Ribbonness of a Stable-Ribbon Surface-Link, II: General Case

Abstract

It is shown that any handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link up to equivalences so that every stable-ribbon surface-link is a ribbon surface-link. This is a generalization of a previously observed result for a stably trivial surface-link. Two observations are given. One observation is that a connected sum of two surface-links is a ribbon surface-link if and only if both the connected summands are ribbon surface-links. The other observation is a characterization of when a surface-link consisting of ribbon surface-knot components becomes a ribbon surface-link.

1. Introduction

This paper generalizes the previous result that a stably trivial surface-link is a trivial surface-link to the result that a stable-ribbon surface-link is a ribbon surface-link [1,2]. A surface-link is a closed oriented (possibly disconnected) surface F which is embedded in the 4-space ${\mathbf{R}}^4$ by a smooth embedding. When F is connected, it is also called a surface-knot. When a fixed (possibly disconnected) closed surface $\mathbf{F}$ is smoothly embedded into ${\mathbf{R}}^4$, it is also called an $\mathbf{F}$-link. If $\mathbf{F}$ is the disjoint union of some copies of the 2-sphere $S^2$, then it is also called an $S^2$-link. When $\mathbf{F}$ is connected, it is also called an $\mathbf{F}$-knot, and an $S^2$-knot for $\mathbf{F}=S^2$. Two surface-links F and $F^{\prime }$ are equivalent by an equivalence f if f is an orientation-preserving diffeomorphism $f:{\mathbf{R}}^4\to {\mathbf{R}}^4$ sending F to $F^{\prime }$ that preserves orientation. A trivial surface-link is a surface-link F which bounds disjoint handlebodies smoothly embedded in ${\mathbf{R}}^4$, where a handlebody is a 3-manifold which is a 3-ball, solid torus, or a disk sum of some number of solid tori. A trivial surface-knot is also called an unknotted surface-knot. A trivial disconnected surface-link is also called an unknotted-unlinked surface-link. For any given closed oriented (possibly disconnected) surface $\mathbf{F}$, a trivial $\mathbf{F}$-link exists uniquely up to equivalences (see [3]). A ribbon surface-link is a surface-link F which is obtained from a trivial $n{S}^2$-link O for some n (where $n{S}^2$ denotes the disjoint union of n copies of the 2-sphere $S^2$) by surgery along an embedded 1-handle system [4,5,6,7]. This object is an old concept in surface-knot theory, but in recent years, it has been considered as a chord diagram, which is a relaxed version of a virtual graph including a virtual knotoid in a plane diagram [8,9,10,11]. A stabilization of a surface-link F is a connected sum $\overline{F}=F{\#}_{k=1}^s{T}_k$ of F and a system of trivial torus-knots $T_k(k=1,2,\cdots ,s)$. By granting $s=0$, a surface-link F itself is regarded as a stabilization of F. The trivial torus-knot system T is called the stabilizer with stabilizer components $T_k(k=1,2,\cdots ,s)$ on the stabilization $\overline{F}$ of F. A stable-ribbon surface-link is a surface-link F such that a stabilization $\overline{F}$ of F is a ribbon surface-link. Every surface-link F is equivalent to a stabilization of a surface-link $F_{\ast }$ with minimal total genus. This surface-link $F_{\ast }$ is called a handle-irreducible summand of F. The following result called Stable-Ribbon Theorem is our main theorem.

Theorem 1. 

Any handle-irreducible summand $F_{\ast }$ of every stable-ribbon surface-link F is a ribbon surface-link that is determined uniquely from F up to equivalences and stabilizations.

Any stabilization of a ribbon surface-link is a ribbon surface-link. So, the following corollary is obtained from Theorem 1.

Corollary 1. 

Every stable-ribbon surface-link is a ribbon surface-link.

A stably trivial surface-link is a surface-link F such that a stabilization $\overline{F}$ of F is a trivial surface-link. Since a trivial surface-link is a ribbon surface-link, Theorem 1 also implies the following corollary, which is used to prove smooth unknotting conjecture for a surface-link. This result leads to 4D smooth and then classical Poincaré conjectures [12,13,14,15].

Corollary 2. 

Any handle-irreducible summand of every stably trivial surface-link is a trivial $S^2$-link, so that every stably trivial surface-link is a trivial surface-link.

The plan for the Proof of Theorem 1 is to show the following two lemmas by using the previous techniques [1,2].

Lemma 1. 

Any handle-irreducible summand of any surface-link is unique up to equivalences and stabilizations.

Lemma 2. 

Any stable-ribbon surface-link is a ribbon surface-link.

The Proof of Theorem 1 is completed by these lemmas as follows:

Proof of Theorem 1 assuming Lemmas 1 and 2. 

By Lemma 2, any handle-irreducible summand of every stable-ribbon surface-link is a ribbon surface-link, which is unique up to equivalences and stabilizations by Lemma 1. This completes the Proof of Theorem 1. □

An idea of the Proof of Lemma 1 is to generalize the unique result of an O2-handle pair on a surface-link earlier established to the case where the restriction on the attaching part is relaxed (see Theorem 4). An idea of the Proof of Lemma 2 is to consider a semi-unknotted multi-punctured handlebody system, simply called a SUPH system, of a ribbon surface-link. Two applications of Theorem 1 are made. One observation on Theorem 1 is the following theorem.

Theorem 2. 

A connected sum $F=F_1\#F_2$ of surface-links $F_i(i=1,2)$ in $S^4$ is a ribbon surface-link if and only if both the surface-links $F_i(i=1,2)$ are ribbon surface-links.

This theorem contrasts with the behavior of classical ribbon knot, because every classical knot is a connected summand of a connected sum ribbon knot. In fact, for every knot k and the inversed mirror image $-k^{\ast }$ of k in the 3-sphere $S^3$, the connected sum $k\#(-k^{\ast })$ is a ribbon knot in $S^3$ [16,17,18,19]. A natural presentation of $k\#(-k^{\ast })$ is seen in a chord diagram of the spun $S^2$-knot of k as a ribbon $S^2$-knot [9].

A surface-knot $F^{\prime }$ in $S^4$ is obtained from a surface-link F of r components in $S^4$ by fusion if $F^{\prime }$ is obtained from F by surgery along $r-1$ disjointedly embedded 1-handles on F in $S^4$. In an earlier preprint of this paper, it is claimed that every $S^2$-link L in $S^4$ consisting of trivial components in $S^4$ is a ribbon $S^2$-link without restrictions. However, the proof contains an error. In fact, there was a non-ribbon $S^2$-link L consisting of two trivial components such that the $S^2$-link obtained from L by a fusion is a non-ribbon $S^2$-knot [20]. As a revised content, the following theorem is shown, giving a characterization of when a surface-link consisting of ribbon surface-knot components is a ribbon surface-link. Since a trivial surface-knot is a ribbon surface-knot, this theorem also gives a characterization of when a surface-link consisting of trivial components is a ribbon surface-link.

Theorem 3. 

The following statements (1)–(3) on a surface-link F consisting of ribbon surface-knot components in $S^4$ are mutually equivalent:

• (1) 

  F is a ribbon surface-link.

• (2) 

  The surface-knot obtained from F by every fusion is a ribbon surface-knot.

• (3) 

  The surface-knot obtained from F by a fusion is a ribbon surface-knot.

There are a lot of classical non-ribbon links consisting of trivial components producing a ribbon-knot by a fusion such as Hopf link, the split sum of Whitehead link and its mirror image, and the split sum of Borromean rings and its mirror image, etc., [16,21]. Thus, this theorem also contrasts with the behavior of a classical ribbon link.

The Proofs of Lemmas 1 and 2 are given in Section 2 and Section 3, respectively. In Section 4, the Proofs of Theorems 2 and 3 are given.

2. Proof of Lemma 1

A 2-handle on a surface-link F in ${\mathbf{R}}^4$ is a 2-handle $D\times I$ on F with D a core disk embedded in ${\mathbf{R}}^4$ such that $D\times I\cap F=\partial D\times I$, where I denotes a closed interval containing 0 and $D\times 0$ is identified with D. Two 2-handles $D\times I$ and $E\times I$ on F are equivalent if there is an equivalence $f:{\mathbf{R}}^4\to {\mathbf{R}}^4$ from F to itself such that the restriction ${f|}_F:F\to F$ is the identity map and $f(D\times I)=E\times I$.

An orthogonal 2-handle pair (or simply, an O2-handle pair) on F is a pair $(D\times I,D^{\prime }\times I)$ of 2-handles $D\times I$, $D^{\prime }\times I$ on F such that

$D\times I\cap D^{\prime }\times I=\partial D\times I\cap \partial D^{\prime }\times I$

and $\partial D\times I$ and $\partial D^{\prime }\times I$ meet orthogonally on F, that is, the boundary circles $\partial D$ and $\partial D^{\prime }$ meet transversely at one point p so that the intersection $\partial D\times I\cap \partial D^{\prime }\times I$ is homeomorphic to the square $Q=p\times I\times I$. Let $(D\times I,D^{\prime }\times I)$ be an O2-handle pair on a surface-link F. Let $F(D\times I)$ and $F(D^{\prime }\times I)$ be the surface-links obtained from F by the surgeries along $D\times I$ and $D^{\prime }\times I$, respectively. Let $F(D\times I,D^{\prime }\times I)$ be the surface-link which is the union $\delta \cup F_{\delta }^c$ of the plumbed disk

$\delta ={\delta }_{D\times I,D^{\prime }\times I}=D\times \partial I\cup Q\cup D^{\prime }\times \partial I$

and the surface

$F_{\delta }^c=\mathrm{cl}(F\setminus (\partial D\times I\cup \partial D^{\prime }\times I)).$

A once-punctured torus $T^o$ in a 3-ball B is trivial if $T^o$ is smoothly and properly embedded in B which splits B into two solid tori. A bump of a surface-link F is a 3-ball B in ${\mathbf{R}}^4$ with $F\cap B=T^o$ a trivial once-punctured torus in B. Let $F\left(B\right)$ be a surface-link $F_B^c\cup {\delta }_B$ which is the union of the surface $F_B^c=\mathrm{cl}(F\setminus T^o)$ and a disk ${\delta }_B$ in the 2-sphere $\partial B$ with $\partial {\delta }_B=\partial T^o$. A cellular move of a compact (possibly, bounded) surface P in ${\mathbf{R}}^4$ is a compact surface $\tilde{P}$ such that the intersection $P^o=P\cap \tilde{P}$ is a once-punctured compact surface of P and $\tilde{P}$ with $d=\mathrm{cl}(P\setminus P^o)$ and $\tilde{d}=(\tilde{P}\setminus P^o)$ disks in the interiors of P and $\tilde{P}$, respectively such that the union $d\cup \tilde{d}$ is a 2-sphere bounding a 3-ball smoothly embedded in ${\mathbf{R}}^4$ and not meeting the interior of $P^o$. Note that $F\left(B\right)$ is uniquely determined up to cellular moves on the disk ${\delta }_B$ keeping $F_B^c$ fixed. For an O2-handle pair $(D\times I,D^{\prime }\times I)$ on a surface-link F, let $\Delta =D\times I\cup D^{\prime }\times I$ is a 3-ball in ${\mathbf{R}}^4$ called the 2-handle union. Consider the 3-ball $\Delta$ as a Seifert hypersurface of the trivial $S^2$-knot $K=\partial \Delta$ in ${\mathbf{R}}^4$ to construct a 3-ball $B_{\Delta }$ obtained from $\Delta$ by adding an outer boundary collar. This 3-ball $B_{\Delta }$ is a bump of F, which we call the associated bump of the O2-handle pair $(D\times I,D^{\prime }\times I)$. When the union of the 3-ball $\Delta$ and a boundary collar of $F_{\delta }^c$ are deformed into the 3-space ${\mathbf{R}}^3\subset {\mathbf{R}}^4$, this associated bump $B_{\Delta }$ is also considered as a regular neighborhood of $\Delta$ in ${\mathbf{R}}^3$. It is observed that an O2-handle unordered pair $(D\times I,D^{\prime }\times I)$ on a surface-link F is constructed uniquely from any given bump B of F in ${\mathbf{R}}^4$ with $F(D\times I,D^{\prime }\times I)\cong F\left(B\right)$ [1]. Further, for any O2-handle pair $(D\times I,D^{\prime }\times I)$ on any surface-link F and the associated bump B, there are identifications

$F\left(B\right)=F(D\times I,D^{\prime }\times I)=F(D\times I)=F(D^{\prime }\times I)$

by equivalences which are attained by cellular moves on the disk $\delta ={\delta }_{D\times I,D^{\prime }\times I}$ keeping $F_{\delta }^c$ fixed [1]. A once-punctured torus $T^o$ in a 4-ball A is trivial if $T^o$ is smoothly and properly embedded in A, and there is a solid torus V in A with $\partial V=T^o\cup {\delta }_A$ for a disk ${\delta }_A$ in the 3-sphere $\partial A$. A 4D bump of a surface-link F is a 4-ball A in ${\mathbf{R}}^4$ with $F\cap A=T^o$ a trivial once-punctured torus in A. A 4D bump A is obtained from a bump B of a surface-link F by taking a bi-collar $c(B\times [-1,1\left]\right)$ of B in ${\mathbf{R}}^4$ with $c(B\times 0)=B$. The following lemma is proved by using a 4D bump A.

Lemma 3. 

Let $(D\times I,D^{\prime }\times I)$ be any O2-handle pair on any surface-link F in ${\mathbf{R}}^4$ and T a trivial torus-knot in ${\mathbf{R}}^4$ with any given spin loop basis $(e,e^{\prime })$. Then, there is an equivalence $f:{\mathbf{R}}^4\to {\mathbf{R}}^4$ from the surface-link F to a connected sum $F(D\times I,D^{\prime }\times I)\#T$ keeping $F_{\delta }^c$ fixed such that $f(\partial D)=e$ and $f(\partial D^{\prime })=e^{\prime }$.

Proof of Lemma 3. 

Let A be a 4D bump associated with the O2-handle pair $(D\times I,D^{\prime }\times I)$ on F. Let ${\delta }_A$ be a disk in the 3-sphere $\partial A$ such that there is a solid torus V in A whose boundary is the union of the trivial once-punctured torus $P=F\cap A$ and the disk ${\delta }_A$. This solid torus V induces an equivalence $f^{\prime }:({\mathbf{R}}^4,F)\to ({\mathbf{R}}^4,F(D\times I,D^{\prime }\times I)\#T)$ sending P to the connected summand $T^o$ of a connected sum $F(D\times I,D^{\prime }\times I)\#T$ in A. Let $(\tilde{e},{\tilde{e}}^{\prime })$ be the spin loop basis of $T^o$ which is the image of the spin loop pair $(\partial D,\partial D^{\prime })$ on F under $f^{\prime }$. There is an orientation-preserving diffeomorphism $g:{\mathbf{R}}^4\to {\mathbf{R}}^4$ with ${g|}_{\mathrm{cl}({\mathbf{R}}^4\setminus A)}=1$ such that $g(\tilde{e},{\tilde{e}}^{\prime })=(e,e^{\prime })$ by the previous techniques [1,22]. The composition $f=g{f}^{\prime }$ is a desired equivalence. This completes the Proof of Lemma 3. □

A surface-link F only has a unique O2-handle pair in the rigid sense if for any O2-handle pairs $(D\times I,D^{\prime }\times I)$ and $(E\times I,E^{\prime }\times I)$ on F with $(\partial D)\times I=(\partial E)\times I$ and $(\partial D^{\prime })\times I=(\partial E^{\prime })\times I$, there is an equivalence $f:{\mathbf{R}}^4\to {\mathbf{R}}^4$ from F to itself keeping $F_{\delta }^c$ fixed such that $f(D\times I)=E\times I$ and $f(D^{\prime }\times I)=E^{\prime }\times I$. It is proved that every surface-link F has only unique O2-handle pair in the rigid sense [1,2]. A surface-link F has only unique O2-handle pair in the soft sense if for any O2-handle pairs $(D\times I,D^{\prime }\times I)$ and $(E\times I,E^{\prime }\times I)$ on F attached to the same connected component, say $F_1$ of F, there is an equivalence $f:{\mathbf{R}}^4\to {\mathbf{R}}^4$ from F to itself keeping $F^{\left(1\right)}=F\setminus F_1$ fixed such that $f(D\times I)=E\times I$ and $f(D^{\prime }\times I)=E^{\prime }\times I$. A surface-link not admitting any O2-handle pair is understood as a surface-link with only a unique O2-handle pair in both the rigid and soft senses. The following uniqueness of an O2-handle pair in the soft sense is essentially a consequence of the uniqueness of an O2-handle pair in the rigid sense.

Theorem 4 (Uniqueness of an O2-handle pair in the soft sense).

Every surface-link only has a unique O2-handle pair in the soft sense.

Proof of Theorem 4. 

Let $(D\times I,D^{\prime }\times I)$ and $(E\times I,E^{\prime }\times I)$ be any two O2-handle pairs on a surface-link F attached to the same connected component $F_1$ of F. Let $F^{\left(1\right)}=F\setminus F_1$. By Lemma 3, there is an equivalence $f:{\mathbf{R}}^4\to {\mathbf{R}}^4$ from F to the connected sum

$F(D\times I,D^{\prime }\times I)\#T=F^{\left(1\right)}\cup {\tilde{F}}_1\#T$

keeping $F^{\left(1\right)}$ fixed and sending $F_1$ to ${\tilde{F}}_1\#T$, where ${\tilde{F}}_1=F_1(D\times I,D^{\prime }\times I)$ and T is a trivial torus-knot in ${\mathbf{R}}^4$. Similarly, there is an equivalence $f^{\prime }:{\mathbf{R}}^4\to {\mathbf{R}}^4$ from F to the connected sum

$F(E\times I,E^{\prime }\times I)\#T^{\prime }=F^{\left(1\right)}\cup {\tilde{F}}_1^{\prime }\#T^{\prime }$

keeping $F^{\left(1\right)}$ fixed and sending $F_1$ to ${\tilde{F}}_1^{\prime }\#T^{\prime }$, where ${\tilde{F}}_1^{\prime }=F_1(E\times I,E^{\prime }\times I)$ and $T^{\prime }$ is a trivial torus-knot in ${\mathbf{R}}^4$. The diffeomorphism $g=f^{\prime }{f}^{-1}:{\mathbf{R}}^4\to {\mathbf{R}}^4$ is an equivalence from $F^{\left(1\right)}\cup {\tilde{F}}_1\#T$ to $F^{\left(1\right)}\cup {\tilde{F}}_1^{\prime }\#T^{\prime }$ keeping $F^{\left(1\right)}$ fixed. The connected sum ${\tilde{F}}_1\#T$ is obtained from the split union ${\tilde{F}}_1+T$ in ${\mathbf{R}}^4$ by surgery along an embedded 1-handle h connecting a disk $d_1\subset {\tilde{F}}_1$ and a disk $d\subset T$, and the connected sum ${\tilde{F}}_1^{\prime }\#T^{\prime }$ is obtained from the split union ${\tilde{F}}_1^{\prime }+T^{\prime }$ in ${\mathbf{R}}^4$ by surgery along an embedded 1-handle $h^{\prime }$ connecting a disk $d_1^{\prime }\subset {\tilde{F}}_1^{\prime }$ and a disk $d^{\prime }\subset \tilde{T}$. Then there is a 4-ball A in ${\mathbf{R}}^4$ such that $T^o=A\cap (F^{\left(1\right)}\cup {\tilde{F}}_1\#T)$ is a trivial once-punctured torus of T in A with $d_1$ a disk bounded by the trivial knot $\partial T^o$ in the 3-sphere $\partial A$. Similarly, there is a 4-ball $A^{\prime }$ in ${\mathbf{R}}^4$ such that ${\left(T^{\prime }\right)}^o=A^{\prime }\cap (F^{\left(1\right)}\cup {\tilde{F}}_1^{\prime }\#T^{\prime })$ is a trivial once-punctured torus of $T^{\prime }$ in $A^{\prime }$ with $d_1^{\prime }$ a disk bounded by the trivial knot $\partial {\left(T^{\prime }\right)}^o$ in the 3-sphere $\partial A^{\prime }$. It may be assumed that $g(\partial d_1)=\partial d_1^{\prime }$ by sliding the attaching loop $g(\partial d_1)$ in $g({\tilde{F}}_1\#T)$ and/or the attaching loop $\partial d_1^{\prime }$ in ${\tilde{F}}_1^{\prime }\#T^{\prime }$. Then, it is assumed that $g\left(T^o\right)={\left(T^{\prime }\right)}^o$ (For a special case that $g\left(T^0\right)=\mathrm{cl}({\tilde{F}}_1^{\prime }\#T^{\prime }\setminus {\left(T^{\prime }\right)}^o)$, there is a deformation from $g\left(A\right)$ into $A^{\prime }$ to obtain $g\left(T^0\right)={\left(T^{\prime }\right)}^o$). Further, by Lemma 3, it is assumed that $g(f(\partial D),f(\partial D^{\prime }))=(f^{\prime }(\partial E),f^{\prime }(\partial E^{\prime }))$. Then

$(f^{\prime }(\partial D),f^{\prime }(\partial D^{\prime }))=g(f(\partial D),f(\partial D^{\prime }))=(f^{\prime }(\partial E),f^{\prime }(\partial E^{\prime })).$

Since every surface-link only has a unique O2-handle pair in the rigid sense, there is an equivalence $g^{\prime }:{\mathbf{R}}^4\to {\mathbf{R}}^4$ from $F^{\left(1\right)}\cup {\tilde{F}}_1^{\prime }\#T^{\prime }$ to itself keeping $F^{\left(1\right)}$ fixed such that $g^{\prime }(f^{\prime }\left(D\right)\times I,f^{\prime }\left(D^{\prime }\right)\times I)=(f^{\prime }\left(E\right)\times I,f^{\prime }\left(E^{\prime }\right)\times I)$. The composite equivalence $g^{\ast }={\left(f^{\prime }\right)}^{-1}{g}^{\prime }gf:{\mathbf{R}}^4\to {\mathbf{R}}^4$ is an equivalence from F to itself keeping $F^{\left(1\right)}$ fixed and sending $(D\times I,D^{\prime }\times I)$ to $(E\times I,E^{\prime }\times I)$. Thus, every surface-link F only has a unique O2-handle pair in the soft sense. This completes the Proof of Theorem 4. □

The following corollary is obtained from the Proof of Theorem 4.

Corollary 3. 

Let F and $F^{\prime }$ be surface-links with ordered components $F_i(i=1,2,\cdots ,r)$ and $F_i^{\prime }(i=1,2,\cdots ,r)$, respectively. Assume that the stabilizations $\overline{F}=F{\#}_{i}T,{\overline{F}}^{\prime }=F^{\prime }{\#}_{i}T$ of $F,F^{\prime }$ with induced ordered components obtained by the connected sums $F_i\#T,F_i^{\prime }\#T$ of the ith components $F_i,F_i^{\prime }$ and a trivial torus-knot T, respectively, are equivalent by a component-order-preserving equivalence ${\mathbf{R}}^4\to {\mathbf{R}}^4$. Then, F is equivalent to $F^{\prime }$ by a component-order-preserving equivalence ${\mathbf{R}}^4\to {\mathbf{R}}^4$.

Remark 1. 

For the case of ribbon surface-links F and $F^{\prime }$, Corollary 3 has a different proof [9,23].

The Proof of Lemma 1 is performed as follows.

Proof of Lemma 1. 

A surface-link F with r ordered components is kth handle-reducible if F is equivalent to a stabilization $F^{\prime }{\#}_k{n}_{k}T$ of a surface-link $F^{\prime }$ for an integer $n_k>0$, where ${\#}_k{n}_{k}T$ denotes the stabilizer components $n_{k}T$ attaching to the kth component of $F^{\prime }$. Otherwise, the surface-link F is said to be kth handle-irreducible. Note that if a surface-link G is equivalent to a kth handle-irreducible surface-link F by component-order-preserving equivalence, then G is also kth handle-irreducible. Let F and G be ribbon surface-links with components $F_i(i=1,2,\cdots ,r)$ and $G_i(i=1,2,\cdots ,r)$, respectively. Let $F_{\ast }$ and $G_{\ast }$ be handle-irreducible summands of F and G, respectively. Assume that there is an equivalence f from F to G. Then, it is shown that $F_{\ast }$ and $G_{\ast }$ are equivalent, as follows. Changing the indexes if necessary, assume that f sends $F_i$ to $G_i$ for every i. □

Let

$\begin{array}{ccc}\hfill F& =& F_{\ast }{\#}_1{n}_{1}T{\#}_2{n}_{2}T{\#}_3\cdots {\#}_r{n}_{r}T,\hfill \\ \hfill G& =& G_{\ast }{\#}_1{n}_1^{\prime }T{\#}_2{n}_2^{\prime }T{\#}_3\cdots {\#}_r{n}_r^{\prime }T.\hfill \end{array}$

If necessary, by taking the inverse equivalence $f^{-1}$ instead of f, assume that $n_1^{\prime }\ge n_1$. If $n_1^{\prime }>n_1$, then there is a component-order-preserving equivalence $f^{\left(1\right)}$ from the first-handle-irreducible surface-link

$F_{\left(1\right)}=F_{\ast }{\#}_2{n}_{2}T{\#}_3\cdots {\#}_r{n}_{r}T$

to the first-handle-reducible surface-link

$G_{\ast }{\#}_1(n_1^{\prime }-n_1)T{\#}_2{n}_2^{\prime }T{\#}_3\cdots {\#}_r{n}_r^{\prime }T,$

by Corollary 3, which contradicts the first handle-irreducibility. Thus, $n_1^{\prime }=n_1$, and the first handle-irreducible surface-link $F_{\left(1\right)}$ is equivalent to the first-handle-irreducible ribbon surface-link

$G_{\left(1\right)}=G_{\ast }{\#}_2{n}_2^{\prime }T{\#}_3\cdots {\#}_r{n}_r^{\prime }T.$

By continuing this process, it is shown that $F_{\ast }$ is equivalent to $G_{\ast }$. This completes the Proof of Lemma 1.

3. Proof of Lemma 2

A chorded loop system is a pair $(o,\alpha )$ of a trivial link o and an arc system $\alpha$ attaching to o in the 3-space ${\mathbf{R}}^3$, where o and $\alpha$ are called a based loop system and a chord system, respectively. A chorded loop diagram or simply a chord diagram is a diagram $C(o,\alpha )$ in the plane ${\mathbf{R}}^2$ of the spatial graph $o\cup \alpha$. Let $D^{+}$ be a proper disk system in the upper half-space ${\mathbf{R}}_{+}^4$ obtained from a disk system $d^{+}$ in ${\mathbf{R}}^3$ bounded by o by pushing the interior into ${\mathbf{R}}_{+}^4$. Similarly, let $D^{-}$ be a proper disk system in the lower half-space ${\mathbf{R}}_{-}^4$ obtained from a disk system $d^{-}$ in ${\mathbf{R}}^3$ bounded by o by pushing the interior into ${\mathbf{R}}_{-}^4$. Let O be the union of $D^{+}$ and $D^{-}$, which is a trivial $n{S}^2$-link in the 4-space ${\mathbf{R}}^4$, where n is the number of components of o. The union $O\cup \alpha$ is called a chorded sphere system constructed from a chorded loop system $(o,\alpha )$. The chorded sphere system $O\cup \alpha$ up to orientation-preserving diffeomorphisms of ${\mathbf{R}}^4$ is independent of choices of $d^{+}$ and $d^{-}$ and uniquely determined by the chorded loop system $(o,\alpha )$ by the Horibe–Yanagawa lemma [19]. Thus, every ribbon surface-link F is uniquely constructed from a chorded loop system $(o,\alpha )$ via the chorded sphere system $O\cup \alpha$ so that $F=F(o,\alpha )$ is obtained from O by surgery along a 1-handle system $N\left(\alpha \right)$ on O with core arc system $\alpha$, where note that the surface-link F up to equivalences is unaffected by choices of any 1-handle system $N\left(\alpha \right)$ fixing $\alpha$ [3]. The moves on a chorded loop system $(o,\alpha )$ giving the same ribbon surface-link up to equivalences are determined [9]. A semi-unknotted multi-punctured handlebody system or simply a SUPH system for a surface-link F in ${\mathbf{R}}^4$ is a multi-punctured handlebody system V (smoothly embedded) in ${\mathbf{R}}^4$ such that $\partial V=F\cup O$ for a trivial $S^2$-link O in ${\mathbf{R}}^4$. Note that the numbers of connected components in F and V are equal. The following lemma makes a characterization of a ribbon surface-link [4,7].

Lemma 4. 

A surface-link F is a ribbon surface-link if and only if F admits a SUPH system V in ${\mathbf{R}}^4$.

Proof of Lemma 4. 

A SUPH system V for a ribbon surface-link F is constructed from a chorded sphere system $O\cup \alpha$ by taking the union of a thickening $O\times [0,1]$ of O in ${\mathbf{R}}^4$ and the 1-handle system $N\left(\alpha \right)$ attaching only to $O\times 0$. Conversely, given a SUPH system V in ${\mathbf{R}}^4$ with $\partial V=F\cup O$ for a trivial $S^2$-link O, then take a chord system $\alpha$ in V attaching to O so that the frontier of the regular neighborhood of $O\cup \alpha$ in V is parallel to F in V. The chorded sphere system $O\cup \alpha$ shows that F is a ribbon surface-link. This completes the Proof of Lemma 4. □

Let F be a surface-link of components $F_i(i=1,2,\cdots ,r)$ in ${\mathbf{R}}^4$. Let $F\#T$ be the connected sum of F and a trivial torus-knot T in ${\mathbf{R}}^4$ consisting of the components $F_1\#T,F_i(i=2,3,\cdots ,r)$. Assume that $F\#T$ is a ribbon surface-link. By Lemma 4, let V be a SUPH system for $F\#T$ in ${\mathbf{R}}^4$. Let $V_1$ be the component of V for $F_1\#T$ and write $V_1=U{\#}_{\partial }W$, a disk sum for a multi-punctured 3-ball U and a handlebody W. The following lemma is needed to prove Lemma 2.

Lemma 5. 

For a suitable spin loop basis $(\ell ,{\ell }^{\prime })$ for $T^o$, there is a spin simple loop ${\tilde{\ell }}^{\prime }$ in the ribbon-surface-link $F_1\#T$ with intersection number $Int(\ell ,{\tilde{\ell }}^{\prime })\ne 0$ in $F_1\#T$ such that the loop ${\tilde{\ell }}^{\prime }$ bounds a disk $D^{\prime }$ in the handlebody W.

Proof of Lemma 5. 

Consider a disk sum decomposition of the handlebody W into solid tori $S^1\times D_j^2(j=1,2,\cdots ,g)$ pasting along mutually disjoint disks. Let $({\ell }_j,m_j)$ be a longitude-meridian pair of the solid torus $S^1\times D_j^2$ for all j. The loop basis $({\ell }_j,m_j)$ for $S^1\times D_j^2$ is chosen to be a spin loop basis in ${\mathbf{R}}^4$ for all j [24]. By a choice of a spin loop basis $(\ell ,{\ell }^{\prime })$ for $T^o$, the loop ℓ meets a meridian loop $m_j$ with a non-zero intersection number in $\partial W$. The loop $m_j$ is taken to be a loop ${\tilde{\ell }}^{\prime }$ in $F_1\#T$ bounding a disk $D^{\prime }$ in W with intersection number $\mathrm{Int}(\ell ,{\tilde{\ell }}^{\prime })\ne 0$ since $m_j$ bounds a meridian disk $1\times D_j^2$ of the solid torus $S^1\times D_j^2\subset W$. This completes the Proof of Lemma 5. □

The following lemma is obtained by using Lemma 5.

Lemma 6. 

There is a stabilization $\overline{F}$ of the ribbon surface-link $F\#T$ in ${\mathbf{R}}^4$ consisting of the components ${\overline{F}}_1,F_i(i=2,3,\cdots ,r)$ where ${\overline{F}}_1$ is the connected sum of $F_1\#T$ and trivial torus-knots $T_i(i=1,2,\cdots ,m)$ for some $m\ge 0$ such that the surface-link $\overline{F}$ has the following conditions (i) and (ii).

• (i). There is an O2-handle pair $(D\times I,D^{\prime }\times I)$ on $\overline{F}$ attached to ${\overline{F}}_1$ such that the surface-link $\overline{F}(D^{\prime }\times I)$ is a ribbon surface-link with trivial 1-handles $h_i^{\prime }(i=1,2,\cdots ,m)$ attached.

• (ii). There is an O2-handle pair $(E\times I,E^{\prime }\times I)$ on $\overline{F}$ attached to ${\overline{F}}_1$ such that the surface-link $\overline{F}(E^{\prime }\times I)$ is F with trivial 1-handles $h_i^{″}(i=1,2,\cdots ,m)$ attached.

Proof of Lemma 6. 

Let $p_i(i=0,1,\cdots ,m)$ be the intersection points of transversely meeting simple loops ℓ and ${\tilde{\ell }}^{\prime }$ in $F_1\#T$ given by Lemma 5. For every $i>0$, let ${\alpha }_i$ be an arc neighborhood of $p_i$ in ℓ, and $h_i$ a 1-handle on $F\#T$ with a core arc ${\widehat{\alpha }}_i$ obtained by pushing the interior of ${\alpha }_i$ outside the SUPH system V. Let $\overline{F}=F\#T{\#}_{i=1}^m{T}_i$ be a stabilization of $F\#T$ with the component ${\overline{F}}_1=F_1\#T{\#}_{i=1}^m{T}_i$ obtained from $F_1\#T$ by surgery along the disjoint trivial 1-handle system $h_i(i=1,2,\cdots ,m)$. Let ${\alpha }_i^{+}={\alpha }_i\cup (h_i\cap \ell )$ be the arc in ℓ extending ${\alpha }_i$. Let ${\tilde{\alpha }}_i$ be a proper arc in the annulus $\mathrm{cl}(\partial h_i\setminus h_i\cap F\#T)$, which is parallel to the core arc ${\widehat{\alpha }}_i$ of $h_j$ in $h_i$ with $\partial {\tilde{\alpha }}_i=\partial {\alpha }_i^{+}$. Let $\tilde{\ell }$ be a simple spin loop in $\overline{F}$ obtained from ℓ by replacing ${\alpha }_i^{+}$ with ${\tilde{\alpha }}_i$ for every $i>0$, which meets ${\tilde{\ell }}^{\prime }$ transversely in just one point. Let $W^{+}\left(D^{\prime }\right)$ be the handlebody obtained from the handlebody $W^{+}=W{\cup }_{i=1}^m{h}_i$ by removing a thickened disk $D^{\prime }\times I$ of $D^{\prime }$. The manifold $V^{+}\left(D^{\prime }\right)$ obtained from the SUPH system $V^{+}=V{\cup }_{i=1}^m{h}_i$ for the ribbon surface-link $\overline{F}$ by replacing $W^{+}$ with $W^{+}\left(D^{\prime }\right)$ is a SUPH system for a surface-link ${\overline{F}}^{\prime }$ in ${\mathbf{R}}^4$ consisting of $F_i(i=2,3,\cdots ,r)$ and a component ${\overline{F}}_1^{\prime }$ with genus reduced by 1 from ${\overline{F}}_1$. By Lemma 4, ${\overline{F}}^{\prime }$ is a ribbon-surface-link in ${\mathbf{R}}^4$. The SUPH system $V^{+}$ for $\overline{F}$ is a disk sum of $V^{+}\left(D^{\prime }\right)$ and a solid torus $W_1$ with the disk $D^{\prime }$ as a meridian disk and the loop $\tilde{\ell }$ as a longitude. Let $d_W=V^{+}\left(D^{\prime }\right)\cap W_1$ be the pasting disk between $V^{+}\left(D^{\prime }\right)$ and $W_1$, which is regarded as a 1-handle $h_W$ joining $V^{+}\left(D^{\prime }\right)$ and $W_1$. Let $(E\times I,E^{\prime }\times I)$ be an O2-handle pair on $F\#T$ in ${\mathbf{R}}^4$ attached to $T^o$ with $(\partial E,\partial E^{\prime })=(\ell ,{\ell }^{\prime })$ in Lemma 5. Let A be a 4D bump of the associated bump B of $(E\times I,E^{\prime }\times I)$. In the case of (i), since there is no need to worry about the intersection of A with $E\cup E^{\prime }$, the 4D ball A is deformed so that $V^{+}\cap A=W_1\cup h_W$ by observing that $V\setminus V_1$ and U are disjoint from A by construction of A and by taking spine graphs of $W^{+}\left(D^{\prime }\right)$, $W_1$ and $h_W$. Then, the loop $\tilde{\ell }$ bounds a disk D in A not meeting the interior of $W_1$ and $h_W$. This means that there is an O2-handle pair $(D\times I,D^{\prime }\times I)$ on the surface-link $\overline{F}$ such that $\overline{F}(D^{\prime }\times I)$ is a ribbon surface-link with trivial 1-handles $h_i^{\prime }(i=1,2,\cdots ,m)$ attached, showing (i). For the case of (ii), note that the 1-handles $h_i(i=1,2,\cdots ,m)$ on $F\#T$ are deformed isotopically in A into 1-handles $h_i^{″}(i=1,2,\cdots ,m)$ on $F\#T$ disjoint from the disk pair $(E,E^{\prime })$ because the core arcs of the 1-handles $h_i(i=1,2,\cdots ,m)$ are deformed to be disjoint from the disk pair $(E,E^{\prime })$ in A. The surface-link $\overline{F}(E^{\prime }\times I)$ that is equivalent to $\overline{F}(E\times I,E^{\prime }\times I)$ is the surface-link F with the trivial 1-handles $h_i^{″}(i=1,2,\cdots ,m)$ attached, showing (ii). Thus, the Proof of Lemma 6 is completed. □

The following lemma is a combination of Lemma 6 and the uniqueness of an O2-handle pair in the soft sense (Theorem 4).

Lemma 7. 

If a connected sum $F\#T$ of a surface-link F and a trivial torus-knot T in ${\mathbf{R}}^4$ is a ribbon surface-link, then F is a ribbon surface-link.

Proof of Lemma 7. 

Let $F\#T=F_1\#T\cup F_2\cup \cdots \cup F_r$ be a ribbon surface-link for a trivial torus-knot T. By Lemma 6 (i), the surface-link $F^{″}=\overline{F}(D\times I,D^{\prime }\times I)$ equivalent to $\overline{F}(D\times I)$ is a ribbon surface-link and further the surface-link $F^{\ast }$ obtained from $F^{″}$ by the surgery on O2-handle pairs of all the trivial 1-handles $h_i^{\prime }(i=1,2,\cdots ,m)$ is a ribbon surface-link. By Lemma 6 (ii), the surface-link $\overline{F}(E\times I,E^{\prime }\times I)$ equivalent to $\overline{F}(E\times I)$ is the surface-link F with the 1-handles $h_i^{″}(i=1,2,\cdots ,m)$ trivially attached. By an inductive use of Theorem 4 (uniqueness of an O2-handle pair in the soft sense), the surface-link F is equivalent to the ribbon surface-link $F^{\ast }$. Thus, F is a ribbon surface-link, and the Proof of Lemma 7 is completed. □

Lemma 2 is a direct consequence of Lemma 7 as follows.

Proof of Lemma 2. 

If a stabilization $\overline{F}$ of a surface-link F is a ribbon surface-link, then F is a ribbon surface-link by an inductive use of Lemma 7. This completes the Proof of Lemma 2. □

4. Proofs of Theorems 2 and 3

The Proof of Theorem 2 is performed as follows.

Proof of Theorem 2. 

The ‘if’ part of Theorem 2 is seen from the definition of a ribbon surface-link. The Proof of the ‘only if’ part of Theorem 2 uses the fact that every surface-link is made a trivial surface-knot by surgery along a finite number of possibly non-trivial 1-handles [3]. The connected summand $F_2$ of $F_1\#F_2$ is made a trivial surface-knot by surgery along 1-handles within the 4-ball defining the connected summand $F_2$ so that the surface-link F changes into a new ribbon surface-link and hence $F_1$ is a stable-ribbon surface-link. By Corollary 1.2, $F_1$ is a ribbon surface-link. By interchanging the roles of $F_1$ and $F_2$, the connected summand $F_2$ is also a ribbon surface-link. This completes the Proof of Theorem 2 □

The following lemma is not used in the present version of Theorem 3, but this lemma remains here since it is an interesting property.

Lemma 8. 

Let K be an $S^2$-knot in $S^4$ obtained from a trivial surface-knot F of genus n in $S^4$ by surgery along disjoint 2-handles $D_i\times I(i=1,2,\cdots ,n)$. Then, there is a disjoint O2-handle pair system $(D_i\times I,D_i^{\prime }\times I)(i=1,2,\cdots ,n)$ on F in $S^4$ if and only if K is a trivial $S^2$-knot in $S^4$.

Proof of Lemma 8. 

If there is a disjoint O2-handle pair system $(D_i\times I,D_i^{\prime }\times I)(i=1,2,\cdots ,n)$ on F in $S^4$, then K is a trivial $S^2$-knot by Corollary 1.3. Note that the 2-handle system $D_i\times I(i=1,2,\cdots ,n)$ on F is a 1-handle system on K. If K is a trivial $S^2$-knot, then the 1-handle system $D_i\times I(i=1,2,\cdots ,n)$ on the trivial $S^2$-knot K is always a trivial 1-handle system on K [3]. Hence, there is a disjoint O2-handle pair system $(D_i\times I,D_i^{\prime }\times I)(i=1,2,\cdots ,n)$ on F in $S^4$. This completes the Proof of Lemma 8. □

The Proof of Theorem 3 is performed as follows.

Proof of Theorem 3. 

The assertions (1) → (2) and (2) → (3) are obvious by definitions. The assertion (3) → (1) is shown as follows: When F is a ribbon surface-knot, there is nothing to prove. By inductive assumption, the result for F of $r-1$ components is assumed. Let F be a surface-link of r ribbon surface-knot components $F_i,(i=1,2,\cdots ,r)$. Since a fusion of F makes a ribbon surface-knot by assumption, let K be a ribbon surface-knot obtained form F by a fusion along a disjoint 1-handle system h on F with only one 1-handle $h_r$ connecting to the component $F_r$. Let $K^{\prime }$ be a surface-knot obtained from the sublink $F\setminus F_r$ by the fusion along the 1-handle system $h\setminus h_r$. Since $F_r$ is a ribbon surface-knot, let $(O_r,{\alpha }_r)$ be a chorded sphere system for the ribbon surface-knot $F_r$. Since $\left[K^{\prime }\right]=0$ in $H_2(S^4\setminus O_r;Z)=0$, there is a compact connected oriented 3-manifold $V^{\prime }$ in $S^4$ with $\partial V^{\prime }=K^{\prime }$ and $V^{\prime }\cap O_r=\varnothing$. Since the arc system ${\alpha }_r$ transversely meets with the interior of $V^{\prime }$, a multi-punctured manifold ${\left(V^{\prime }\right)}^{\left(0\right)}$ of $V^{\prime }$ does not meet the chorded sphere system $O_r\cup {\alpha }_r$ so that $W_r\cap {\left(V^{\prime }\right)}^{\left(0\right)}=\varnothing$ for a SUPH system $W_r$ for $F_r$ with $\partial W_r=F_r\cup O_r$, constructed from $O_r\cup {\alpha }_r$. Let g be a disjoint 1-handle system on $F^{\prime }$ embedded in ${\left(V^{\prime }\right)}^{\left(0\right)}$ such that the closed complement $H^{\left(0\right)}=\mathrm{cl}({(V^{\prime })}^{\left(0\right)}\setminus g)$ is a multi-punctured handlebody of a genus, say n. Let $\partial H^{\left(0\right)}={\left(K^{\prime }\right)}^{+}\cup O^H$ where ${\left(K^{\prime }\right)}^{+}$ is the surface-knot obtained from $K^{\prime }$ by surgery along the 1-handle system g. The union $W=H^{\left(0\right)}\cup W_r$ is a SUPH system for the surface-link ${\left(K^{\prime }\right)}^{+}\cup F_r$ in $S^4$, so that ${\left(K^{\prime }\right)}^{+}\cup F_r$ is a ribbon surface-link in $S^4$ by Lemma 4. By replacing W with a multi-puncture manifold of W, the union $W^{+}=W\cup h_r$ is a SUPH system for the ribbon surface-knot $K^{+}$ in $S^4$ obtained from the surface-link ${\left(K^{\prime }\right)}^{+}\cup F_r$ by fusion along $h_r$. Note that the ribbon surface-knot $K^{+}$ is also obtained from the ribbon surface-knot K by surgery along g. Let $W_K$ be a SUPH system for the ribbon surface-knot K in $S^4$. By replacing $W_K$ with a multi-punctured manifold of $W_K$, the union $W_K^{+}=W_K\cup g$ is a SUPH system for the ribbon surface-knot $K^{+}$ in $S^4$. Equivalent ribbon surface-links are faithfully equivalent and they are moved into each other by the moves M0, M1, M2, [23]. This means that there is an orientation-preserving diffeomorphism f of $S^4$ sending a multi-punctured manifold of the SUPH system ${\left(W^{+}\right)}^{\left(0\right)}$ of $W^{+}$ to a multi-punctured manifold ${\left(W_K^{+}\right)}^{\left(0\right)}$ of the SUPH system $W_K^{+}$ and keeping $K^{+}$ set-wise fixed. Let $D\left(h_r\right)$ and $D\left(g\right)$ be a proper disk and a proper disk system in $h_r$ and g parallel to the attaching disks with one disk for every 1-handle, respectively. The proper disk $f\left(D\left(h_r\right)\right)$ and the proper disk system $D\left(g\right)$ may meet transversely with simple loops in the interior of the punctured handlebody ${\left(W_K^{+}\right)}^{\left(0\right)}$. Let $f{\left(D\left(h_r\right)\right)}^{\prime }$ be a proper disk disjoint from $D\left(g\right)$ with $\partial f{\left(D\left(h_r\right)\right)}^{\prime }=\partial f\left(D\left(h_r\right)\right)$ in ${\left(W_K^{+}\right)}^{\left(0\right)}$ obtained from $f\left(D\left(h_r\right)\right)$ by using a cutting technique along an innermost loop of $f\left(D\left(h_r\right)\right)\cap D\left(g\right)$ in $D\left(g\right)$ inductively. By splitting ${\left(W^{+}\right)}^{\left(0\right)}$ along the disk system $f{\left(D\left(h_r\right)\right)}^{\prime }\cup D\left(g\right)$, a SUPH system $W\left(K^{\prime }\right)\cup W\left(F_r\right)$ for a ribbon surface-link with $K^{\prime }\cup F_r$ as a fusion with a trivial $S^2$-link is obtained, meaning that the surface-link $K^{\prime }\cup F_r$ is a ribbon surface-link. In particular, $K^{\prime }$ is a ribbon surface-knot in $S^4$. By inductive assumption, $F\setminus F_r$ is a ribbon surface-link. Let $W(F\setminus F_r)$ be a SUPH system for the ribbon surface-link $F\setminus F_r$ in $S^4$. A multi-punctured manifold of the SUPH system $W(F\setminus F_r)\cup (h\setminus h_r)$ for the ribbon surface-knot $K^{\prime }$ is sent to a multi-punctured manifold of the SUPH system $W\left(K^{\prime }\right)$ for $K^{\prime }$ by an orientation-preserving diffeomorphism $f^{\prime }$ of $S^4$. After replacing $W(F\setminus F_r)\cup (h\setminus h_r)$ and $W\left(K^{\prime }\right)\cup W\left(F_r\right)$ with multi-punctured manifolds, respectively, the preimage ${\left(f^{\prime }\right)}^{-1}(W\left(K^{\prime }\right)\cup W\left(F_r\right))$ is the union of $W(F\setminus F_r)\cup (h\setminus h_r)$ and ${\left(f^{\prime }\right)}^{-1}\left(W\left(F_r\right)\right)$, showing that $W(F\setminus F_r)\cup {\left(f^{\prime }\right)}^{-1}\left(W\left(F_r\right)\right)$ is a SUPH system for the surface-link F. Thus, F is a ribbon surface-link. By induction on r, (3) → (1) is obtained and the Proof of Theorem 3 is completed. □

5. Conclusions

The ribbonness of a stable-ribbon surface-link shown in Theorem 1 is applied to determine the Ribbonness of some classes of surface-links. Theorem 3 appears unrelated to Theorem 1, but the result that equivalent ribbon surface-links are faithfully equivalent used in the Proof of Theorem 3 comes from a prior result of Theorem 1 [23]. The ribbonness of a surface-link relates not only to smooth unknotting conjecture for a surface-link leading to classical and 4D smooth Poincaré conjectures but also to J. H. C. Whitehead asphericity conjecture for aspherical 2-complex [25,26,27,28] as well as Kervaire conjecture on group weight [29,30]. In another direction, it may be an interesting problem to investigate a canonical relationship between a chorded loop diagram of a ribbon surface-knot and a knot diagram. In conclusion, ribbon surface-knot theory will be a tool for studies of low-dimensional topology.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Acknowledgments

The author would like to thank O. Chterental (a student of D. Bar-Natan) for making him aware of the reference [20] and comments.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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