Alternative Proof of the Ribbonness on Classical Link

1. Introduction

For a set A in the 3-space ${\mathbf{R}}^3=\left\{(x,y,z)\right|-\infty <x,y,z<+\infty \}$ and an interval $J\subset \mathbf{R}$, let

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

The upper-half 4-space ${\mathbf{R}}_{+}^4$ is denoted by ${\mathbf{R}}^3[0,+\infty )$. Let k be a link in the 3-space ${\mathbf{R}}^3$, which always bounds a proper oriented surface F embedded smoothly in the upper-half 4-space ${\mathbf{R}}_{+}^4$, where ${\mathbf{R}}^3\left[0\right]$ is canonically identified with ${\mathbf{R}}^3$. Two proper oriented surfaces F and $F^{\prime }$ in ${\mathbf{R}}_{+}^4$ are equivalent if there is an orientation-preserving diffeomorphism f of ${\mathbf{R}}_{+}^4$ sending F to $F^{\prime }$, where f is called an equivalence. Let $\mathbf{b}$ be a band system spanning the link k in ${\mathbf{R}}^3$, namely a system of finitely many disjoint oriented bands $b_i,(i=1,2\cdots ,m)$ spanning the link k in ${\mathbf{R}}^3$. Let $k^{\prime }$ be a link in ${\mathbf{R}}^3$ obtained from k by surgery along this band system $\mathbf{b}$. This band surgery operation is denoted by $k\to k^{\prime }$. If the link $k^{\prime }$ has $r-m$ or $r+m$ knot components for a link k of r knot components, then the band surgery operation $k\to k^{\prime }$ is called a fusion or fission, respectively. These terminologies are used in [6]. Assume that a band surgery operation $k\to k^{\prime }$ consists of a band surgery operation $k_i\to k_i^{\prime }$ for the knot components $k_i(i=1,2,\cdots ,n)$ of k. Then if the link $k_i^{\prime }$ is a knot for every i, then the band surgery operation $k\to k^{\prime }$ is called a genus addition. Every band surgery operation $k\to k^{\prime }$ along a band system $\mathbf{b}$ is realized as a proper surface $F_s^u$ in ${\mathbf{R}}^3[s,u]$ for any interval $[s,u]$ as follows.

$$F_s^u\cap {\mathbf{R}}^3\left[t\right]=\left\{\begin{array}{cc}\hfill k^{\prime }\left[t\right],& \mathrm{f}\mathrm{o}\mathrm{r}{\displaystyle \frac{s+u}{2}}<t\le u,\hfill \\ \hfill (k\cup \mathbf{b})\left[t\right],& \mathrm{f}\mathrm{o}\mathrm{r} t={\displaystyle \frac{s+u}{2}},\hfill \\ \hfill k\left[t\right],& \mathrm{f}\mathrm{o}\mathrm{r} s\le t<{\displaystyle \frac{s+u}{2}}.\hfill \end{array}\right.$$

For every band surgery sequence $k_0\to k_1\to k_2\to \cdots \to k_n$, the realizing surface $F_s^u$ in ${\mathbf{R}}^3[s,u]$ is given by the union

$$F_{s_0}^{s_1}\cup F_{s_1}^{s_2}\cup \cdots \cup F_{s_{n-1}}^{s_n}$$

for any division

$$s=s_0<s_1<s_2<\cdots <s_n=u$$

of the interval $[s,u]$. For a band surgery sequence $k_0\to k_1\to k_2\to \cdots \to o_n$ with ${\mathbf{o}}_n$ a trivial link, the upper-closed realizing surface in ${\mathbf{R}}^3[s,t]$ is the surface

$$\mathrm{u}\mathrm{c}\mathrm{l}\left(F_s^u\right)=F_s^u\cup \mathbf{\delta }\left[u\right]$$

in ${\mathbf{R}}^3[s,t]$ with boundary $\partial ucl\left(F_s^u\right)=k_0\left[s\right]$ in ${\mathbf{R}}^3\left[s\right]$, where $\mathbf{\delta }$ denotes a disk system in ${\mathbf{R}}^3$ of mutually disjoint disks with $\partial \mathbf{\delta }={\mathbf{o}}_n$. Further, if the link $k_0$ is the split sum $k+\mathbf{o}$ of a link k and a trivial link $\mathbf{o}$ in ${\mathbf{R}}^3$, then a bounded realizing surface for the link k $p\left(F_s^u\right)$ in ${\mathbf{R}}^3[s,u]$ is defined to be the surface

$$\mathrm{p}\left(F_s^u\right)=F_s^u\cup \mathbf{d}\left[s\right]\cup \mathbf{\delta }\left[u\right]$$

in ${\mathbf{R}}^3[s,u]$ with $\partial p\left(F_s^u\right)=k\left[s\right]$ in ${\mathbf{R}}^3\left[s\right]$, where $\mathbf{d}$ is a disk system in ${\mathbf{R}}^3$ with $\partial \mathbf{d}=\mathbf{o}$ and $\mathbf{d}\cap k=\varnothing$. A proper realizing surface for the link k is a proper surface $p{\left(F_s^u\right)}^{+}$ in ${\mathbf{R}}^3[s,+\infty )$ with $\partial p{\left(F_s^u\right)}^{+}=k\left[s\right]$ in ${\mathbf{R}}^3\left[s\right]$ which is obtained from $p\left(F_s^u\right)$ by raising the level s of the disk system d into the level $s+\epsilon$ for a sufficiently small $\epsilon >0$. The upper-closed proper surface $ucl\left(F_s^u\right)$ for $k_0$ in ${\mathbf{R}}_{+}^4$ does not depend on choices of $\mathbf{\delta }$ and is determined up to equivalences only by the band surgery sequence $k_0\to k_1\to k_2\to \cdots \to {\mathbf{o}}_n$ with ${\mathbf{o}}_n$ a trivial link by Horibe-Yanagawa’s lemma, [6]. Also, the proper realizing surface $p{\left(F_s^u\right)}^{+}$ for k in ${\mathbf{R}}_{+}^4$ does not depend on choices of $\mathbf{\delta }$, $\mathbf{d}$, $\epsilon$ and is determined up to equivalences only by the band surgery sequence $k+\mathbf{o}\to k_1\to k_2\to \cdots \to {\mathbf{o}}_n$ with $\mathbf{o}$, ${\mathbf{o}}_n$ trivial links. A stable-exact band surgery sequence for a link k in ${\mathbf{R}}^3$ is a band surgery sequence

$$(\#)k+\mathbf{o}\to k_1\to k_2\to k_3\to {\mathbf{o}}_4$$

for trivial links $\mathbf{o},{\mathbf{o}}_4$ in ${\mathbf{R}}^3$ such that

(0)

the operation $k+\mathbf{o}\to k_1$ is a fusion along a band system ${\mathbf{b}}_1$ connecting every component of $\mathbf{o}$ to k with just one band,

(1)

the operation $k_1\to k_2$ is a fusion along a band system ${\mathbf{b}}_2$,

(2)

the operation $k_2\to k_3$ is a genus addition along a band system ${\mathbf{b}}_3$, and

(2)

the operation $k_3\to {\mathbf{o}}_4$ is a fission along a band system ${\mathbf{b}}_4$.

In (0), the link $k_1$ is called a band sum of the link k and the trivial link $\mathbf{o}$. By band slides, assume that the band systems ${\mathbf{b}}_i(i=2,3,4)$ do not meet with the trivial link $\mathbf{o}$. For every stable-exact band surgery sequence (#) for a link k in ${\mathbf{R}}^3$, a proper realizing surface $p\left(F_0^1\right)$ for k in ${\mathbf{R}}_{+}^4$ with $\partial p\left(F_0^1\right)=k$ is constructed for any division $0=s_0<s_1<s_2<s_3<s_4=1$ of the interval $[0,1]$. The following theorem is known, [6].

Normal form theorem. For every proper oriented surface F without closed component in the upper-half 4-space ${\mathbf{R}}_{+}^4$, there is a stable-exact band surgery sequence (#) for the link $k=\partial F$ in ${\mathbf{R}}^3$ such that the proper realizing surface $p{\left(F_0^1\right)}^{+}$ is equivalent to F in ${\mathbf{R}}_{+}^4$.

The surface $p{\left(F_0^1\right)}^{+}$ in ${\mathbf{R}}_{+}^4$ is called a normal form of the proper surface F in ${\mathbf{R}}_{+}^4$. In the stable-exact band surgery sequence (#), if the trivial link o is taken the empty set ∅, then the item (0) is omitted and the stable-exact band surgery sequence (#) is reduced to the band surgery sequence

$$(\#\#)k=k_1\to k_2\to k_3\to {\mathbf{o}}_4$$

This band surgery sequence (##) is called an exact band surgery sequence for the link k. In classical knot theory, a proper surface F in ${\mathbf{R}}_{+}^4$ is a ribbon surface in ${\mathbf{R}}_{+}^4$ for the link $k=\partial F$ in ${\mathbf{R}}^3$ if it is equivalent to the upper-closed realizing surface $ucl\left(F_0^1\right)$ of an exact band surgery sequence (##) for the link k. In the following example, it is observed that there are lots of proper oriented surfaces without closed component in ${\mathbf{R}}_{+}^4$ which is not equivalent to any ribbon surface.

Example 1.

For every link k, let $F^{\prime }$ be any ribbon surface in ${\mathbf{R}}_{+}^4$ with $k=\partial F^{\prime }$. For example, let $F^{\prime }$ be a proper surface in ${\mathbf{R}}_{+}^4$ obtained from a Seifert surface for k in ${\mathbf{R}}^3$ by an interior push into ${\mathbf{R}}_{+}^4$. Take a connected sum $F=F^{\prime }\#K$ of $F^{\prime }$ and a non-trivial $S^2$-knot K in ${\mathbf{R}}^4$ with non-abelian fundamental group. The ribbon surface $F^{\prime }$ is a renewal embedding of F into ${\mathbf{R}}_{+}^4$ with $k=\partial F^{\prime }=\partial F$. The fundamental groups of $k,F^{\prime },F,K$ are denoted as follows.

$$\pi \left(k\right)={\pi }_1({\mathbf{R}}^3\setminus k,x_0),\pi \left(F^{\prime }\right)={\pi }_1({\mathbf{R}}^4\setminus F^{\prime },x_0),$$

$$\pi \left(F\right)={\pi }_1({\mathbf{R}}^4\setminus F,x_0),\pi \left(K\right)={\pi }_1(S^4\setminus K,x_0).$$

Let $\pi {\left(k\right)}^{*},\pi {\left(F^{\prime }\right)}^{*}\pi {\left(F\right)}^{*},\pi {\left(K\right)}^{*}$ be the kernels of the canonical epimorphisms from the groups $\pi \left(k\right),\pi \left(F^{\prime }\right),\pi \left(F\right),$$\pi \left(K\right)$ to the infinite cyclic group sending every meridian element to the generator, respectively. It is a special feature of a ribbon surface $F^{\prime }$ that the canonical homomorphism $\pi \left(k\right)\to \pi \left(F^{\prime }\right)$ is an epimorphism, so that the induced homomorphism $\pi {\left(k\right)}^{*}\to \pi {\left(F^{\prime }\right)}^{*}$ is onto. On the other hand, the canonical homomorphism $\pi \left(k\right)\to \pi \left(F\right)$ is not onto, because the group $\pi {\left(F\right)}^{*}$ is the free product $\pi {\left(F^{\prime }\right)}^{*}*\pi {\left(K\right)}^{*}$ and $\pi {\left(K\right)}^{*}\ne 0$ and the image of the induced homomorphism $\pi {\left(k\right)}^{*}\to \pi {\left(F\right)}^{*}$ is just the free product summand $\pi {\left(F^{\prime }\right)}^{*}$. Thus, the proper surface F in ${\mathbf{R}}_{+}^4$ is not equivalent to any ribbon surface, in particular to $F^{\prime }$.

A proper surface $F^{\prime }$ in ${\mathbf{R}}_{+}^4$ is a renewal embedding of a proper surface F into ${\mathbf{R}}_{+}^4$ if $\partial F^{\prime }=\partial F$ in ${\mathbf{R}}^3$ and there is an orientation-preserving surface-diffeomorphism $F^{\prime }\to F$ keeping the boundary fixed. The proof of the following theorem is given, [4]. In this paper, an alternative proof of this theorem is given from a viewpoint of deformations of a ribbon surface-link in ${\mathbf{R}}^4$.

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

A link k in ${\mathbf{R}}^3$ is a ribbon link if there is a fission $k\to {\mathbf{o}}_1$ for a trivial link ${\mathbf{o}}_1$. A link k in ${\mathbf{R}}^3$ is a slice link in the strong sense if k bounds a proper disk system embedded smoothly in ${\mathbf{R}}_{+}^4$. Then there is a stable-exact band surgery sequence (#) with $k_1=k_2=k_3$ for k. The following corollary is a spacial case of Classical ribbon theorem.

Corollary 1.

Every slice link in the strong sense in ${\mathbf{R}}^3$ is a ribbon link.

Thus, Classical ribbon theorem solves Slice-Ribbon Problem,1,2]. If there is a fusion $k+\mathbf{o}\to k_1$ for a trivial link $\mathbf{o}$ and a ribbon link $k_1$, then k is a slice link in the strong sense, for k bounds a proper disk system as a proper realizing surface $p{\left(F_0^1\right)}^{+}$ of a band surgery sequence $k+\mathbf{o}\to k_1\to {\mathbf{o}}_2$ with a fission $k_1\to {\mathbf{o}}_2$ for a trivial link ${\mathbf{o}}_2$. Hence, the following corollary is obtained from Corollary 1.

Corollary 2.

A link k in ${\mathbf{R}}^3$ is a ribbon link if if there is a fusion $k+\mathbf{o}\to k_1$ for a trivial link $\mathbf{o}$ and a ribbon link $k_1$.

An idea of the present proof of this theorem is to consider the ribbon surface-link $cl\left(F_{-1}^1\right)$ in the 4-space ${\mathbf{R}}^4$ obtained by doubling from the upper-closed realizing surface $ucl\left(F_0^1\right)$ in ${\mathbf{R}}_{+}^4$ of a stable-exact band surgery sequence (#) for a link k in ${\mathbf{R}}^3$ obtained from F by the normal form theorem. The effort is to remove the interior intersection between the 2-handle systems on $cl\left(F_{-1}^1\right)$ arising from the band systems of a stable-exact band surgery sequence (#) and the 2-handle system on $cl\left(F_{-1}^1\right)$ arising from the disk system $\mathbf{d}$ at the expense of type of the ribbon surface-link $cl\left(F_{-1}^1\right)$.

2. Proof of Classical Ribbon Theorem

Throughout this section, the proof of the classical ribbon theorem is done. Let F be a proper oriented surface without closed component in ${\mathbf{R}}_{+}^4$, and $\partial F=k$ a link in ${\mathbf{R}}^3$. By the normal form theorem, consider a stable-exact band surgery sequence (#) for k such that $p{\left(F_0^1\right)}^{+}$ is equivalent to F in ${\mathbf{R}}_{+}^4$. Also, consider the ribbon surface-link $cl\left(F_{-1}^1\right)$ in the 4-space ${\mathbf{R}}^4$ constructed by doubling from the upper-closed realizing surface $ucl\left(F_0^1\right)$ in ${\mathbf{R}}_{+}^4$ of the stable-exact band surgery sequence (#) , [6]. Let ${\mathbf{b}}_i,(i=1,2,3,4)$ be the band system used for the operations $k+o\to k_1$, $k_1\to k_2$, $k_2\to k_3$ and $k_3\to o_4$, respectively in (#), which are taken disjoint. Further, the band systems ${\mathbf{b}}_i,(i=1,2,3,4)$ are taken to be attached to k. Then the disjoint 2-handle systems ${\mathbf{b}}_i[-t_i,t_i],(i=1,2,3,4)$ on the ribbon surface-link $cl\left(F_{-1}^1\right)$ are obtained, where $t_i=(s_{i-1}+s_i)/2(i=1,2,3,4)$. The disk system $\mathbf{d}$ with $\partial \mathbf{d}=\mathbf{o}$ and $\mathbf{d}\cap k=\varnothing$ for the split sum $k+\mathbf{o}$ also constructs the 2-handle system $\mathbf{d}[-\epsilon ,\epsilon ]$ on the ribbon surface-link $cl\left(F_{-1}^1\right)$ for a sufficiently small $\epsilon >0$. Let $\iota$ be the reflection of ${\mathbf{R}}^4$ sending every point $(x,y,z,t)$ to the point $(x,y,z,-t)$. The ribbon surface-link $cl\left(F_{-1}^1\right)$ and the 2-handles ${\mathbf{b}}_i[-t_i,t_i],(i=1,2,3,4)$, $\mathbf{d}[-\epsilon ,\epsilon ]$ are $\iota$-invariant. Let $\mathbf{\delta }$ be a disk system in ${\mathbf{R}}^3$ bounded by the trivial link $\mathbf{\lambda }={\mathbf{o}}_4$. The band systems ${\mathbf{\alpha }}_i,(i=1,2,3,4)$ in ${\mathbf{R}}^3$ spanning the trivial link $\mathbf{\lambda }$ are obtained, as a dual viewpoint, from the band systems ${\mathbf{b}}_i,(i=1,2,3,4)$ spanning the link $k+\mathbf{o}$ in ${\mathbf{R}}^3$, respectively. It is considered that the ribbon surface-link $cl\left(F_{-1}^1\right)$ is obtained from the trivial $S^2$-link $O=\partial \left(\mathbf{\delta }\right[-1,1\left]\right)$ by surgery along the $\iota$-invariant 1-handle systems ${\mathbf{\alpha }}_i[-t_i,t_i],(i=1,2,3,4)$ on O which are the duals of the 2-handles ${\mathbf{b}}_i[-t_i,t_i],(i=1,2,3,4)$ on $cl\left(F_{-1}^1\right)$, respectively, [6]. In recent terms, the ribbon surface-link $cl\left(F_{-1}^1\right)$ is presented by the pair $(\mathbf{\lambda },\mathbf{\alpha })$ of a based loop system $\mathbf{\lambda }$ and a chord system $\mathbf{\alpha }$ consisting of the band systems ${\mathbf{\alpha }}_i(i=1,2,3,4)$ spanning $\mathbf{\lambda }$ in ${\mathbf{R}}^3$,3]. The chord system $\alpha$ is generally understood as a system of spanning strings, but here it is a system of spanning bands. Since every component o of $\mathbf{o}$ meets with a based loop $\lambda$ in $\mathbf{\lambda }$ in an arc I not meeting ${\mathbf{\alpha }}_1$, let $\mathbf{I}$ be an arc system obtained by choosing one such arc I for every component o of $\mathbf{o}$. Then there is a disk system ${\mathbf{d}}_0$ in $\mathbf{d}$ not meeting the band system $\mathbf{\alpha }$ such that $\mathbf{I}\subset \partial {\mathbf{d}}_0$ and the complement ${\mathbf{d}}^{\prime }=cl(\mathbf{d}\setminus {\mathbf{d}}_0)$ is a disk system which is a strong deformation retract of $\mathbf{d}$. Note that every band $\alpha$ in the band system $\mathbf{\alpha }$ meets the interior of the disk system $\mathbf{d}$ in an arc system consisting of proper arcs parallel to the centerline of $\alpha$. The following claim (2.1) is obtained.

(2.1) There is a band system ${\mathbf{\alpha }}^{\prime }$ spanning $\mathbf{\lambda }$ isotopic to the band system $\mathbf{\alpha }$ by band slide moves on $\mathbf{d}$ keeping $\mathbf{\lambda }$ fixed and keeping $\mathbf{d}$ setwise fixed such that every band ${\alpha }^{\prime }$ of ${\mathbf{\alpha }}^{\prime }$ meets $\mathbf{d}$ only in the disk system ${\mathbf{d}}_0$.

After the claim (2.1), let

$${\mathbf{\lambda }}^{\prime }=cl(\mathbf{\lambda }\setminus \mathbf{I})\cup cl(\partial {\mathbf{d}}_0\setminus \mathbf{I})$$

be a trivial link spanned by the band system $\mathbf{\alpha }$, so that the pair $({\mathbf{\lambda }}^{\prime },{\mathbf{\alpha }}^{\prime })$ is a chord system in ${\mathbf{R}}^3$. Let $cl{\left(F_{-1}^1\right)}^{\prime }$ be the $\iota$-invariant surface-link obtained from the chord system $({\mathbf{\lambda }}^{\prime },{\mathbf{\alpha }}^{\prime })$ in ${\mathbf{R}}^3$. Then the middle cross-sectional link $cl{\left(F_{-1}^1\right)}^{\prime }\cap {\mathbf{R}}^3\left[0\right]$ is the split sum $k+{\mathbf{o}}^{\prime }$ for the trivial link ${\mathbf{o}}^{\prime }=\partial {\mathbf{d}}^{\prime }$. In fact, the link obtained from $\mathbf{\lambda }$ by surgery along $\mathbf{\alpha }$ is the split sum $k+\mathbf{o}$ and the link obtained from $\mathbf{\lambda }$ by surgery along ${\mathbf{\alpha }}_1$ is a link $k^{\prime }\cup \mathbf{o}$ for a link $k^{\prime }$, so that k is obtaine from $k^{\prime }$ by surgery along ${\mathbf{\alpha }}_i(i=2,3,4)$. On the other hand, the link obtained from ${\mathbf{\lambda }}^{\prime }$ by surgery along ${\mathbf{\alpha }}_1^{\prime }$ is the split sum $k^{\prime }+{\mathbf{o}}^{\prime }$, so that the link obtained from ${\mathbf{\lambda }}^{\prime }$ by surgery along ${\mathbf{\alpha }}^{\prime }$ is the split sum $k+{\mathbf{o}}^{\prime }$. Note that the surface-link $cl{\left(F_{-1}^1\right)}^{\prime }$ is obtained by sacrificing an equivalence to the surface-link $cl\left(F_{-1}^1\right)$ although they are the same surface. By replacing ${\mathbf{d}}^{\prime }$, ${\mathbf{\lambda }}^{\prime }$ and ${\mathbf{o}}^{\prime }$ with $\mathbf{d}$, $\mathbf{\lambda }$ and $\mathbf{o}$, respectively, the following claim (2.2) is obtained.

(2.2) There is a stable-exact band surgery sequence (#) for the link k with $p{\left(F_0^1\right)}^{+}$ a renewal embedding of F in ${\mathbf{R}}_{+}^4$ such that the band system $\mathbf{\alpha }$ does not meet the interior of the disk system $\mathbf{d}$.

Take a stable-exact band surgery sequence (#) for the link k of (2.2). Then the link $k_1$ is isotopic to the link k in ${\mathbf{R}}^3$. Thus, there is an exact band surgery sequence (##) for the knot k such that the upper-closed realizing surface $ucl\left(F_0^1\right)$ is a renewal embedding of F. This completes the proof of the classical ribbon theorem.

In (2.2), the ribbon surface-link $cl\left(F_{-1}^1\right)$ in the 4-space ${\mathbf{R}}^4$ constructed by doubling from the upper-closed realizing surface $ucl\left(F_0^1\right)$ in ${\mathbf{R}}_{+}^4$ of the stable-exact band surgery sequence (#) admits the O2-handle pair system $({\mathbf{\alpha }}_1[-t_1,t_1],\mathbf{d}[-\epsilon ,\epsilon ])$,5]. The last explanation above is related to the surgery of $cl\left(F_{-1}^1\right)$ along the O2-handle pair system.