such that F\cap B is theunion of a pair of mutually disjoint 2‐disks in \partial B, F\cap B misses the double points of F,

and the orientation of F\cap B induced from \partial B is opposite to the orientation induced fromFPut.

\mathrm{h}(F;B) :=\mathrm{C}1(F\cup\partial B-F\cap B),

which we call the surface‐link obtained from F by a 1‐handle surgery along B.

Cl means the closure.)are

Two 1‐handles attached to F are said to be(Here,equivalent if theyambient isotopic by an isotopy of \mathb {R}^{4} keeping F setwise fixed. It is known [1, 3] that1‐handles attached to F are equivalent if and only if their cores are equivalent as chords

attached to F.

Forachord $\gamma$

attaching toF,

wedenote by \mathrm{h}(F; $\gamma$) the surface‐link

obtained from F by a 1‐handle surgery along a 1‐handle whose core is $\gamma$
a local.

Figure 5 showsmodel of a 1‐handle surgery along $\gamma$.

Figure 5; A local model of a 1‐handle surgery along $\gamma$
A finger move is the inverse operation of the Whitney trick; for details, see [2, 7]. We33give an alternative definition of a finger move by using a Montesinos twin and 1‐handlesurgeries as follows.
Let F be a surface‐link and U a 4‐disk in \mathbb{R}^{4} disjoint from F Put a Montesinos twin
T=S_{1}\cup S_{2} in U and let B_{1} and B_{2} be embedded 3‐disks in U with a single clasp singularity
with S_{i}=\partial B_{i}(i\in\{1,2Take a point p_{1}\in S_{1}-S_{1}\cap S_{2} a point p_{2}\in S_{2}-S_{1}\cap S_{2} andtwo distinct points q_{1}, q_{2} in F missing the double points of F Let $\gamma$_{1} and $\gamma$_{2} be oriented.

,.

chords attached to F\cup T such that for i\in\{1 , 2 \}, $\gamma$_{i} starts from p_{i} and terminates at q_{i}and $\gamma$_{i}\cap(B_{1}\cup B_{2})= {pi}. Let F' be the surface‐link obtained from F\cup T by 1‐handle

surgeries along two 1‐handles whose cores are $\gamma$_{1} and $\gamma$_{2} Let $\gamma$ be a chord attached to F
which is the concatenation of $\gamma$_{1}^{-1} , a simple arc from p_{1} to p_{2} in U and $\gamma$_{2} We say that.

.

F' is obtained from F by a finger move along $\gamma$ , which we denote by \mathrm{f}(F; $\gamma$).

It is seenthat if $\gamma$ and $\gamma$' are equivalent chords attached to F then \mathrm{f}(F; $\gamma$) is equivalent to \mathrm{f}(F; $\gamma$Figure 6 shows a local model of a finger move along $\gamma$ where two 1‐handles attached toF\cup T whose cores are $\gamma$_{1} and $\gamma$_{2} are omitted for simplicity.
,

Figure 6: A local model of a finger move along $\gamma$
In terms of 1‐handle surgeries and finger moves, ribbon surface‐links and ribbon‐clasp

surface‐links are characterized as follows.
Theorem 3.1([6, 8]) A surface‐link is ribbon if and only if it is obtained from a trivial

2‐link by 1‐handle surgeries.
Theorem 3.2([4]) For a surface‐link F,

the following conditions are equivalent.

(1) F is a ribbon‐clasp surface‐link.
(2) F is obtained from a ribbon surface‐link by finger moves.

(3) F is obtained from a trivial 2‐link by 1‐handle surgeries and finger moves,(4) F is obtained from an M ‐trivial 2‐link by 1‐handle surgeries.
It is seen that Theorem 3.2 is a generalization of Theorem 3.1.

34

Normal forms for surface‐links

4
4.1

Normal forms for embedded surface‐links

Let L be an oriented link in \mathbb{R}^{3}. A band attached to L means an oriented 2‐disk B in \mathbb{R}^{3}such that L\cap B is the union of a pair of mutually disjoint arcs in \partial B and the orientationsof L\cap\partial B induced from \partial B and L are opposite. We say that a link L' is obtained from L

by a band surgery along B if L'=\mathrm{C}1(L\cup\partial B-L\cap\partial B),

see

Figure 7. Let \mathcal{B}=B_{1}\cup\ldots\cup B_{ $\tau$ r $\iota$}\rightarrow) (\rightar ow\rightarrow$\Gamma$ \mathrm{V}LL

Figure 7: A band surgery from L to L' along a band B
be mutually disjoint oriented 2‐disks in \mathbb{R}^{3} such that each B_{i} is a band attached to L. \mathrm{A}band surgery from L to L' along \mathcal{B} is denoted by L\rightarrow BL' or simply L\rightarrow L'.

For a band surgery L\rightarrow BL' the realizing surface is a compact oriented surface, say \mathrm{F},,properly embedded in \mathbb{R}^{3}\times[a, b] defined by:

F\cap\mathb{R}^3\times\{t}=\left{\begin{ar y}{l L'\times\{t}&\mathrm{f}\mathrm{o}\mathrm{}t\in(a+b)/2,b]\ L\cup\mathcl{B}\times\{t}&\mathrm{f}\mathrm{o}\mathrm{}t=(a+b)/2\ L\times\{t}&\mathrm{f}\mathrm{o}\mathrm{}t\in[a,(+b)/2. \end{ar y}\right.
This realizing surface is denoted by
Let \mathcal{L}F(L\rightarrow BL')_{[a,b]}.

L_{1}\rightarrow L_{2}\rightarrow\ldots\rightarrow L_{m} be a band surgery sequence. The realizing surfaceofF(\mathcal{L})_{[a,b]} L in \mathbb{R}^{3}\times[a, b] with a division a=t_{1}<t_{2}<\ldots<t_{m}=b is the union of them-1 see the left of Figure 8. (Noterealizing surfaces F(L_{i}\rightarrow L_{i+1})_{[t_{i},t_{i+1}]} for i=1,
that the ambient isotopy class of the realizing surface F(\mathcal{L})_{[a,b]} does not depend on thechoice of a division.) If the links L_{1} and L_{m} are trivial links, then there exist disk systems\mathcal{D} and \mathcal{D}' in \mathb {R}^{3} with \partial D=L_{1} and \partial D'=L_{m} Then we obtain a closed oriented surface:,.

\overline{F}(\mathcal{L})_{[a,b]}:=\mathcal{D}\times\{a\}\cup F(\mathcal{L})_{[a,b]}\cup \mathcal{D}'\times\{b\}in \mathbb{R}^{3}\times[a, b] , which we call the ctosed realizing surface of \mathcal{L} , see the right of Figure 8. Notethat by Horibe‐Yanagawas lemma shown in [5], the equivalence class of\overline{F}(L)_{[a,b]} does not

choices of disk systems \mathcal{D} and \mathcal{D}' We say that an embedded surface‐link is independa normal form if it is a closed realizing surface
of a band surgery sequence \mathcal{L}.
on.

\overline{F}(\mathcal{L})_{[a,b]}([5]) Every embedded surface‐link with $\mu$ components and g total genus isequivalent to the closed realizing surface of a band surgery sequence

Theorem 4.1

O\rightarrow L_{-}\rightarrow L_{0}\rightarrow L_{+}\rightarrow O',
where O and O'aretrivial links, L_{-} and L_{+} are $\mu$ ‐component links and L_{0} is a( $\mu$+g)-component link.
A ribbon surface‐link is characterized in terms of normal forms as follows.

35

Figure 8: The realizing surface F(L_{1^{-}}^{B}3L_{2^{-}}^{B}SL_{3})_{[-2,2]} and the closed realizing surface \overline{F}(L_{1}\rightar ow^{B}L_{2^{-} ^{B}3,L_{3})_{[-2,2]}

Theorem 4.2([5]) An embedded surface‐link is ribbon if and only if it is equivalent tothe closed realizing surface of a band surgery sequence

O\rightarrow L\rightarrow O,
where O is a trivial link and the band surgery L\rightarrow O is the inverse of O\rightarrow L.
4.2

Normal forms for immersed surface‐links

Let L be a link and L' a link obtained from L by applying some crossing changes. Thereis a homotopy(g_{s} : M^{1}\rightarrow \mathbb{R}^{3}|s\in[0,1]) of the source circles M^{1} of the link into \mathbb{R}^{3}with g_{0}(M^{1})=L and g_{1}(M^{1})=L' such that each g_{s} , except s=1/2 , is an embeddingof M^{1} and at s=1/2 intersections occur. We call such a homotopy a crossing change

deformation. A crossing change deformation from L to L' is denoted by L\rightarrow L'.
For a crossing change deformation L\rightarrow L' by a homotopy (g_{s}|s\in[0,1the realizing
isaoriented\mathrm{F}immersed
insurfacecompact
surface, sayproperly\mathbb{R}^{3}\times[a, b] defined by:,F\cap \mathbb{R}^{3}\times\{t\}=g_{s}(L)\times\{t\} for t\in[a, b],
where s=(t-a)/(b-a).

This realizing surface is denoted by F(L\rightarrow L')_{[a,b]}.

A link is called an H ‐trivial link with k Hopf links if it is a split union of a trivial link

and k Hopf links for some k\geq 0.

An H ‐trivial link with k Hopf links can be transformed

into a trivial link by k crossing changes. We call a crossing change deformation determinedby the crossing changes a Hopf‐splitting deformation.
Let \mathcal{L} : L_{1}\rightarrow L_{2}\rightarrow\ldots\rightarrow L_{m} be a band surgery sequence with H‐trivial links L_{1} andL_{m} The realizing surface F(\mathcal{L})_{[a+ $\epsilon$,b-$\epsilon$']} in \mathbb{R}^{3}\times[a+ $\epsilon$, b- $\epsilon$ for some small $\epsilon$, $\epsilon$'>0 is,.

extended to an oriented surface

F(\mathcal{L})_{[a,b]}^{\mathrm{x} :=F(L_{1}'\rightar ow L_{1})_{[a,a+ $\epsilon$]}\cup F(\mathcal{L})_{1- $\xi$ j'}a+ $\epsilon$,b]\cup F(L_{m}\rightar ow L_{m}')_{[b-$\epsilon$',b]}in \mathbb{R}^{3}\times[a, b] , where Lí \rightarrow Ll is the inverse operation of a Hopf‐splitting deformation andL_{m}\rightarrow L_{m}' is a Hopf‐splitting deformation. Since links Lí and L_{m}' are trivial links, there

exist disk systems \mathcal{D} and \mathcal{D}' in \mathbb{R}^{3} with \partial \mathcal{D}=L\'{i} and \partial \mathcal{D}'=L_{m}'.

Then we obtain a closedoriented surface\overline{F}(\mathcal{L})_{[a,b]}^{\times}:=\mathcal{D}\times\{a\}\cup F(\mathcal{L})_{[a,b]}^{\times}\cup \mathcal{D}\times\{b\}36Figure 9: The closed realizing surface\overline{F}(L_{1^{-3L_{2}-3L_{3})_{[-3,3]}^{\times} }^{B\mathcal{B}in \mathbb{R}^{3}\times[a, b] , which we call the closed realizing surface of \mathcal{L}an.

See Figure 9. We say thatimmersed surface‐link is in a normal form if it is a closed realizing surface

a bandsurgery sequence \mathcal{L}.

\overline{F}(\mathcal{L})_{[a,b]}^{\times} ofTheorem 4.3 ([4])

Every immersed surface‐link with $\mu$ components and g total genus isequivalent to the closed realizing surface of a band surgery sequence

O\rightarrow L_{-}\rightarrow L_{0}\rightarrow L_{+}\rightarrow O',
where O and O' are H ‐trivial links, L_{-} and L+ are $\mu$ ‐component links and L_{0} is a( $\mu$+g)-component link.

A ribbon‐clasp surface‐link is characterized in terms of normal forms as follows.
Theorem 4.4 ([4]) An immersed surface‐link is ribbon‐clasp if and only if it is equivalent
to the closed realizing surface of a band surgery sequence

O\rightarrow L\rightarrow O,
where O is an H ‐trivial link and the band surgery L\rightarrow O is the inverse of O\rightarrow L.
These theorems are generalization of Theorem 4.1 and Theorem 4.2.
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306 (1988), 475‐487.
[2] A. J. Casson, Three lectures on new‐infinite constructions in 4‐dimensional manifolds,
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[3] F. Hosokawa and A. Kawauchi, Proposals for unknotted surfaces in four‐spaces, Osaka
J. Math. 16 (1979), 233‐248.

37[4] S. Kamada and K. Kawamura, On ribbon‐clasp surface‐links and normal forms of
singular surface‐links, arXiv:1602.07855.

[5] A. Kawauchi, T.Shibuya and S.Suzuki, Descriptions on surfaces in four‐space, I:
Normal forms, Math. Sem. Notes, Kobe Univ. 10 (1982) 75‐125.
[6] A. Kawauchi, T.Shibuya and S.Suzuki, Descriptions on surfaces in four‐space, II:Singularities and cross‐sectional links, Math. Sem. Notes, Kobe Univ. 11 (1983) 31‐
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[7] R. Kirby, The topology of 4‐manifolds, Lecture Notes in Mathematics 1374, Springer‐
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Osaka City University Advanced Mathematical Institute

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