# Ribbonness on classical link

Akio Kawauchi
Osaka Central Advanced Mathematical Institute, Osaka Metropolitan University
Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
<EMAIL_ADDRESS>
###### ABSTRACT

It is shown that if a link in 3-space bounds a proper oriented surface
(without closed component) in the upper half 4-space, then the link bounds a
proper oriented ribbon surface in the upper half 4-space which is a renewal
embedding of the original surface. In particular, every slice knot is a ribbon
knot, answering an old question by R. H. Fox affirmatively.

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Keywords: Ribbon surface, Slice knot, Ribbon knot.

2020 Mathematics Subject Classification: Primary 57K10; Secondary 57K45

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1\. Introduction

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For a long time, the author has considered the (2,1)-cable of the figure-eight
knot, which is not ribbon but rationally slice, as a candidate for a non-
ribbon knot which might be slice (see [4, 5]). However, in [1], I. Dai, S.
Kang, A. Mallick, J. Park and M. Stoffregen showed that it is not a slice
knot. In this paper, the author comes back to elementary research beginning
point of [10] on the difference between a slice knot and a ribbon knot. Then
it is concluded that every slice knot is a ribbon knot. More generally, it is
shown that if a link in 3-space bounds a proper oriented surface (without
closed component) in the upper half 4-space, then the link bounds a proper
oriented ribbon surface in the upper half 4-space which is a renewal embedding
of the original surface.

This detailed explanation is done as follows. For a set $A$ in the 3-space
${\mathbf{R}}^{3}=\\{(x,y,z)|\,-\infty<x,y,z<+\infty\\}$ and an interval
$J\subset{\mathbf{R}}$, let

$AJ=\\{(x,y,z,t)|\,(x,y,z)\in A,\,t\in J\\}.$

The upper-half 4-space ${\mathbf{R}}^{4}_{+}$ is denoted by
${\mathbf{R}}^{3}[0,+\infty)$. Let $k$ be a link in the 3-space
${\mathbf{R}}^{3}$, and $F$ a proper oriented surface in the upper-half
4-space ${\mathbf{R}}^{4}_{+}$ with $\partial F=k$. Let
$b_{j}\,(j=1,2,\dots,m)$ be finitely many disjoint oriented bands spanning the
link $k$ in ${\mathbf{R}}^{3}$, which are regarded as framed arcs spanning $k$
in ${\mathbf{R}}^{3}$. Let $k^{\prime}$ be a link in ${\mathbf{R}}^{3}$
obtained from $k$ by surgery along these bands. Then this band surgery
operation is denoted by $k\to k^{\prime}$. Let $k$ have $r$ knot components.
If the link $k^{\prime}$ has $r-m$ components, then the band surgery operation
$k\to k^{\prime}$ is called a fusion. If the link $k^{\prime}$ has $r+m$
components, then the band surgery operation $k\to k^{\prime}$ is called a
fission. These terminologies are used in [10].

A band sum $k\\#_{b}o$ of a link $k$ and a trivial link $o$ of components
$o_{i}\,(i=1,2,\dots,r)$ is a special fusion of the split sum $k+o$ along a
disjoint band system $b_{i}\,(i=1,2,\dots,r)$ spanning $k$ and $o_{i}$ for
every $i$. For the knot components $k_{i}\,(i=1,2,\dots,n)$ of $k$, the band
surgery operation $k\to k^{\prime}$ induces the band surgery operation
$k_{i}\to k^{\prime}_{i}$ for all $i$. If the link $k^{\prime}_{i}$ is a knot
for all $i$, then the band surgery operation $k\to k^{\prime}$ is called a
genus addition.

Every band surgery operation $k\to k^{\prime}$ along a band system $b$ is
realized as a proper surface $F_{s}^{u}$ in ${\mathbf{R}}^{3}[s,u]$ for any
interval $[s,u]$, as follows (see [10]):

$F_{s}^{u}\cap{\mathbf{R}}^{3}[t]=\left\\{\begin{array}[]{rl}k^{\prime}[t],&\quad\mbox{for
$\frac{s+u}{2}<t\leq u$},\\\ (k\cup b)[t],&\quad\mbox{for
$t=\frac{s+u}{2}$},\\\ k[t],&\quad\mbox{for $s\leq
t<\frac{s+u}{2}$}.\end{array}\right.$

For every band surgery sequence $k_{1}\to k_{2}\to\dots\to k_{n-1}\to k_{n}$,
the realizing surface $F_{s}^{u}$ in ${\mathbf{R}}^{3}[s,t]$ is given by the
union

$F_{s_{0}}^{s_{1}}\cup F_{s_{1}}^{s_{2}}\cup\dots\cup
F_{s_{m-2}}^{s_{m-1}}\cup F_{s_{m-1}}^{s_{m}}$

for any division

$s=s_{0}<s_{1}<s_{2}<\dots<s_{m-1}<s_{m}=u$

of the interval $[s,u]$. Note that the realizing surface $F_{s}^{u}$ in
${\mathbf{R}}^{3}[s,t]$ is uniquely determined up to smooth isotopies of
${\mathbf{R}}^{3}[s,t]$ keeping ${\mathbf{R}}^{3}[s]\cup{\mathbf{R}}^{3}[t]$
fixed. For a band surgery sequence $k_{1}\to k_{2}\to\dots\to k_{n-1}\to
k_{n}$ where $k_{1}$ is a split sum $k^{\prime}_{1}+o$ for a link
$k^{\prime}_{1}$ and a trivial link $o$ and $k_{n}$ is a trivial link
$o^{\prime}$, a semi-closed realizing surface $\mbox{scl}(F_{s}^{u})$ in
${\mathbf{R}}^{3}[s,t]$ bounded by the link $k^{\prime}_{1}$ in
${\mathbf{R}}^{3}$ is constructed as follows.

$\mbox{scl}(F_{s}^{u})=F_{s}^{u}\cup d[s]\cup d^{\prime}[u]$

for disk systems $d,d^{\prime}$ in ${\mathbf{R}}^{3}$ with $\partial d=o$ and
$\partial d^{\prime}=o^{\prime}$. A modified semi-closed realizing surface
$\mbox{scl}(F_{s}^{u})^{+}$ of the band surgery sequence
$k_{1}=k^{\prime}_{1}+o\to k_{2}\to\dots\to k_{n-1}\to k_{n}=o^{\prime}$ is a
proper surface in ${\mathbf{R}}^{3}[s,+\infty)$ bounded by the link
$k^{\prime}_{1}$ obtained from $\mbox{scl}(F_{s}^{u})$ by raising the level
$s$ of the disk $d$ into the level $d+\varepsilon$ for a sufficiently small
$\varepsilon>0$.

Let $F$ be an $r$-component proper surface without closed component in the
upper-half 4-space ${\mathbf{R}}^{4}_{+}$ which bounds a link $k$ in
${\mathbf{R}}^{3}$. By [10], the proper surface $F$ in ${\mathbf{R}}^{4}_{+}$
is equivalent to a modified semi-closed realizing surface
$\mbox{scl}(F^{1}_{0})^{+}$ of a band surgery $k+o\to o^{\prime}$ in
${\mathbf{R}}^{4}_{+}$. Since the band system used for $k+o\to o^{\prime}$ is
made disjoint, the modified semi-closed realizing surface
$\mbox{scl}(F^{1}_{0})^{+}$ is further equivalent to a modified semi-closed
realizing surface $\mbox{scl}(F^{1}_{0})^{+}$ of a band surgery sequence

$\mbox{(*)}\qquad\qquad k+o\to k_{1}\cup o\to k_{2}\cup o\to k_{3}\to
o_{4}=o^{\prime},$

where

(0) $k_{1}$ is a link of $r$ components and the operation $k+o\to k_{1}\cup o$
is a fusion fixing $o$,

(1) the operation $k_{1}\cup o\to k_{2}\cup o^{\prime}$ is a genus addition
fixing $o$,

(2) the operation $k_{2}\cup o\to k_{3}$ is a fusion along a band system
connecting every component of $o$ to $k_{2}$ so that $k_{3}$ is a link with
$r$ components,

(3) the operation $k_{3}\to o_{4}=o^{\prime}$ is a fission (cf. [10]).

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In particular, in the band surgery sequence (*) above, if the trivial link $o$
is taken the empty set $\emptyset$, then the step (2) is omitted and we have
$k_{2}=k_{3}$. A proper surface $F$ in ${\mathbf{R}}^{4}_{+}$ is said to be
ribbon if it is equivalent to a semi-closed realizing surface of a band
surgery sequence (*) with $o=\emptyset$.

The purpose of this paper is to show the following theorem.

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Theorem 1.1. Assume that a link $k$ in the 3-space ${\mathbf{R}}^{3}$ bounds a
proper oriented surface $F$ without closed component in the upper-half 4-space
${\mathbf{R}}^{4}_{+}$. Then the link $k$ in ${\mathbf{R}}^{3}$ bounds a
ribbon surface $F^{*}$ in ${\mathbf{R}}^{4}_{+}$ which is a renewal embedding
of $F$.

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For a link $k$ in ${\mathbf{R}}^{3}$, let $g^{*}(k)$ be the minimal genus of a
smoothly embedded connected proper surface in ${\mathbf{R}}^{4}_{+}$ bounded
by $k$, and $g^{*}_{r}(k)$ the minimal genus of a connected ribbon surface in
${\mathbf{R}}^{4}_{+}$ bounded by $k$. The following corollary is a direct
consequence of Theorem 1.1.

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Corollary 1.2. $g^{*}(k)=g^{*}_{r}(k)$ for every link $k$.

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Since a slice knot in ${\mathbf{R}}^{3}$ is the boundary knot of a smoothly
embedded proper disk in ${\mathbf{R}}^{4}_{+}$ and a ribbon knot in
${\mathbf{R}}^{3}$ is the boundary knot of a ribbon disk in
${\mathbf{R}}^{4}_{+}$, Corollary 1.2 contains an affirmative answer to Fox
Problem 25 [2].

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Corollary 1.3. Every slice knot is a ribbon knot.

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2\. Proof of Theorem 1.1

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The following lemma is a starting point of the proof of Theorem 1.1.

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Lemma 2.1. For a knot $k$ in ${\mathbf{R}}^{3}$, assume that a band sum
$o^{\prime}=k\\#_{b}o$ of $k$ and a trivial link $o$ is a trivial knot in
${\mathbf{R}}^{3}$. Then the knot $k$ is a ribbon knot in ${\mathbf{R}}^{3}$.

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Proof of Lemma 2.1. Let $-k^{*}$ be the reflected inverse knot of a knot $k$
in ${\mathbf{R}}^{3}$. Then the connected sum $(-k^{*})\\#k$ is a ribbon knot
in ${\mathbf{R}}^{3}$ (see [3]). Since the band sum $o^{\prime}=k\\#_{b}o$ is
a trivial knot, the connected sum $(-k^{*})\\#(k\\#_{b}o)$ obtained by locally
tying $-k^{*}$ to a string of $k$ in $k\\#_{b}o$ is equivalent to the knot
$(-k^{*})\\#o^{\prime}=-k^{*}$. On the other hand, the knot
$(-k^{*})\\#(k\\#_{b}o)$ is a ribbon knot because it is a band sum of the
ribbon knot $(-k^{*})\\#k$ and the trivial link $o$. Thus, the knot $-k^{*}$
is a ribbon knot. Since the reflected inverse knot of a ribbon knot is a
ribbon knot, the knot $k$ is a ribbon knot. This completes the proof of Lemma
2.1. $\square$

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Remark 2.2. A ribbon presentation of the connected sum $(-k^{*})\\#k$ for a
knot $k$ in ${\mathbf{R}}^{3}$ can be obtained from the chord diagram of any
given diagram $D(k)$ of $k$ by [6, 7, 8, 9]. In fact, by [9], let $D$ be an
inbound diagram of $D(k)$ (namely, an arc diagram obtained from $D(k)$ by
removing an open arc not containing a crossing point) with the end points in
the infinite region of the plane ${\mathbf{R}}^{3}$, and $C$ a chord diagram
of $D$. The diagram obtained from the based loop system of $C$ by surgery
along a band system thickening the chord system is a ribbon presentation of
the connected sum $(-k^{*})\\#k$. This is because the connected sum
$(-k^{*})\\#k$ is the middle cross-section of the spun knot $S(k)$ of $k$ in
${\mathbf{R}}^{4}$ and the chord diagram $C$ canonically represents the spun
knot $S(k)$ as a ribbon $S^{2}$-knot (see [6, 9, 11]).

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Lemma 2.1 is generalized as follows.

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Lemma 2.3. For a link $k$ of $n$ knot components in ${\mathbf{R}}^{3}$, assume
that a band sum $k\\#_{b}o$ of $k$ and a trivial link $o$ is a ribbon link in
${\mathbf{R}}^{3}$. Then the link $k$ is a ribbon knot in ${\mathbf{R}}^{3}$.

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Proof of Lemma 2.3. For the components $k_{i}\,(i=1,2,\dots,n)$ of $k$, the
band sum $k^{\prime}=k\\#_{b}o$ is the union of band sums
$k^{\prime}_{i}=k_{i}\\#_{b}o_{i}\,(i=1,2,\dots,n)$. Let
$o_{ij}\,(j=1,2,\dots,n_{i})$ be the components of the trivial link $o_{i}$,
and $b_{ij}$ the band spanning $k_{i}$ and $o_{ij}$ used for the band sum
$k^{\prime}_{i}=k_{i}\\#_{b}o_{i}$ for all $j\,(j=1,2,\dots,n_{i})$. The link
$k^{\prime}$ is a ribbon link with components
$k^{\prime}_{i}\,(i=1,2,\dots,n)$, there is a fusion $o^{\prime}\to
k^{\prime}$ with a trivial link $o^{\prime}$ consisting of fusions
$o^{\prime}_{i}\to k^{\prime}_{i}\,(i=1,2,\dots,n)$. Let
$o^{\prime}_{ih}\,(h=1,2,\dots,m_{i})$ be the components of $o^{\prime}_{i}$,
and $b^{\prime}_{ih}\,(h=1,2,\dots,m_{i})$ the bands used for the fusion
$o^{\prime}_{i}\to k^{\prime}_{i}$. By band slides and by regarding bands as
framed arcs, the bands $b_{ij}(i=1,2,\dots,n;\,j=1,2,\dots,n_{i})$,
$b^{\prime}_{ih}\,(i=1,2,\dots,n;\,h=1,2,\dots,m_{i})$ are made disjoint.
Further, the bands $b_{ij}\,(j=1,2,\dots,n_{i})$ are taken to be attached only
to the component $o^{\prime}_{i1}$. Let
$B^{\prime}_{ij}\,(i=1,2,\dots,n;j=1,2,\dots,m_{i})$ be disjoint 3-balls in
${\mathbf{R}}^{3}$ containing the component $o^{\prime}_{ij}$ in the interior.
Let $d_{ij}\,(i=1,2,\dots,n;\,j=1,2,\dots,n_{i})$ be a disjoint disk system
bounded by the trivial loop system
$o_{ij}\,(i=1,2,\dots,n;\,j=1,2,\dots,n_{i})$ in ${\mathbf{R}}^{3}$. Let
$a^{\prime}_{ih}\,(i=1,2,\dots,n;\,h=1,2,\dots,m_{i})$ be a core arc system of
the band system $b^{\prime}_{ih}\,(i=1,2,\dots,n;\,h=1,2,\dots,m_{i})$, and
$a^{\prime\prime}_{ih}\,(i=1,2,\dots,n;\,h=1,2,\dots,m_{i})$ an arc system
obtained from $a^{\prime}_{ih}\,(i=1,2,\dots,n;\,h=1,2,\dots,m_{i})$ by
deforming not to meet the disjoint disk system
$d_{ij}\,(i=1,2,\dots,n;\,j=1,2,\dots,n_{i})$. The deformation should be taken
so that the arc system
$a^{\prime\prime}_{ih}\,(i=1,2,\dots,n;\,h=1,2,\dots,m_{i})$ is isotopic to
the arc system $a^{\prime}_{ih}\,(i=1,2,\dots,n;\,h=1,2,\dots,m_{i})$ when the
disk system $d_{ij}\,(i=1,2,\dots,n;\,j=1,2,\dots,n_{i})$ is forgotten. Let
$b^{\prime\prime}_{ih}\,(i=1,2,\dots,n;\,h=1,2,\dots,m_{i})$ be the band
system thickening the core arc system
$a^{\prime\prime}_{ih}\,(i=1,2,\dots,n;\,h=1,2,\dots,m_{i})$. Then the
disjoint disk system $d_{ij}\,(i=1,2,\dots,n;\,j=1,2,\dots,n_{i})$ can be
moved into $B^{\prime}_{i1}$ while keeping the band system
$b^{\prime\prime}_{ih}\,(i=1,2,\dots,n;\,h=1,2,\dots,m_{i})$ fixed. In this
move, some parts of the band system
$b_{ij}\,(i=1,2,\dots,n;\,j=1,2,\dots,n_{i})$ may be moved. Since $o_{i1}$ and
$d_{ij}\,(j=1,2,\dots,n_{i})$ are disjoint except for the meeting part of the
band system $b_{ij}\,(j=1,2,\dots,n_{i})$, there is a knot
$k^{\prime\prime}_{i}$ such that the trivial knot $o_{i1}$ is the band sum
$k^{\prime\prime}_{i}\\#_{b}o_{i}$ using the bands
$b_{ij}\,(j=1,2,\dots,n_{i})$. By Lemma 2.1, the knot $k^{\prime\prime}_{i}$
is a ribbon knot and thus there is a fusion $o^{\prime\prime}_{i}\to
k^{\prime\prime}_{i}$ for a trivial link $o^{\prime\prime}_{i}$ in
${\mathbf{R}}^{3}$. Note that the knot $k^{\prime\prime}_{i}$ is disjoint from
$B^{\prime}_{ij}\,(i=2,3,\dots,m_{i})$, so that the trivial link
$o^{\prime\prime}_{i}$ is movable into $B^{\prime}_{i1}$ although some parts
of the bands used for the fusion $o^{\prime\prime}\to k^{\prime\prime}$ may
not be in $B^{\prime}_{i1}$. The link $k$ is a fusion of the trivial link
consisting of the split sum of
$o^{\prime}_{i},\,(i=,2,3,\dots,n);\,o^{\prime\prime}_{i}\,(i=1,2,\dots,n)$,
meaning that the knot $k$ is a ribbon link. This completes the proof of Lemma
2.3. $\square$

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Proof of Corollary 2.3. For the knot components $k_{i}\,(i=1,2,\dots,r)$ of
$k$, the band sum $k^{\prime}=k\\#_{b}o$ is the union of band sums
$k^{\prime}_{i}=k_{i}\\#_{b}o_{i}\,(i=1,2,\dots,r)$ for each of which the
proof of Lemma 2.3 can be applied although a band used for the band sum
$k^{\prime}_{i}=k_{i}\\#_{b}o_{i}$ may pass through the other loop system
$o\setminus o_{i}$. Thus, the link $k$ is a ribbon link. This completes the
proof of Corollary 2.3. $\square$

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The proof of Theorem 1.1 is done as follows.

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Proof of Theorem 1.1. Consider that a proper oriented surface $F$ is given by
the sequence

$k+o\to k_{1}\cup o\to k_{2}\cup o\to k_{3},$

where $k_{3}\to k_{2}\cup o$ is a fission, $k_{2}\cup o\to k_{1}\cup o$ is a
genus addition fixing $o$ and $k_{1}\cup o\to k+o$ is a fission fixing $o$.
Consider the inverse sequence

$k_{3}\to k_{2}\cup o\to k_{1}\cup o\to k+o$

of the sequence $k+o\to k_{1}\cup o\to k_{2}\cup o\to k_{3}$. Replace the
bands used for the genus addition $k_{2}\cup o\to k_{1}\cup o$ and the fission
$k_{1}\cup o\to k+o$ by bands such that

(i) every band does not change the attaching parts, and

(ii) every band does not pass the trivial link $o$, and

(iii) every band is deformable into the original band if the trivial link $o$
is forgotten.

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Since the bands used for the fission $k_{3}\to k_{2}\cup o$ cut the bands of
the fusion $k_{2}\cup o\to k_{3}$, the genus addition $k_{2}\cup o\to
k_{1}\cup o$ changes into a genus addition $k_{2}+o\to k^{\prime}_{1}+o$
fixing $o$ for a new link $k^{\prime}_{1}$ and the fission $k_{1}\cup o\to
k+o$ changes into a fission $k^{\prime}_{1}+o\to k+o$ fixing $o$,
respectively. Note that the link $k^{\prime}_{1}$ has the same number of
components as $k_{1}$. Then the sequence

$k+o\to k_{1}\cup o\to k_{2}\cup o\to k_{3}$

changes a sequence

$k+o\to k^{\prime}_{1}+o\to k_{2}+o\to k_{3},$

where the operation $k+o\to k^{\prime}_{1}+o$ is a fusion fixing $o$ with
$k^{\prime}_{1}$ a link of $r$ components, the operation $k^{\prime}_{1}+o\to
k_{2}+o$ is a genus addition fixing $o$, and the operation
$k^{\prime}_{2}+o\to k_{3}$ is a fusion meaning that $k_{3}$ is a bund sum
$k^{\prime}_{2}\\#_{b}o$ of $k^{\prime}_{2}$ and $o$. Since $k_{3}$ is a
ribbon link, $k^{\prime}_{2}$ is a ribbon link by Lemma 2.3. Thus, there is a
sequence a sequence

$k\to k^{\prime}_{1}\to k_{2}\to o^{\prime}_{3},$

where the operation $k\to k^{\prime}_{1}$ is a fusion with $k^{\prime}_{1}$ a
link of $r$ components, the operation $k^{\prime}_{1}\to k^{\prime}_{2}$ is a
genus addition and the operation $k^{\prime}_{2}\to o^{\prime}_{3}$ is a
fission with $o^{\prime}_{3}$ a trivial link. This means that the link $k$ in
${\mathbf{R}}^{3}$ bounds a ribbon surface $F^{\prime}$ in
${\mathbf{R}}^{4}_{+}$ which is a renewal embedding of $F$. This completes the
proof of Theorem 1.1. $\square$

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Acknowledgements. This paper was completed during the author’s stay at Dalian
University of Technology, China from July 2, 2023 to July 21, 2023. The author
would like to thank Feng Chun Lei and Fengling Li for their kind
hospitalities. This work was partly supported by JSPS KAKENHI Grant Numbers
JP19H01788, JP21H00978 and Osaka Central Advanced Mathematical Institute,
Osaka Metropolitan University (MEXT Joint Usage/ Research Center on
Mathematics and Theoretical Physics JPMXP 0619217849).

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