Abstract

It is shown that a surface-link of ribbon surface-knot components is a ribbon surfacelink if and only if it is a surface-link producing a ribbon surface-link by surgery along some self-trivial 1-handle system. This corrects an earlier statement. This result makes a corrected proof for the claim that every surface-link of trivial surface-knot components with at most one aspheric component is a ribbon surface-link. For non-ribbon surfacelinks of trivial components with at least two aspheric components constructed in a previous note, the new property that non-ribbonability continues through surgery along any self-trivial 1-handle system is added.

Keywords: Ribbon surface-link, Surgery, Self 1-handle, Self-trivial 1-handle.

Introduction

Let \({\mathbf F}\) be a (possibly disconnected) closed oriented surface. An \({\mathbf F}\)-link in the 4-sphere \(S^4\) is the image of a smooth embedding \({\mathbf F}\to S^4\). When \({\mathbf F}\) is connected, it is also called an \({\mathbf F}\)-knot. An \({\mathbf F}\)-link or \({\mathbf F}\)-knot for an \({\mathbf F}\) is called a surface-link or surface-knot in \(S^4\), respectively. If \({\mathbf F}\) consists of some copies of the 2-sphere \(S^2\), then it is also called an \(S^2\)-link and an \(S^2\)-knot for \({\mathbf F}=S^2\). A trivial surface-link is a surface-link \(F\) in \(S^4\) which bounds disjoint handlebodies smoothly embedded in \(S^4\). A 1-handle \(h\) on a surface-link \(F\) in \(S^4\) is a 1-handle on \(F\) embedded smoothly in \(S^4\), which is a trivial 1-handle on \(F\) if the core arc of \(h\) is an interior push of a simple arc in \(F\) into \(S^4\), a self 1-handle on \(F\) if the number of connected components of the union \(F\cup h\) is equal to the number of connected components of \(F\), and a self-trivial 1-handle on \(F\) if \(h\) is a self 1-handle on \(F\) such that \(h\) is a trivial 1-handle on the attaching component of \(F\) with the other components of \(F\) ignored. A 1-handle system on a surface-link \(F\) in \(S^4\) consists of disjoint 1-handles on \(F\) unless otherwise specified, where a trivial 1-handle system consists of trivial 1-handles on \(F\) and a self-trivial 1-handle system consists of self-trivial 1-handles on \(F\). A 1-handle system \(h\) of 1-handles \(h_j\,(j=1,2,\dots, s (\geq 1))\) on a surface-link \(F\) of \(r(\geq 2)\) components is a fusion 1-handle system if the number of connected components of the union \(F\cup h\) has just \(r-s\) connected components, where the inequality \(s\leq r-1\) must hold. The surface-link obtained from a surface-link \(F\) in \(S^4\) by surgery along a 1-handle system \(h\) on \(F\) is denoted by \(F(h)\). The surface-link \(F(h)\) is called a fusion of \(F\). A ribbon surface-link is the surface-link \(F=O(h)\) in \(S^4\) obtained from a trivial \(S^2\)-link \(O\) by surgery along a 1-handle system \(h\) on \(O\), [5, 12], . A semi-unknotted multi-punctured handlebody system or simply a SUPH system for a surface-link \(F\) in \(S^4\) is a compact oriented 3-manifold \(W\) smoothly embedded in \(S^4\) such that \(W\) is a handlebody system with a finite number of open 3-balls removed and the boundary \(\partial W\) of \(W\) is given by \(\partial W = F \cup O\) for a trivial \(S^2\)-link \(O\) in \(S^4\). A typical SUPH system \(W\) is constructed from a ribbon surface-link \(F\) defined from a trivial \(S^2\)-link \(O\) and a 1-handle system \(h\) on \(F\) as the union \(O\times[0,1]\cup h\) for a normal collar \(O\times [0,1]\) of \(O\) in \(S^4\) with \(O\times \{0\}=O\) not meeting \(h\) except for the attaching part to \(O\) and the 1-handle system \(h\) attaching to \(O\times \{0\}\), where \(\partial W=F\cup O\times\{1\}\)→ For a SUPH system \(W\) with \(\partial W = F \cup O\), there is a proper arc system \(\alpha\) in \(W\) spanning \(O\) such that a regular neighborhood \(N(O\cup\alpha)\) of the union \(O\cup\alpha\) in \(W\) is diffeomorphic to the closed complement \(\mbox{cl}(W\setminus c(F\times[0,1]))\) of a boundary collar \(c(F\times[0,1])\) of \(F\) in \(W\). This pair \((O,\alpha)\) is called a sphere-chord system of the SUPH system \(W\). By replacing \(\alpha\) with a 1-handle system \(h\) attaching to \(O\) with core arc system \(\alpha\), the surface-link \(F\) is a ribbon surface-link defined by \(O\) and \(h\). In other words, to give a SUPH system \(W\) with \(\partial W = F \cup O\) is the same as to say that the surface-link \(F\) is a ribbon surface-link with sphere system \(O\). A multi-fusion SUPH system of a SUPH system \(W\) with \(\partial W = F \cup O\) in \(S^4\) is a SUPH system for \(F\) in \(S^4\) obtained from \(W\) by deleting an open regular neighborhood of a disjoint simple proper arc system in \(W\) spanning \(O\). A multi-fission SUPH system of a SUPH system \(W\) with \(\partial W = F \cup O\) in \(S^4\) is a SUPH system for \(F\) in \(S^4\) obtained from \(W\) by adding a disjoint 2-handle system on \(O\) disjoint from \(W\) except for the attaching part in \(O\). As it is explained at the end of this section, an earlier claimed characterization of when a surface-link \(F\) of ribbon surface-knot components is a ribbon surface-link is not true in general, [10, Theorem 1.4]. The following theorem gives a new characterization.

Theorem 1.1. Let \(F\) be a surface-link in \(S^4\) of ribbon surface-knot components \(F_i\, (i=1,2,\dots, r)\). Then the following statements on (1)-(3) on \(F\) are mutually equivalent.

(1) \(F\) is a ribbon surface-link.

(2) The surface-link obtained from \(F(h)\) by surgery along every 1-handle system \(h\) is a ribbon surface-link.

(3) The surface-link \(F(h)\) obtained from \(F\) by surgery along some self-trivial 1-handle system \(h\) on \(F\) is a ribbon surface-link.

There are many surface-links \(F\) with non-ribbon surface-knot components such that the surgery surface-link \(F(h)\) for a self 1-handle system \(h\) on \(F\) is a ribbon surface-link. The following lemma is implicitely used in the proofs of [9, Theorem 1] and [10, Theorem 1.4] although the full proof is given in this paper for convenience.

Lemma 1.2. For every surface-link \(F\) in \(S^4\) with at most one aspheric component, there is a self 1-handle system \(h\) on \(F\) such that the surface-link \(F(h)\) obtained from \(F\) by surgery along \(h\) is a ribbon surface-link in \(S^4\).

On the other hand, there exist non-ribbon surface-links \(F\) with at least two trivial aspheric components such that the surface-link \(F(h)\) obtained from \(F\) by surgery along any self 1-handle system \(h\) on \(F\) is a non-ribbon surface-link (see Theorem 1.4 later). The following corollary to Theorem 1.1 and Lemma 1.2 can be viewed as providing a corrected proof of [9, Theorem 1]¹

Corollary 1.3. Every surface-link \(F\) of trivial surface-knot components with at most one aspheric component is a ribbon surface-link.

Proof of Corollary 1.3. Every self 1-handle on every surface-link \(F\) of trivial surface-knot components is a self-trivial 1-handle on \(F\), [3]. Thus, if \(F\) has at most one aspheric component, then there is a self 1-handle system \(h\) on \(F\) such that the surface-link \(F(h)\) is a ribbon surface-link by Lemma 1.2. Hence there is a self-trivial 1-handle system \(h\) on \(F\) such that the surgery surface-link \(F(h)\) is a ribbon surface-link. By Theorem 1.1, \(F\) is a ribbon surface-link. This completes the proof of Corollary 1.3.

The following result slightly strengthens an earlier result, [9, Theorem 2].

Theorem 1.4. Let \({\mathbf F}\) be any closed oriented disconnected surface with at least two aspheric components. Then there is a pair \((K,K')\) of \({\mathbf F}\)-links \(K, K'\) in \(S^4\) both of trivial components with the same fundamental groups up to meridian-preserving isomorphisms such that \(K\) is a ribbon surface-link and \(K'\) is a non-ribbon surface-link. Further, there is a canonical correspondence between the self-trivial 1-handle systems on \(K\) and the self-trivial 1-handle systems on \(K'\) so that every self-trivial 1-handle surgery transforms the pair \((K, K')\) into a pair \((L,L')\) of a ribbon \({\mathbf F}^+\)-link \(L\) and a non-ribbon \({\mathbf F}^+\)-link \(L'\) with the same fundamental group up to meridian-preserving isomorphisms for the closed oriented surface \({\mathbf F}^+\) induced from \({\mathbf F}\).

The continuation of non-ribbonability from \(K'\) to \(L'\) in Theorem 1.4 explains why non-ribbonability continues through surgery along every self 1-handle system. A non-ribbon surface-link \(K'\) of two components in Theorem 1.4 has the free abelian fundamental group of rank \(2\), as it is observed after the proof of Theorem 1.4. Then by van Kampen theorem, the surface-knot \(K'(h)\) obtained from \(K'\) by surgery along any fusion 1-handle \(h\) on \(K'\) has the infinite cyclic fundamental group, so that \(K'(h)\) is a trivial surface-knot in \(S^4\) by smooth unknotting result of a surface-knot, [7,8]. This example shows that there is a non-ribbon surface-link \(F\) of ribbon surface-knot components such that the surface-knot obtained from \(F\) by any fusion is a ribbon surface-knot, giving counterexamples to [10, Theorem 1.4]. For a positive result for a boundary surface-link, see [11].

Proofs of Theorem 1.1, Lemma 1.2 and Theorem 1.4

An O2-handle pair on a surface-link \(F\) in \(S^4\) is a pair \((D\times I, D' \times I)\) of 2-handles \(D\times I\), \(D' \times I\) on \(F\) in \(S^4\) which intersect orthogonally only with the attaching parts \((\partial D)\times I\), \((\partial D')\times I\) to \(F\), so that the intersection \(Q=(\partial D)\times I\cap (\partial D')\times I\) is a square, [7].

The proof of Theorem 1.1 is done as follows.

Proof of Theorem 1.1. The assertions (1) \(\to\) (2) and (2) \(\to\) (3) are obvious by definitions. The assertion (3) \(\to\) (1) is shown as follows. For a surface-link \(F\) in \(S^4\) of ribbon surface-knot components \(F_i\, (i=1,2,\dots, r)\), assume that \(F(h)\) is a ribbon surface-link for a system \(h\) of self-trivial 1-handles \(h_j\, (j=1,2,\dots,s)\) on \(F\). Assume that the 1-handle \(h_1\) attaches to \(F_1\). Let \(h'=h\setminus\{h_1\}\). Then \(h'\) is a self-trivial 1-handle system on \(F\) and \(h_1\) is a self-trivial 1-handle on the surface-link \(F(h')\). If it is shown that the ribbonness of \(F(h_1)\) implies the ribbonness of \(F\), then the ribbonness of \(F(h)\) implies the ribbonness of \(F(h')\) by replacing \(F\) with \(F(h')\). By inductive argument, the assertion (3) \(\to\) (1) will be obtained. Thus, it sufficies to show that if \(F(h_1)\) is a ribbon surface-link, then the surface-link \(F\) is a ribbon surface-link. Assume that \(F(h_1)\) is a ribbon surface-link. Since \(h_1\) is a trivial 1-handle on \(F_1\), the punctured torus in the ribbon surface-knot \(F_1(h_1)\) arising from \(h_1\) admits an O2-handle pair \((D\times I,D'\times I)\) where the core disk \(D\) of \(D\times I\) is a transverse disk of the 1-handle \(h_1\) and the interior of the core disk \(D'\) of \(D'\times I\) may transversely meet the ribbon surface-link \(F'=F\setminus F_1\) with finite points. Since \(F_1\) is a ribbon surface-knot in \(S^4\), let \(W_1\) be a SUPH system for \(F_1\) with \(\partial W_1=F_1\cup O_1\) for a trivial \(S^2\)-link \(O_1\) in \(S^4\). Since \(h_1\) is a trivial 1-handle on \(F_1\), the union \(W_1^+=W_1\cup h_1\) with the trivial 1-handle \(h_1\) attached to \(F_1\) is considered as a SUPH system for the surface-knot \(F_1(h_1)\) with \(\partial W_1^+=F_1(h_1)\cup O_1\). On the other hand, since \(F(h_1)=F_1(h_1)\cup F'\) is a ribbon surface-link in \(S^4\), let \(W_1^h\cup W'\) be a SUPH system for the ribbon surface-link \(F(h_1)\) with \(W_1^h\) a SUPH system for \(F_1(h_1)\) and \(W'\) a SUPH system for \(F'\). Let \(\partial W_1^h=F(h_1)\cup O_1^h\) and \(\partial W'=F'\cup O'\) where \(O_1^h\cup O'\) is a trivial \(S^2\)-link in \(S^4\). It is known that two ribbon structures of equivalent surface-links are moved into each other by a finite number of the moves \(M_0, M_1, M_2\), [6]. This means that there is an orientation-preserving diffeomorphism \(f\) of \(S^4\) sending a multi-fusion SUPH system \(W_1^{**}\) of a multi-fission SUPH system \(W_1^*\) of the SUPH system \(W_1^+\) to the SUPH system \(W_1^h\) (see Appendix of [11]). Note that the multi-fission SUPH system \(W_1^*\) is the union \(W_1^+\cup h^2\) for a disjoint 2-handle system \(h^2\) on \(O_1\) not meeting \(W_1^+\) except for the attaching part of \(h^2\) to \(O_1\), so that \(\partial W_1^*=F_1(h_1)\cup O_1^*\) and \(\partial W_1^{**}=F_1(h_1)\cup O_1^{**}\) for trivial \(S^2\)-links \(O_1^*\) and \(O_1^{**}\) in \(S^4\). After moves of \(O_1^*\) and the arc system used for the multi-fusion SUPH system in \(W_1^*\) keeping \(F_1(h_1)\) fixed, the 2-handle \(h_1\) on \(F_1(h_1)\) is assumed to be in \(W_1^{**}\) with \(W^{***}=\mbox{cl}(W_1^{**}\setminus h_1)\) a SUPH system for \(F_1\). In other words, the SUPH system \(W_1^{**}\) for \(F_1(h_1)\) is given by the union \(W_1^{**}=W^{***} \cup h_1\) for the SUPH system \(W^{***}\) for \(F_1\) and the trivial 1-handle \(h_1\) on \(F_1\). The image \((fD\times I,fD'\times I)\) of the O2-handle pair \((D\times I,D'\times I)\) on \(F_1(h_1)\) under \(f\) is an O2-handle pair on \(F_1(h_1)\) whose core disk pair is denoted by \((fD,fD')\). By replacing the 1-handle \(h_1\) with a thinner 1-handle and by a slight move of \(f\), the image 1-handle \(fh_1\) of \(h_1\) on the image \(fF_1\) of \(F_1\) under \(f\) is disjoint from \(h_1\). Thus, the transverse disk \(D\) of \(h_1\) is made disjoint from the transverse disk \(fD\) of \(fh_1\), and any intersection point between the disks \(D'\) and \(fD'\) is an intersection point between \(F_1\) and \(fF_1\) or an isolated intersection point betwen the disk interiors. By construction, the surface-link \(F'\) does not meet the 2-handles \(D\times I\) and \(fD\times I\) and transversely meet the disks \(D'\) and \(fD'\) with finite interior points, which are assumed by general position to be different from the intersection points between \(D'\) and \(fD'\). Consider that the ribbon surface-link \(F'\) is constructed from a sphere-chord system \((O',\alpha')\) of the SUPH system \(W'\) and the intersection points between \(F'\) and \(D'\cup fD'\) belong to the trivial \(S^2\)-link \(O'\). Then there is a surface-link \(F(h_1)'=F_1(h_1)\cup F''\) with the pairs \((D\times I,D'\times I)\) and \((fD\times I,fD'\times I)\) as O2-handle pairs which is constructed from \(F(h_1)=F_1(h_1)\cup F'\) such that the surgery surface-links \(F(h_1)'(D\times I)\) and \(F(h_1)'(fD\times I)\) are respectively equivalent to the original surface-link \(F\) and the ribbon surface-link \(F(h_1)(fD\times I)\) with SUPH system \(fW^{***}\cup W'\) for the image \(fW^{***}\) of \(W^{***}\) under \(f\) by Finger Move Canceling, [8]. The surgery surface-link \(F(h_1)'(D\times I,D'\times I)\) is equivalent to the surgery surface-link \(F(h_1)'(fD\times I,fD'\times I)\), [10]. Further, the surgery surface-links \(F(h_1)'(D\times I,D'\times I)\) and \(F(h_1)'(fD\times I,fD'\times I)\) are respectively equivalent to the surgery surface-links \(F(h_1)'(D\times I)\) and \(F(h_1)'(fD\times I)\), [7]. Thus, the surface-link \(F\) is equivalent to the ribbon surface-link, \(F(h_1)(fD\times I)\). This completes the proof of Theorem 1.1.

The proof of Lemma 1.2 is done as follows.

Proof of Lemma 1.2. Let \(F\) be a surface-link in \(S^4\) of a possibly non-sphere surface-knot component \(K\) and the remaining \(S^2\)-link \(L=F\setminus K\). Since the second homology class \([K]=0\) in \(H_2(S^4\setminus L;Z)=0\), there is a compact connected oriented 3-manifold \(V_K\) smoothly embedded in \(S^4\) with \(\partial V_K=K\) and \(V_K\cap L=\emptyset\). Let \(h_K\) be a 1-handle system on \(K\) in \(V_K\) such that the closed complement \(H_K=\mbox{cl}(V_K\setminus h_K)\) is a handlebody given by a decomposition into a 3-ball \(B_K\) and an attaching 1-handle system \(H^1_K\). Let \(S\) be any \(S^2\)-knot component in \(L\), which bounds a compact connected oriented 3-manifold \(V_S\) smoothly embedded in \(S^4\) such that \(V_S\cap (L\setminus S)=\emptyset\). The 3-ball \(B_K\) and the 1-handle system \(H^1_K\) are deformed in \(S^4\) by shrinking \(B_K\) into a smaller 3-ball and the 1-handle system \(H^1_K\) into a thinner 1-handle system so that \(V_S\cap B_K=\emptyset\) and the 1-handle system \(H^1_K\) transversely meets \(V_S\) with transversal disks in the interior of \(V_S\). Then there is a 1-handle system \(h_S\) on \(S\) in \(V_S\) such that the closed complement \(H_S=\mbox{cl}(V_S\setminus h_S)\) is a handlebody given by a decomposition into a 3-ball \(B_S\) and an attaching 1-handle system \(H^1_S\) so that the transversal disks of \(H^1_S\) in the interior of \(V_S\) are in the interior of \(B_S\). Then the surface-link \(K(h_K)\cup S(h_S)\) is a ribbon surface-link given by the trivial \(S^2\)-link \(\partial B_K\cup\partial B_S\) and the 1-handle system \(H^1_K\cup H^1_S\). Because \(h_S\) is a 1-handle system on the surface-link \(K(h_K)\cup S\) and the core arc system of \(h_S\) transversely meets the interior of \(h_S\) with finite points by general position, the 1-handle systems \(h_K\) and \(h_S\) are made disjoint by isotopic deformations of \(h_S\) keeping \(K(h_K)\cup S\) fixed which are changing \(h_S\) into a thinner 1-handle system and then sliding \(h_S\) along \(h_K\). Next, let \(T\) be any \(S^2\)-knot component in \(L\setminus S\), which bounds a compact connected oriented 3-manifold \(V_T\) smoothly embedded in \(S^4\) such that \(V_T\cap (L\setminus (S\cup T))=\emptyset\). The 3-balls \(B_K\), \(B_S\) and the 1-handle systems \(H^1_K\) and \(H^1_S\) are deformed in \(S^4\) so that \(V_T\cap (B_K\cup B_S)=\emptyset\) and the 1-handle systems \(H^1_K\) and \(H^1_S\) transversely meet \(V_T\) with transversal disks in the interior of \(V_T\). Then there is a 1-handle system \(h_T\) on \(T\) in \(V_T\) such that the closed complement \(H(T)=\mbox{cl}(V_T\setminus h_T)\) is a handlebody given by a decomposition into a 3-ball \(B(T)\) and an attaching 1-handle system \(H^1_T\) such that the transversal disks of \(H^1_K\) and \(H^1_S\) in the interior of \(V_T\) are in the interior of \(B_T\). Then the surface-link \(K(h_K)\cup S(h_S)\cup T(h_S)\) is a ribbon surface-link given by the trivial \(S^2\)-link \(\partial B_K\cup\partial B_S\cup\partial B_T\) and the 1-handle system \(H^1_K\cup H^1_S\cup H^1_T\). Because \(h_K\) and \(h_S\) are disjoint and \(h_T\) is a 1-handle system on the surface-link \(K(h_K)\cup S(h_S)\cup T\) and the core arc system of \(h_T\) transversely meets the interior of \(h_K\cup h_S\) with finite points by general position, the 1-handle systems \(h_K\), \(h_S\) and \(h_T\) are made disjoint by isotopic deformations of \(h_T\) keeping \(K(h_K)\cup S(h_S)\cup T\) fixed. By continuing this process, it is shown that there is a self 1-handle system \(\bar h\) on \(F\) such that the surface-link \(F(\bar h)\) obtained from \(F\) by surgery along \(\bar h\) is a ribbon surface link. This completes the proof of Lemma 1.2.

The proof of Lemma 1.2 is also modified as follows. Regard \(V_K\) as a spine 2-dimensional complex \(\kappa(V_K)\) of \(V_K\), consisting of the vertex \(\kappa(B_K)\), the 1-cell system \(\kappa(H^1_K)\) and the 2-cell system \(\kappa(h_K)\). By general position in \(S^4\), the vertex \(\kappa(B_K)\) is disjoint from \(V_S\), the 1-cell system \(\kappa(H^1_K)\) transversely meets \(V_S\) with finite points and the 2-cell system \(\kappa(h_K)\) meets \(V_S\) with a graph, so that the intersection \(a_K=\kappa(B_K)\cap V_S\) is a graph. Choose a self 1-handle system \(h_S\) in \(V_S\) disjoint from the graph \(a_K\) to construct \(H^1_S\). Regard \(V_S\) as a spine 2-dimensional complex \(\kappa(V_S)\) consisting of the vertex \(\kappa(B_S)\), the 1-cell system \(\kappa(H^1_S)\) and the 2-cell system \(\kappa(h_S)\). Consider a 2-dimensional complex \(\kappa(V_S)\cup \kappa(V_K)^{\wedge}\) which is the union of 2-dimensional complex \(\kappa(V_S)\) and a 2-dimensional complex \(\kappa(V_K)^{\wedge}\) arising from \(\kappa(V_K)\) by transforming the graph \(a_K\) into the image graph of \(a_K\) by the projection \(H^1_S\to \kappa(H^1_S)\). The 2-dimensional complex \(\kappa(V_S)\cup \kappa(V_K)^{\wedge}\) meets \(V_T\) in \(S^4\) with a graph \(a_{K,S}\) disjoint from the vertexes \(\kappa(B_K), \kappa(B_S)\) and the new introduced vertex set of \(\kappa(V_K)^{\wedge}\). Choose a self 1-handle system \(h_T\) in \(V_T\) disjoint from the graph \(a_{K,S}\) to construct \(H^1_T\). Consider \(B_S\), \(H^1_S\), \(h_S\) as small thicker versions of \(\kappa(B_S)\), \(\kappa(H^1_S)\), \(\kappa(h_S)\), respectively and then choose \(B_K\), \(H^1_K\), \(h_K\) as small thicker versions of \(\kappa(B_K)\), \(\kappa(H^1_K)\), \(\kappa(h_K)\), respectively. By continuing this procedure, a self 1-handle system \(\bar h\) on \(F\) increasing the self 1-handle systems \(h_K\), \(h_S\), \(h_T\) on \(F\) is constructed so that the surface-link \(F(\bar h)\) obtained from \(F\) by surgery along \(\bar h\) is a ribbon surface-link.

Before proving Theorem 1.4, a generalization of the null-homotopic Gauss sum invariant of a surface-knot to a surface-link is discussed, [4]. The quadratic function \(\eta:H_1(K;Z_2)\to Z_2\) of a surface-knot \(K\) in \(S^4\) is defined as follows. For a loop \(\ell\) on \(K\), let \(d\) be a compact (possibly non-orientable) surface in \(S^4\) with \(d\cap K=\partial d=\ell\). The value \(\eta([\ell])\) is defined by the \(Z_2\)-self-intersection number \(\mbox{Int}(d,d) \mod{2}\) with respect to the framing of the surface \(K\) which is independent of a choice of \(d\) by a calculation. The function \(\eta:H_1(K;Z_2)\to Z_2\) is a \(Z_2\)-quadratic function with the identity \[\eta(x+y)=\eta(x)+\eta(y) + x\cdot y\quad (x,y \in H_1(K;Z_2)),\] where \(x\cdot y\) denotes the \(Z_2\)-intersection number of \(x\) and \(y\) in \(K\). A loop \(\ell\) on \(K\)is spin or non-spin according to whether \(\eta([\ell])\) is \(0\) or \(1\), respectively. For a surface-link \(F\) in \(S^4\), the quadratic function \(\eta:H_1(F;Z_2)\to Z_2\) of \(F\) is defined to be the split sum of the quadratic functions \(\eta_K:H_1(K;Z_2)\to Z_2\) for all the surface-knot components \(K\) of \(F\). This quadratic function may be identified with the quadratic function \(\eta^{\#}:H_1(F^{\#};Z_2)\to Z_2\) of a surface-knot \(F^{\#}\) in \(S^4\) which is a fusion of \(F\) along a fusion 1-handle system on \(F\) under a canonical isomorphism \(\iota:H_1(F;Z_2)\cong H_1(F^{\#};Z_2)\). To see this, let \(\ell\) be a loop in a surface-knot components \(K\) of \(F\), \(d\) a compact surface in \(S^4\) with \(d\cap K=\partial d=\ell\), and \(F^{\#}\) a fusion of \(F\) along a fusion 1-handle system on \(F\) with \(D_F\) the attaching disk system of \(h\) to \(F\). By modifications of \(d\) using the piping technique, the compact surface \(d\) is modified into a compact surface \(d_F\) in \(S^4\) with \(d_F\cap F= \partial d_F=\ell \cup m_F\) for a loop subsystem \(m_F\) of the boundary loop system \(\partial D_F\) of \(D_F\). Since every loop of \(m_F\) is a spin loop in \(F^{\#}\), the identity \[\eta^{\#}(\iota([\ell]))=\eta^{\#}(\iota([\ell])+ \iota([m_F]))=\eta([\ell]),\] holds, showing the identification of \(\eta\) to \(\eta^{\#}\). Let \(\Delta(F;Z_2)\) be the subgroup of \(H_1(F;Z_2)\) consisting of an element represented by a loop \(\ell\) in \(F\) which bounds an immersed disk \(d\) in \(S^4\) with \(d\cap F=\ell\). The restriction \(\xi:\Delta(F;Z_2)\to Z_2\) of the quadratic function \(\eta\) on \(H_1(F;Z_2)\) is called the null-homotopic quadratic function of the surface-link \(F\). The null-homotopic Gauss sum of \(F\) is the Gauss sum \(GS_0(F)\) of \(\xi\) defined by \[GS_0(F)=\sum _{x\in \Delta(F;Z_2)} \exp(2\pi\sqrt{-1}\frac{\xi(x)}{2}).\] This number \(GS_0(F)\) is an invariant of a surface-link \(F\), which is calculable as it is shown for the case of a surface-knot, [4]. In particular, it is known that if \(F\) is a ribbon surface-link of total genus \(g\), then \(GS_0(F)=2^g\). By using this invariant \(GS_0(F)\), a proof of Theorem 1.4 strengthening an earlier result of [9, Theorem 2] is obtained, as it is shown below.

Proof of Theorem 1.4. Let \(k\cup k'\) be a non-splittable link in the interior of a 3-ball \(B\) such that \(k\) and \(k'\) are trivial knots. For the boundary 2-sphere \(S=\partial B\) and the disk \(D^2\) with the boundary circle \(S^1\), let \(K\) be the torus-link of the torus-components \(T= k\times S^1\) and \(T'= k'\times S^1\) in the 4-sphere \(S^4\) with \(S^4=B\times S^1 \cup S\times D^2\), which is a ribbon torus-link in \(S^4\), [5]. In particular, \(GS_0(K)=2^2\). Since \(k\) and \(k'\) are trivial knots in \(B\), the torus-knots \(T\) and \(T'\) are trivial torus-knots in \(S^4\) by construction. Since \(k\cup k'\) is non-splittable in \(B\), there is a simple loop \(t(k)\) in \(T\) coming from the longitude of \(k\) in \(B\) such that \(t(k)\) does not bound any disk not meeting \(T'\) in \(S^4\), meaning that there is a simple loop \(c\) in \(T\) unique up to isotopies of \(T\) which bounds a disk \(d\) in \(S^4\) not meeting \(T'\), where \(c\) and \(d\) are given by \(c=\{p\}\times S^1\) and \(d=a\times S^1\cup \{q\} \times D^2\) for a simple arc \(a\) in \(B\) joining a point \(p\) of \(k\) to a point \(q\) in \(S\) with \(a\cap(k\cup k')=\{p\}\) and \(a\cap S=\{q\}\). Regard the 3-ball \(B\) as the product \(B=B_1\times [0,1]\) for a disk \(B_1\). Let \(\tau_1\) is a diffeomorphism of the solid torus \(B_1\times S^1\) given by one full-twist rounding the meridian disk \(B_1\) one time along the \(S^1\)-direction. Let \(\tau=\tau_1\times 1\) be the product diffeomorphism of \((B_1\times S^1)\times[0,1]=B\times S^1\) for the identity map \(1\) of \([0,1]\). Let \(\partial\tau\) be the diffeomorphism of the boundary \(S\times S^1\) of \(B\times S^1\) obtained from \(\tau\) by restricting to the boundary, and the 4-manifold \(M\) obtained from \(B\times S^1\) and \(S\times D^2\) by pasting the boundaries \(\partial (B\times S^1)=S\times S^1\) and \(\partial (S\times D^2)=S\times S^1\) by the diffeomorphism \(\partial\tau\). Since the diffeomorphism \(\partial\tau\) of \(S\times S^1\) extends to the diffeomorphism \(\tau\) of \(B\times S^1\), the 4-manifold \(M\) is diffeomorphic to \(S^4\). Let \(K_M=T_M\cup T'_M\) be the torus-link in the 4-sphere \(M\) arising from \(K=T\cup T'\) in \(B\times S^1\). There is a meridian-preserving isomorphism \(\pi_1(S^4\setminus K, x)\to \pi_1(M\setminus K_M, x)\) by van Kampen theorem. The loop \(t(k)\) in \(T_M\) does not bound any disk not meeting \(T'_M\) in \(M\), so that the loop \(c\) in \(T_M\) is a unique simple loop up to isotopies of \(T_M\) which bounds a disk \(d_M=a\times S^1\cup D^2_M\) in \(M\) not meeting \(T'_M\), where \(D^2_M\) denotes a proper disk in \(S\times D^2\) bounded by the loop \(\partial\tau(\{q\}\times S^1)\). An important observation is that the self-intersection number \(\mbox{Int}(d_M, d_M)\) in \(M\) with respect to the surface-framing on \(K_M\) is \(\pm1\). This means that the loop \(c\) in \(T_M\) is a non-spin loop. Similarly, there is a unique non-spin loop \(c'\) in \(T'_M\) which bounds a disk \(d'_M\) with the self-intersection number \(\mbox{Int}(d'_M, d'_M)=\pm 1\) with respect to the surface-framing on \(K_M\). Then it is calculated that \(GS_0(K_M)=0\) and the torus-link \(K_M\) in \(M\) is a non-ribbon torus-link, . Let \((S^4,K')=(M,K_M)\). If \({\mathbf F}\) consists of two tori, then the pair \((K,K')\) forms a desired pair. If \({\mathbf F}\) is any surface of two aspheric components, then a desired \({\mathbf F}\)-link pair is obtained from the pair \((K,K')\) by taking connected sums of some trivial surface-knots, because every stabilization of a ribbon surface-link is a ribbon surface-link and every stable-ribbon surface-link is a ribbon surface-link, [9]. If \({\mathbf F}\) has some other surface \({\mathbf F}_1\) in addition to a surface \({\mathbf F}_0\) of two aspheric components, then a desired \({\mathbf F}\)-link pair is obtained from a desired \({\mathbf F}_0\)-link pair by adding the trivial \({\mathbf F}_1\)-link as a split sum. Thus, a desired \({\mathbf F}\)-link pair \((K,K')\) is obtained. In particular, if \({\mathbf F}\) has total genus \(g\geq 2\), then \(GS_0(K)=2^g\) and \(GS_0(K')=2^{g-2}\). Let \(A\) be a 4-ball in \(S^4\) such that \(A\cap L=A\cap K'\) is a trivial disk system in \(A\) with one disk component from one component of \(K\) and of \(K'\). The self-trivial 1-handle system \(h\) used for every surgery of \(K\) in \(S^4\) is deformed into \(A\), so that \(h\) is also considered as a self-trivial 1-handle system used for the surgery of \(K'\) in \(S^4\). Thus, the surface-links \(L\) and \(L'\) obtained from \(K\) and \(K'\) by surgery along the same self-trivial 1-handle system \(h\) in \(A\) are \({\mathbf F}^+\)-links for the same surface \({\mathbf F}^+\) induced from \({\mathbf F}\) by the surgery along \(h\). By van Kampen theorem, the fundamental groups of the \({\mathbf F}^+\)-links \(L\) and \(L'\) are the same group up to meridian-preserving isomorphisms. Since the boundary of a transverse disk of a 1-handle is a spin loop, the null-homotopic Gauss sum invariant is shown to be independent of choices of a self 1-handle by a calculation of the \(Z_2\)-quadratic function identity, [4]. Thus, if the self-trivial 1-handle system \(h\) consists of \(s\) self-trivial 1-handles, then \(L\) is a ribbon \({\mathbf F}^+\)-link of total genus \(g+s\) with \(GS_0(L)=2^{g+s}\) and \(L'\) is a non-ribbon \({\mathbf F}^+\)-link with \(GS_0(L')=2^{g-2+s}\). This completes the proof of Theorem 1.4.

Note that the non-ribbon surface-link \(L'\) of two components starting from the Hopf link \(k\cup k'\) in the interior of a 3-ball \(B\) has the free abelian fundamental group of rank \(2\). The diffeomorphism \(\partial \tau\) of \(S\times S^1\) in the proof of Theorem 1.4 coincides with Gluck’s non-spin diffeomorphism of \(S^2\times S^1\), [2]. The torus-link \((M,K_M)\) called a turned torus-link of a link \(k\cup k'\) in \(B\) is an analogy of a turned torus-knot of a knot in \(B\), [1].

Acknowledgements

The author is grateful to a referee for pointing out the many flaws in the early version of this paper. This work was partly supported by JSPS KAKENHI Grant Number JP21H00978 and MEXT Promotion of Distinctive Joint Research Center Program JPMXP0723833165 and Osaka Metropolitan University Strategic Research Promotion Project (Development of International Research Hubs).

Conflicts of interest

The author declares no competing interests.

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