Abstract

Quasi-ribbon .surface-links are introduced as a generalization of the concept of ribbon surface -links: surface-links that are transformed into ribbon surface-links (without changing the components) by surgery along a locally standard self (1, 2)-handle pair system. For every disconnected closed oriented surface F with at most one aspheric component, every F-link of trivial components is a quasi-ribbon surface-link. For every disconnected closed oriented surface F, there are non-ribbon quasi-ribbon F-links of trivial components. For every closed oriented surface F with at least two aspheric components, there are non-quasi-ribbon F-links of trivial components.

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 \({\mathbf F} \to S^4\). A surface-link in \(S^4\) is an \({\mathbf F}\)-link for some \({\mathbf F}\). If \({\mathbf F}\) consists of some copies of the 2-sphere \(S^2\), then it is also called an \(S^2\)-link. When \({\mathbf F}\) is connected, they are also called an \({\mathbf F}\)-knot, surface-knot and \(S^2\)-knot, respectively. A trivial surface-link is a surface-link \(F\) in \(S^4\) which bounds disjoint handlebodies smoothly embedded in \(S^4\). A 1-handle on a surface-link \(F\) in \(S^4\) is a 1-handle \(h\) on \(F\) embedded smoothly in \(S^4\), which is called a self 1-handle on \(F\) if the attaching part of \(h\) belongs to the same component of \(F\). A self-trivial 1-handle on \(F\) is a self 1-handle \(h\) on \(F\) such that the core arc \(\alpha\) of \(h\) is an interior push of an arc \(\alpha_0\) in the connected component \(F_0\) of \(F\) containing the attaching disk system of \(h\) into \(S^4\setminus F_0\), where note that the disk bounded by the loop \(\alpha\cup \alpha_0\) may meet the sublink \(F\setminus F_0\). A 1-handle system on a surface-link \(F\) in \(S^4\) consists of disjoint 1-handles on \(F\). A self or self-trivial 1-handle system on \(F\) is a 1-handle system consisting of self or self-trivial 1-handles on \(F\), respectively. Let \(F(h)\) be the surface-link obtained from \(F\) by surgery along a 1-handle system \(h\). A ribbon surface-link is the surface-link \(F=O(h^O)\) in \(S^4\) obtained from a trivial \(S^2\)-link \(O\) by surgery along a 1-handle system \(h^O\) on \(O\), [1, 2]. Let \(D_h\) be a transverse disk system of the 1-handle system \(h\) with one disk for each 1-handle. Let \(D\times I\) be a 2-handle system on \(F\), namely a disjoint 2-handle system smoothly embedded in \(S^4\) with core disk system \(D\). The surface-link obtained from \(F\) by surgery along a 2-handle system \(D\times I\) on \(F\) is also denoted by \(F(D\times I)\). For a 1-handle system \(h\) on \(F\), let \(D'_h\times I\) be a 2-handle system on \(F(h)\) with \(\partial D'_h=\partial D_h\). The pair \((h,D'_h\times I)\) is called a \((1,2)\)-handle pair system on \(F\), and a self \((1,2)\)-handle pair system on \(F\) if \(h\) is a self 1-handle system. A self \((1,2)\)-handle pair system \((h,D'_h\times I)\) on a surface-knot \(F\) in \(S^4\) is standard if \(h\) is a self-trivial 1-handle system on \(F\) and the 2-handle core disk system \(D'_h\) is \(F\)-relatively isotopic to the 2-handle core disk system \(D_h\). For a surface-link \(F\) of surface-knot components \(F_i\, (i=1,2,\dots,r)\) in \(S^4\), a self \((1,2)\)-handle pair system \((h,D'_h\times I)\) on \(F\) is locally standard if the subsystem \((h_i,D'_{h_i}\times I)\) on \(F_i\) is a standard \((1,2)\)-handle pair system by forgetting the sublink \(F\setminus F_i\) for every \(i\). A quasi-ribbon surface-link is a surface-link \(F\) in \(S^4\) such that the surface-link \(F(h;D'_h)\) defined by \(F(h;D'_h)=F(h)(D'_h\times I)\) for a locally standard self \((1,2)\)-handle pair system \((h,D'_h\times I)\) is a ribbon surface-link in \(S^4\). By taking \(D'_h=D_h\), every ribbon surface-link is seen to be a quasi-ribbon surface-link. If \(F\) is a quasi-ribbon surface-link, then \(F(h)\) is a ribbon surface-link, because \(F(h)\) is obtained from the ribbon surface-link \(F(h;D'_h)\) by surgery along the 1-handle given by \(D'_h\times I\), and the components \(F_i\,(i=1,2,\dots,r)\) of \(F\) are ribbon surface-knots because \(F_i(h_i)(D'_{h_i})=F_i(h_i)(D_{h_i})=F_i\) for every \(i\). The following theorem, giving a characterization on quasi-ribbon surface-links corrects an earlier claimed characterization of when a surface-link of ribbon components is a ribbon surface-link, [3, Theorem 1.4].

Theorem 1.1. A surface-link \(F\) in \(S^4\) is a quasi-ribbon surface-link if and only if the surface-link \(F(h)\) for a self-trivial 1-handle system \(h\) on \(F\) is a ribbon surface-link in \(S^4\).

The following lemma is implicitly used in the proofs of [3, Theorem 1.4] and 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 \(F(h)\) is a ribbon surface-link in \(S^4\).

Note that every self 1-handle system on every surface-link \(F\) of trivial components is a self-trivial 1-handle system on \(F\), [5]. Thus, the following corollary is a direct consequence of obtained from Theorem 1.1 and Lemma 1.2, which corrects the author’s earlier statement, [4, Theorem 1].

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

This corollary leads to the following examples of quasi-ribbon surface-links that are not ribbon.

Example 1.4. For every disconnected closed oriented surface \({\mathbf F}\), there is a non-ribbon quasi-ribbon \({\mathbf F}\)-link of trivial compthat are not ribbononents . In fact, for every disconnected closed oriented surface \({\mathbf F}\) with total genus \(0\) or with just one aspheric component of any even genus, non-ribbon \({\mathbf F}\)-links of trivial components are constructed, [6, 7]. Let \(K\) be such a non-ribbon \(S^2\)-link of trivial components, which is quasi-ribbon by Corollary 1.3. For every disconnected closed oriented surface \({\mathbf F}\), a non-ribbon quasi-ribbon \({\mathbf F}\)-link \(F\) is constructed by connected-summing trivial surface-knots with \(K\) and/or adding a trivial surface-link component to \(K\) as a splitting component. Non-ribbonness of \(F\) is seen from that if \(F\) is a ribbon surface-link, then \(K\) must be a ribbon \(S^2\)-link, [3].

The following theorem shows that there are surface-links of trivial components with at least two aspheric components that are not quasi-ribbon. This result is obtained by slightly strengthening an earlier result, [4, Theorem 2].

Theorem 1.5. Let \({\mathbf F}\) be any closed oriented disconnected surface with at least two aspheric components. Then there is a pair \((F,F')\) of \({\mathbf F}\)-links \(F, F'\) in \(S^4\) both of trivial components with the same fundamental group up to meridian-preserving isomorphisms such that \(F\) is a ribbon surface-link and \(F'\) is not a quasi-ribbon surface-link.

Proofs of Theorem 1.1, Lemma 1.2 and Theorem 1.5

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 \(S^4\), [3]. 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 \(h^O\) on \(F\) as the union \(O\times[0,1]\cup h^O\) for a normal collar \(O\times [0,1]\) of \(O\) in \(S^4\) with \(O\times \{0\}=O\) where \(h^O\) does not meet \(O\times [0,1]\) except for the attaching part to \(O\) and \(\partial W = F \cup O \times\{1\}\). 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 chorded sphere system of the SUPH system \(W\). By replacing \(\alpha\) with a 1-handle system \(h^O\) attaching to \(O\) with core arc system \(\alpha\), the surface-link \(F\) is a ribbon surface-link defined by \(O\) and \(h^O\). In other words, giving a SUPH system \(W\) with \(\partial W = F \cup O\) is the same as saying 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 2-handle system on \(O\) disjoint from \(W\) except for the attaching part in \(O\) where the 2-handle system can be taken in the complement \(B\setminus D_W\) for a disjoint 3-ball system \(B\) bounded by \(O\) in \(S^4\) whose interior meets \(W\) with a disjoint 2-disk system \(D_W\), [7, Appendix]. Two ribbon structures of equivalent surface-links are moved into each other by a finite number of the moves \(M_0, M_1, M_2\), [8]. This means that for any two SUPH systems \(W\) and \(W'\) for a ribbon surface-link \(F\) in \(S^4\), there is an orientation-preserving diffeomorphism \(f\) of \(S^4\) sending \(W\) to a multi-fusion SUPH system \(W^{**}\) of a multi-fission SUPH system \(W^*\) of the SUPH system \(W'\), [7, Appendix]. An O2-handle pair on a surface-link \(F\) in \(S^4\) is a pair \((D\times I, E \times I)\) of 2-handles \(D\times I\), \(E\times I\) on \(F\) in \(S^4\) which intersect orthogonally only with the attaching parts \((\partial D)\times I\), \((\partial E)\times I\) to \(F\), so that the intersection \(Q=(\partial D)\times I\cap (\partial E)\times I\) is a square, [9]. The proof of Theorem 1.1 is done as follows.

Proof of Theorem 1.1. If there is a locally standard self \((1,2)\)-handle pair system \((h,D'_h\times I)\) on a surface-link \(F\) in \(S^4\) such that \(F(h;D'_h)\) is a ribbon surface-link, then \(h\) is a self-trivial 1-handle system on \(F\) and \(F(h)\) is a ribbon surface-link. Conversely, assume that \(F(h)\) is a ribbon surface-link for a self-trivial 1-handle system \(h\) on \(F\). The ribbon surface-link \(F(h)\) consisting of ribbon surface-knots \(F_i(h_i)\) for self-trivial 1-handle system \(h_i\) on the components \(F_i\, (i=1,2,\dots, r)\) of \(F\). By the ribbonness of \(F_i(h_i)\) and the self-triviality of \(h_i\), the surface-knot \(F_i\, (i=1,2,\dots, r)\) are ribbon surface-knots, [3]. Let \(W\) be a SUPH system for \(F(h)\) which is the union of SUPH systems \(W_i\) for \(F_i(h_i)\, (i=1,2,\dots, r)\). Since \(F_i\) is a ribbon surface-knot and \(h_i\) is a self-trivial 1-handle system on \(F_i\), let \(W^0_i\) be a SUPH system for \(F_i\), and \(W'_i=W^0_i \cup h_i\) a SUPH system for \(F_i(h_i)\) such that the core arc \(\alpha_i\) of \(h_i\) is obtained from an arc \(\alpha^0_i\) in \(F_i(h_i)\) by pushing the interior of \(\alpha^0_i\) outside of \(W^0_i\). Then there is an orientation-preserving diffeomorphism \(f_i\) of \(S^4\) sending \(W_i\) to a multi-fusion SUPH system \(W^{**}_i\) of a multi-fission SUPH system \(W^*\) of the SUPH system \(W'_i\). The disk system \(D_{h_i}\) is in \(W'_i\) and hence in \(W^*_i\), which is modified into a disk system \(D'_{h_i}\) in \(W^{**}_i\) with \(\partial D'_{h_i}=\partial D_{h_i}\). The disk system \(D'_h=\cup_{i=1}^r f^{-1}_i(D'_{h_i})\) is a 2-handle core disk system on \(F(h)\) in \(W\) with \(\partial D'_{h}=\partial D_{h}\) and \((h,D'_h\times I)\) is a self\((1, 2)\)-handle pair system on \(F\). The surface-link \(F(h;D'_h)\) is a ribbon surface-link because the 3-manifold obtained from \(W\) by splitting along the disk system \(D'_h\) is a SUPH system for \(F(h;D'_h)\). To show that the self \((1,2)\)-handle pair system \((h,D'_h\times I)\) on \(F\) is locally standard, let \(h\) be a self-trivial 1-handle system on a surface-knot \(F\) in \(S^4\). By definition, there is an O2-handle pair system \((D_h\times I, E\times I)\) on \(F(h)\) where \(E\) is made from a 2-handle core disk system bounded by the simple loop system \(\alpha_0\cup\alpha\) for an arc system \(\alpha_0\) in \(F\) and the core arc system \(\alpha\) of \(h\). There is an O2-handle pair system \((D'_h\times I, E\times I)\) on \(F(h)\) obtained by replacing the 2-handle system \(D_h\times I\) with a 2-handle system \(D'_h\times I\). Then the 2-handle system \(D'_h\times I\) is equivalent to the 2-handle system \(D_h\times I\) under 3-cell moves keeping \(F(h)\) fixed by the common 2-handle property, [9, 10]. Thus, the self \((1,2)\)-handle pair system \((h,D'_h\times I)\) on \(F\) is locally standard. 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 \(K(h_K)\cup S(h_S)\cup T(h_T)\) 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 \(h\) on \(F\) such that the surface-link \(F(h)\) is a ribbon .surface-link. This completes the proof of Lemma 1.2.

Before proving Theorem 1.5, a generalization of the null-homotopic Gauss sum invariant of a surface-knot to a surface-link is discussed, [11]. 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 calculation. The function \(\eta:H_1(K;Z_2)\to Z_2\) is a \(Z_2\)-quadratic function with the identity \[\eta(x+y)=\eta(x)+\eta(y) + x\cdot y,\] where \(x,y \in H_1(K;Z_2)\) and \(x\cdot y\) 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 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 \(\iota:H_1(F;Z_2)\to H_1(F_{\#};Z_2)\). To see this, let \(\ell\) be a loop in a component \(K\) of \(F\), \(d\) a compact surface in \(S^4\) with \(d\cap K=\partial d=\ell\), and \(F_{\#}\) the surface obtained from \(F\) by surgery along a fusion 1-handle system \(h\) on \(F\). Every transverse intersection point between \(d\) and \(F\setminus K\) in \(S^4\) can be moved into \(K\setminus h\cap K\) through \(F_{\#}\), so that the compact surface \(d\) is modified into a compact surface \(d_K\) in \(S^4\) with \(d_K\cap F= \partial d_K=\ell \cup o_K\) for a trivial loop system \(o_K\) in \(K\). Since every loop of \(o_K\) is a spin loop in \(F_{\#}\), the identity \[\eta_{\#}(\iota([\ell]))=\eta_{\#}(\iota([\ell])+ \iota([o_K]))=\eta([\ell]),\] 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 \(\xi:\Delta=\Delta(F;Z_2)\to Z_2\) 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 \[GS_0(F)=\sum _{x\in \Delta} \exp(\xi(x)\pi\sqrt{-1}),\] where \(\xi:\Delta=\Delta(F;Z_2)\). This number \(GS_0(F)\) is an invariant of a surface-link \(F\), which is calculable as shown for the case of a surface-knot, [11]. Then 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 \(GS_0(F)\), the proof of Theorem 1.5 strengthening an earlier result of [4, Theorem 2] is obtained as shown below.

Let \(k\cup k'\) be a non-split link in the interior of a 3-ball \(B\) such that \(k\) and \(k'\) are trivial knots. For the boundary 2-sphere \(S^B=\partial B\) and the disk \(D^2\) with boundary circle \(S^1\), let \(K\) be the surface-link of 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 surface-link in \(S^4\), [2]. Then, \(GS_0(K)=2^2\). 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-split 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 \(\tau_1\) be 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 \(\tau_{\partial}\) be the diffeomorphism of the boundary \(S^B\times S^1\) 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 \(S^B\times D^2\) by pasting the boundaries \(\partial (B\times S^1)=S^B\times S^1\) and \(\partial (S^B\times D^2)=S^B\times S^1\) by the diffeomorphism \(\tau_{\partial}\). Since the diffeomorphism \(\tau_{\partial}\) of \(S^B\times S^1\) 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 surface-link of torus components \(T_M\) and \(T'_M\) 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 \(S^B\times D^2\) bounded by the loop \(\tau_{\partial}(\{q\}\times S^1)\). 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 surface-link \(K_M\) in \(M\) is a non-ribbon torus-link, [11]. Let \((S^4,K')=(M,K_M)\). For every closed oriented disconnected surface \({\mathbf F}\) with at least two aspheric components, the pair \((F, F')\) of a ribbon \({\mathbf F}\)-link \(F\) and a non-ribbon \({\mathbf F}\)-link \(F'\) is constructed from the pair \((K, K')\) by connected-summing a trivial surface-knot with both \(K\) and \(K'\) and/or adding a trivial surface-link component to both \(K\) and \(K'\) as a splitting component, because if \(F'\) is a ribbon surface-link then \(K'\) must be a ribbon \(S^2\)-link, [3]. For the total genus \(g (\geq 2)\) of \(F\), it is calculated that \(GS_0(F) = 2^g\) and \(GS_0(F') = 2^{g-2}\). Let \(A\) be a 4-ball in \(S^4\) such that \(A\cap K=A\cap K'\) is a trivial disk system in \(A\) taking one disk from each component of \(K\) and from each component of \(K'\). Assume that the connected sum and the addition operations used to construct \((F, F')\) from \((K, K')\) are done in the 4-ball \(A\). By van Kampen theorem, it is shown that the fundamental groups \(\pi_1(S^4\setminus F,x)\) and \(\pi_1(S^4\setminus F',x)\) are the same group up to meridian-preserving isomorphisms. Since the boundary loop 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, [11]. Thus, if the self-trivial 1-handle system \(h\) of \(s\) members are done for \(F\) and \(F'\), then \(F(h)\) is a ribbon surface-link of total genus \(g+s\) with \(GS_0(F(h)) = 2^{g+s}\) and \(F'(h)\) is a non-ribbon surface-link with \(GS_0(F'(h)) = 2^{g-2+s}\), so that \(F'\) is not a quasi-ribbon surface-link. This completes the proof of Theorem 1.5.

In the proof of Theorem 1.5, note that the non-ribbon surface-link\(K'\) 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 \(2\). 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, [9, 10]. This example shows that there is a non-ribbon surface-link \(F'\) of ribbon components such that the surface-knot obtained from \(F'\) by surgery along any fusion 1-handle is a ribbon surface-knot meaning that non-ribbonness of a surface-link cannot detect by surgery along any fusion 1-handle, giving a strong counterexample to . A positive result for a boundary surface-link, [7]. The diffeomorphism \(\tau_{\partial}\) of \(S^B\times S^1\) in the proof of Theorem 1.4 coincides with Gluck’s non-spin diffeomorphism of \(S^2 \times S^1\), [12]. The surface-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 \(B\), [13].

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.

References

1. Kawauchi, A., Shibuya, T., Suzuki, S. (1982). Descriptions on surfaces in four-space, II: Singularities and cross-sectional links, Math Sem Notes Kobe Univ, 11: 31-69.

2. Kawauchi, A. (2015). A chord diagram of a ribbon surface-link, J Knot Theory Ramifications, 24: 1540002 (24 pages).

3. Kawauchi, A. (2025). Ribbonness of a stable-ribbon surface-link, II: General case, (MDPI) Mathematics, 13 (3): 402 (1-11).

4. Kawauchi, A. (2024). Note on surface-link of trivial components, Journal of Comprehensive Pure and Applied Mathematics, 2 (1) : 1 - 05.

5. Hosokawa, F. and Kawauchi, A. (1979). Proposals for unknotted surfaces in four-space, Osaka J. Math, 16: 233-248.

6. Ogasa, E. (2001). Nonribbon 2-links all of whose components are trivial knots and some of whose band-sums are nonribbon knots, J. Knot Theory Ramifications, 10 : 913-922.

7. Kawauchi, A. (2025). Ribbonness on boundary surface-link. arXiv:2507.18154

8. Kawauchi, A. (2018). Faithful equivalence of equivalent ribbon surface-links, Journal of Knot Theory and Its Ramifications, 27: 1843003 (23 pages).

9. Kawauchi, A. (2021). Ribbonness of a stable-ribbon surface-link, I. A stably trivial surface-link, Topology and its Applications, 301: 107522 (16pages).

10. Kawauchi, A. (2023). Uniqueness of an orthogonal 2-handle pair on a surface-link, Contemporary Mathematics (UWP), 4: 182-188.

11. Kawauchi, A. (2002). On pseudo-ribbon surface-links, J Knot Theory Ramifications, 11: 1043-1062.

12. Gluck, H. (1962). The embedding of two-spheres in the four-sphere, Trans Amer Math Soc, 104: 308-333.

13. Boyle, J. (1993). The turned torus knot in \(S^4\), J Knot Theory Ramifications, 2: 239-249.