Abstract

As a previous result, it has shown that every sphere-link consisting of trivial components is a ribbon sphere-link. In this note, it is shown that for every closed oriented disconnected surface F with just one non-sphere component, every F-link consisting of trivial components is a ribbon surface-link. Further, it is shown that for every closed oriented disconnected surface F containing at least two non-sphere components, there exist a pair of a ribbon F-link and a non-ribbon F-link that consist of trivial components and have meridian-preservingly isomorphic fundamental groups.

Let \({\mathbf F}\) be a (possibly disconnected) closed 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 \(F\) which bounds disjoint handlebodies smoothly embedded in \(S^4\). 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 \(S^2\)) by surgery along an embedded 1-handle system, [6-9 ,11]. It was shown that every \(S^2\)-link consisting of trivial components is a ribbon \(S^2\)-link, [10]. The following result is a generalization of this result.

Theorem 1. Let \({\mathbf F}\) be a closed oriented disconnected surface with at most one non-sphere component. Then every \({\mathbf F}\)-link \(L\) in \(S^4\) consisting of trivial components is a ribbon \({\mathbf F}\)-link in \(S^4\).

Proof of Theorem 1. The case that \({\mathbf F}\) consists of only \(S^2\)-components has been given, [10]. This proof is done by a similar method. Let \({\mathbf F}\) have \(S^2\)-components and only one non-sphere component, and \(L\) an \({\mathbf F}\)-link in \(S^4\) consisting of trivial components. Let \(F\) be the trivial non-sphere component of \(L\) and \(L'=L\setminus F\) the remaining \(S^2\)-link consists of trivial components. Since the second homology class \([F]=0\) in \(H_2(S^4\setminus L';Z)=0\), there is a compact connected oriented 3-manifold \(V_F\) smoothly embedded in \(S^4\) with \(\partial V_F=F\) and \(V_F\cap L'=\emptyset\). The \(S^2\)-link \(L'\) is a ribbon \(S^2\)-link in \(S^4\), [10]. Let \(W\) be a SUPH system in \(S^4\) with \(\partial W=L'\cup O\) for a trivial \(S^2\)-link \(O\), [10]. Let \(\alpha\) be an arc system in \(W\) spanning \(O\) such that the closed complement \(\mbox{cl}(W\setminus N(\alpha))\) is identified with a boundary collar \(L'\times[0,1]\) of \(L'\) in \(W\) with \(L'\times 0=L'\) where \(N(\alpha)\) is a regular neighborhood of \(\alpha\) in \(W\) which is written as a trivial disk fiber bundle \(d\times \alpha\) over \(\alpha\). Since \(V_F\cap L'=\emptyset\), it can be assumed that \((L'\times[0,1])\cap V_F=\emptyset\) and the interior of the disk fiber \(d\times x\) for a point \(x\) of \(\alpha\) meets \(V_F\) transversely with a simple proper arc system and a simple loop system. By deforming \(V_F\), the interior of the disk fiber \(d\times x\) meets \(V_F\) transversely only with a simple proper arc system \(\beta\). As a result, the intersection \(N(\alpha)\cap V_F\) is assumed to be a thickening \(\beta\times [0,1]\) of \(\beta\). A regular neighborhood of \(\beta\times [0,1]\) in \(V_F\) is a 1-handl system \(h(\beta)\) on \(F\). By adding a disjoint 1-handle system \(h^+\) on \(F\) embedded in \(V_F\), the closed complement \(H=\mbox{cl}(V_F\setminus (h(\beta)\cup h^+))\) is a handlebody of genus, say \(n\). Let \(F^+=\partial H\). Let \(H^0\) be a once-punctured handlebody obtained from \(H\) by removing a 3-ball with \(\partial H^0=F^+\cup O^H\). The union \(W\cup H^0\) is a SUPH system for the surface-link \(L'\cup F^+\) in \(S^4\), so that \(L'\cup F^+\) is a ribbon surface-link in \(S^4\) by [10, Lemma 3.1]. The 1-handle system \(h(\beta)\cup h^+\) on \(F\) is a 2-handle system \(D_i\times I \, (i=1,2, \dots , n)\) on \(F^+\), where \(I\) denotes an interval containing \(0\) in the interior. Then there is a disjoint O2-handle pair system \((D_i \times I, D'_i\times I) \, (i = 1, 2, \dots, n)\) on \(F^+\) because any disjoint 1-handle system on the trivial surface-knot \(F\) is a disjoint trivial 1-handle system on \(F\), [4, 10, Lemman 4.1]. Let \((O\cup O^H, \alpha\cup \alpha^H)\) be a chorded sphere system for the ribbon surface-link \(L'\cup F^+\) constructed in the SUPH system \(W\cup H^0\). Let \(B(O)\cup B(O^H)\) be a disjoint 3-ball system bounded by the trivial \(S^2\)-link \(O\cup O^H\) in \(S^4\). The intersections \(B(O)\cap D_i = B(O)\cap D'_i = \emptyset (i =1, 2, \dots , n)\) can be assumed by moving the 3-ball system \(B(O)\cup B(O^H)\) in \(S^4\). The intersections \((\alpha\cup \alpha^H)\cap D_i = (\alpha\cup \alpha^H)\cap D'_i = \emptyset\, (i =1, 2, \dots , n)\) can be also assumed by general position. Thus, the disjoint O2-handle pair system \((D_i \times I, D'_i\times I) \, (i = 1, 2, \dots , n)\) on \(F\) is deformed into a disjoint O2-handle pair system on the ribbon surface-link \(L'\cup F^+\) in \(S^4\), whose surgery surface-link \(L = L'\cup F\) is a ribbon surface-link, [10]. This completes the proof of Theorem1.

In the case that \({\mathbf F}\) has at least two non-sphere components, the following result is obtained.

Theorem 2. Let \({\mathbf F}\) be any closed oriented disconnected surface with at least two non-sphere components. Then there exist a pair of a ribbon \({\mathbf F}\)-link \(L\) and a non-ribbon \({\mathbf F}\)-link \(L'\) in \(S^4\) that consist of trivial components and have meridian-preservingly isomorphic fundamental groups.

Proof of Theorem 2. Let \(k\cup k'\) be a non-splitable 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 \(L\) be the torus-link consisting 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 \(S^4\), [6]. 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-splitable 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 bound 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 along the meridian disk \(B_1\), and \(\tau=\tau_1\times 1\) the product diffeomorphism of \((B_1\times S^1)\times[0,1]=B\times S^1\). 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 \(L_M=T_M\cup T'_M\) be the torus-link in the 4-sphere \(M\) arising from \(L=T\cup T'\) in \(B\times S^1\). The fundamental groups \(\pi_1(S^4\setminus L, x)\) and \(\pi_1(M\setminus L_M, x)\) are meridian-preservingly isomorphic 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 \(L_M\) is \(\pm1\). This means that the loop \(c\) in \(T_M\) is a non-spin loop and thus, the torus-link \(L_M\) in \(M\) is not any ribbon torus-link, [3, 5]. Let \((S^4,L')=(M,L_M)\). If \({\mathbf F}\) consists of two tori, then the pair \((L,L')\) forms a desired pair. If \({\mathbf F}\) is any surface consisting of two non-sphere components, then a desired \({\mathbf F}\)-link pair is obtained from the pair \((L,L')\) 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, [10]. If \({\mathbf F}\) has some other surface \({\mathbf F}_1\) in addition to a surface \({\mathbf F}_0\) of two non-sphere 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 \((L,L')\) is obtained. This completes the proof of Theorem2.

In the proof of Theorem2, the diffeomorphism \(\partial \tau\) of \(S\times S^1\) coincides with Gluck’s non-spin diffeomorphism of \(S^2\times S^1\), [2]. The torus-link \((M,T_M)\) 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]. There is an invariant of a surface-knot used to confirm a non-ribbon surface-knot, [5]. This invariant is easily generalized to an invariant of a surface-link, which can be also applied to confirm that \(L'\) is a non-ribbon surface-link.

Acknowledgements

This work was partly supported by JSPS KAKENHI Grant Number JP21H00978 and MEXT Promotion of Distinctive Joint Research Center Program JPMXP0723833165.

Conflicts of interest

The author declares no competing interests.

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