Alternative Proof of the Ribbonness on Classical Link

Alternative proof is given for an earlier presented result that if a link in 3-space bounds a compact oriented proper surface (without closed component) in the upper half 4-space, then the link bounds a ribbon surface in the upper half 4-space which is a boundary-relative renewal embedding of the original surface.

2020 Mathematics Subject Classification: Primary 57K45; Secondary 57K40

For a set A in the 3-space R³ = {(x, y, z)|−∞ < x, y, z < +∞} and an interval

J ⊂ R, let

AJ = {(x, y, z, t)|(x, y, z) ∈ A, t ∈ J}.

The upper-half 4-space $R_{+}^4$ is denoted by R³[0,+∞). Let k be a link in the 3- space R³, which always bounds a compact oriented proper surface F embedded smoothly in the upper-half 4-space $k_0^{\text{'}}$ , where R³[0] is canonically identified with R³. Two such surfaces F and F′ in $R_{+}^4$ are equivalent if there is an orientation-preserving diffeomorphism f of $R_{+}^4$ sending F to F′, where f is called an equivalence. For a link k₀ in R³, let b be a band system spanning k₀, namely a system of finitely many disjoint oriented bands spanning the link k₀ in R³. The pair (k₀, b) is called a banded link. The surgery link of (k₀, b) is the link obtained from k₀ by surgery along b. Assume that the surgery link of a banded link (k₀, b) is a trivial link κ in R³. Then the band system b is considered as a band system β spanning κ. The pair (κ, β) is called a banded loop system with loop system κ and surgery link k₀. Throughout the paper, the surgery link k₀ will be a union k ∪ o of a link k in question and a trivial link o called an extra trivial link. Here, it is assumed that there is a band sub-system b₁ of the band system b such that b₁ connects to o with just one band b₁ ∈ b₁ for every component o ∈ o and every band spans the link k. Let α₁ be the arc system of the attaching arc α₁ of every band b₁ ∈ b₁ to o ∈ o, and ${\alpha }_1^c$ the complementary arc system of α₁ in o consisting of every complementary arc ${\alpha }_1^c=cl(0\backslash {\alpha }_1)$ . Any disk system d in R³ bounded by the extra trivial link o is called an extra disk system, which is fixed and the argument proceeds. Let δ be a disk system consisting of disjoint disks in R³ with ∂δ = κ, which is called a based disk system for a loop system κ. A ribbon surface-link cl( $F_{-1}^1$ ) in R⁴ is constructed from a banded loop system (κ, β) by taking the surgery of the trivial S²- link

O = ∂(δ[−1, 1]) = δ[−1] ∪ (∂δ)[−1, 1] ∪ δ [1]

along the 1-handle system β[−t, t] in R⁴ for any t with 0 < t < 1. The proper surface ucl $\left(F_0^1\right)=cl(F_{-1}^1)\cap R_{+}^4$ in $R_{+}^4$ is called the upper-closed realizing surface of a banded loop system (κ, β) with surgery link k₀. Note that choices of the based disk systems δ are independent of the equivalences of ucl( $F_0^1$ ) and cl( $F_{-\text{1}}^1$ ) by Horibe-Yanagawa’s lemma [1]. The reason for dealing with a banded loop system (κ, β) rather than a banded link (k₀, b) is because not only can a based disk system δ be chosen freely, but it also makes a band deformation of the band system β easier. Actually, an isotopic deformation of β respecting the arc system α₁ and the loop system κ does not change the ribbon surface-link cl( $F_{-\text{1}}^1$ ) in R⁴ and the proper surface ucl( $F_0^1$ ) in $R_{+}^4$ , up to equivalences.

Let cl( $F_{-\text{1}}^1$ )d be the surface-link in R⁴ obtained from the ribbon surface-link cl( $F_{-\text{1}}^1$ ) by surgery along the 2-handle d[−ε, ε] on cl( $F_{-\text{1}}^1$ ) where 0 < ε < t < 1.

The proper surface $P\left(F_0^1\right)=cl(F_{-1}^1)d\cap R_{+}^4$ in $R_{+}^4$ with $\partial P=F_0^1=k$ is called a proper realizing surface of a banded loop system (κ, β) with surgery link k₀ = k ∪ o. The following theorem is known [1].

Normal form theorem: Every compact oriented proper surface F without closed component in the upper-half 4-space $R_{+}^4$ with ∂F = k in R³ is equivalent to a proper realizing surface $P\left(F_0^1\right)$ in $R_{+}^4$ with $\partial P=\left(F_0^1\right)=k$ of a banded loop system (κ, β) with surgery link k₀ = k + o which is a split sum of k and an extra trivial link o.

The proper realizing surface $P\left(F_0^1\right)$ in $R_{+}^4$ is called a normal form of the proper surface F in $R_{+}^4$ . If the extra trivial link o is taken the empty link, namely $P\left(F_0^1\right)=ucl\left(F_0^1\right)$ , then the proper surface F in $R_{+}^4$ is called a ribbon surface. In the following example, it is observed that there are lots of compact oriented proper surfaces without closed component in $R_{+}^4$ which are not equivalent to any ribbon surface in $R_{+}^4$ .

Example. For every link k in R³, let F ′ be any ribbon surface in with k = ∂F ′. For example, let F ′ be a proper surface in $R_{+}^4$ obtained from a Seifert surface for k in R³ by an interior push into $R_{+}^4$ . Take a connected sum F = F ′#K of F ′ and a nontrivial S²-knot K in R⁴ with non-abelian fundamental group. Then k = ∂F ′ = ∂F. It is shown that F is not equivalent to any ribbon surface in $R_{+}^4$ . The fundamental groups of k, F ′, F, K are denoted as follows.

π(k) = π₁(R³ \ k, x₀), π(F′) = π₁(R⁴ \ F′, x₀), π(F) = π₁(R⁴ \ F, x₀), π(K) = π₁(S⁴ \ K, x₀).

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

A compact oriented proper surface F′ in $R_{+}^4$ is a renewal embedding of a compact oriented proper surface F in $R_{+}^4$ if there is an orientation-preserving surfacediffeomorphism F′ → F keeping the boundary fixed. A renewal embedding F ′of F is boundary-relative if the link k′ = ∂F ′ in R³ is equivalent to the link k = ∂F in R³. The proof of the following theorem is given [2]. In this paper, an alternative proof of this theorem is given from a viewpoint of deformations of a ribbon surface-link in R⁴.

Classical ribbon theorem: Assume that a link k in the 3-space R³ bounds a compact oriented proper surface F without closed component in the upper-half 4space $R_{+}^4$ . Then the link k in R³ bounds a ribbon surface F ′ in $R_{+}^4$ which is a boundary-relative renewal embedding of F.

A link k in R³ is a slice link in the strong sense if k bounds a proper disk system embedded smoothly in $R_{+}^4$ . A link k in R³ is a ribbon link if k bounds a ribbon disk system in $R_{+}^4$ . The following corollary is a special case of Classical ribbon theorem.

Corollary 1: Every slice link in the strong sense in R³ is a ribbon link.

Thus, Classical ribbon theorem solves Slice-Ribbon Problem, [3,4]. The following corollary is obtained from Corollary 1.

Corollary 2: A link k in R³ is a ribbon link if a ribbon link is obtained from the split sum k + o of k and a trivial link o by a band sum of k and every component of o.

The proof of the classical ribbon theorem is done throughout the section 2. An idea of the proof is to consider the 2-handle pair system (D × I, D′ × I) on the ribbon surface-link cl( $F_{-\text{1}}^1$ ) with k + o as the middle-cross sectional link such that is $P\left(F_0^1\right)$ equivalent to a previously given surface F in $R_{+}^4$ , where the 2-handle system D × I is constructed from the band system b₁ and the 2-handle system D′ × I is constructed from the extra disk system d. The interior intersections of (D × I, D′ × I) will be eliminated and (D × I, D′ × I) becomes an O2-handle pair system on a new ribbon surface-link cl( $F_{-\text{1}}^1$ ) with k + o as the middle-cross sectional link obtained by sacrificing equivalences. Then $P\left(F_0^1\right)$ is a ribbon surface that is a boundary-relative renewal embedding of F, which will complete the proof.

Throughout this section, the proof of the classical ribbon theorem is done. Let F be a compact oriented proper surface without closed component in $R_{+}^4$ , and ∂F = k a link in R³. By the normal form theorem, there is a banded loop system (κ, β) with surgery link k₀ = k + o such that $P\left(F_0^1\right)$ is equivalent to F. The extra trivial link o is uniquely specified by the banded loop system (κ, β), which is the union of the arc system α₁ and the complementary arc system ${\alpha }_1^c$ , where the interior of α₁ transversely meets the interior of a based disk system δ with finite points and is disjoint from the based loop system κ and ${\alpha }_1^c$ belongs to the loop system κ.

A renewal embedding of a banded loop system (κ, β) with surgery link k₀ = k ∪ o is a banded loop system (κ′, β′) with surgery link such that there is a homeomorphism κ ∪ β →κ′ ∪ β′ with restrictios κ → κ′ and β → β′ orientation preserved.

The following observation is directly obtained by definition.

(2.1) If a banded loop system system (κ′, β′) with surgery link k′ ∪ o is a renewal embedding of a banded loop system (κ, β) with surgery link k ∪ o, then the upper-closed realizing surface ucl ${\left(F_0^1\right)}^{\text{'}}$ constructed from (κ′, β′) is a renewal embedding of the upper-closed realizing surface ucl ${\left(F_0^1\right)}^{\text{'}}$ constructed from (κ, β) such that $\partial ucl=\left(F_0^1\right)=k\cup o$ and $\partial ucl=\left(F_0^1\right)\text{'}=k^{\prime }\cup o$ .

A transversal arc of a band spanning a link is a simple proper arc in the band which is parallel to an attaching arc. For a band b ∈ b transversely meeting the interior of an extra disk d ∈ d, the d-arc system of b is the arc system d(b) of every transversal arc a of b in the interior of d. The d-arc system of a band system b is the collection d(b) of d(b) for every d ∈ d and every b ∈ b. For a based disk δ ∈ δ, the δ-arc system of a band β ∈ β is the arc system δ(β) of every transversal arc c of β in the interior of δ. The δ-arc system of β is the collection δ(β) of δ(β) for every δ ∈ δ and every β ∈ β. A normal proper arc in the extra disk system d is a simple proper arc in d with the endpoints in the interior of the arc system α₁. The following assertion is shown.

(2.2) By isotopic deformations in R³, the banded loop system (κ, β) in R³ with surgery link k₀ = k + o is deformed so that a based disk system δ transversely meets the extra disk system d with interior simple arcs or normal proper arcs in d except for the complementary arc system ${\alpha }_1^c$ .

Proof of (2.2). By transverse regularity, the intersection d ∩ δ for every d ∈ d and every δ ∈ δ is made interior simple loops, interior simple arcs, clasp type simple arcs or simple proper arcs in d except for the complementary arc system ${\alpha }_1^c$ . A simple loop is changed into a normal proper arc by a pushing out deformation to α₁, Figure 1(1). A clasp type simple arc is changed into a simple proper arc by moving out the interior point to α₁, Figure 1(2). A simple proper arc which is not normal is also changed into a normal proper arc by a pushing out deformation of the arc system of δ meeting a boundary collar of ${\alpha }_1^c$ in d, Figure 1(3). Thus, a deformed based disk system δ transversely meets d with interior simple arcs or normal proper arcs in d except for the complementary arc system ${\alpha }_1^c$ . This completes the proof of (2.2).

In the proof of (2.2), there is no need to worry about the intersection of the based disk system δ and the interior of the arc system α₁ in R³, because δ is taken and deformed in the 3-spaces R³[±1] and the extra disk system d and the arc system α₁ are taken and fixed in the 3-space R³[0] for the ribbon surface-link cl( $F_{-\text{1}}^1$ ) in R⁴. The following operation gives a standard renewal embedding of a banded loop system.

Band move operation: In the banded loop system (κ, β) with surgery link k₀ = k ∪ o, assume that there is a transversal arc c of a band β ∈ β in the interior of an extra disk d ∈ d and there is a simple path ω in d from a point p ∈ c to an interior point of the arc ${\alpha }_1^C=\partial d\cap {\alpha }_1^C$ which avoids meeting β other than c. Let β′be a band obtained from β by sliding the arc c off the disk d along the path ω. Replace the banded loop system (κ, β) with the banded loop system (κ, β′) obtained by replacing β with β′, Figure 2.

By this operation, the new banded loop system (κ, β′) is a renewal embedding of the original banded loop system (κ, β) and has as the surgery link a new union $k_0^{\text{'}}$ of the same links k and o, not necessarily the split sum k + o, because the band system β′is isotopic to β if ${\alpha }_1^c$ is forgotten. In the final stage of this paper, the surgery link $k_0^{\text{'}}=k+o$ will have k ∩ d = ∅, so that $k_0^{\text{'}}$ will be the split sum k + o, because o = ∂d.

To achieve a situation where the Band Move Operation can be applied, the follwing concept is needed. A splitting of a banded loop system (κ, β) is a banded loop system (k*, β*) such that a based disk system δ* for κ* is obtained from a based disk system δ for κ by splitting along a disjoint proper arc system γ in δ not meeting o and β, and the band system β* is obtained from the band system β by adding the band system β_γ thickening γ. This splitting operation comes from Fission-Fusion move of a banded loop system, [5,6]. After some splittings of a banded loop system, a situation where the Band Move Operation can be applied is realized by a replacement of the based disk system and an isotopic deformation of the band system.

The following assertion is used.

(2.3) If there is a splitting (k*, β*) of a banded loop system (κ, β) with surgery knot k₀ a union of k and o such that κ* does not meet the interior of the extra disk system d, then there is a renewal embedding (k′, β′) of (k, β) such that (k′, β′) does not meet the interior of d and has the surgery knot $k_0^{\text{'}}=k+o$ .

Proof of (2.3). Since κ* does not meet the interior of d, there is a based disk system δ* for κ* not meeting the interior of d. The band system β* transversely meets the interior of d with transverse arc system A. Let ${\delta }_1^{*}$ be the sub-system of δ* containing the complementary arc system ${\alpha }_1^c$ in the boundary, and $N({\alpha }_1^C)$ a boundary collar disk system of ${\alpha }_1^c$ in ${\delta }_1^{*}$ . The Band Move Operation means that the band system β* is deformed so that the transverse arc system A moves from the interior of d into the interior of $N({\alpha }_1^C)$ . Then by changing the band system β_γ back into the arc system γ, the banded loop system (k*, β*) is changed back to a pair (k′, β′), where the loop system κ′bounds an immersed disk system δ′obtained from the based disk system δ by moving a transverse arc system of β_γ into the interior of $N({\alpha }_1^C)$ . The immersed disk system δ′is deformed into a disjoint disk system by repeatedly pulling the band in β_γ connecting to an outer most disk of δ* or passing the outer most disk of δ* through $N({\alpha }_1^C)$ in order to eliminate the nearest transverse arc of the band. This means that the loop system κ′is a trivial link and (k′, β′) is a banded loop system. Thus, there is a renewal embedding (k′, β′) of (k, β) which does not meet the interior of d. The surgery knot $k_0^{\text{'}}$ is necessarily the split sum k + o since ∂d = o. This completes the proof of (2.3).

By using (2.2) and (2.3), the following assertion is shown.

(2.4) There is a renewal embedding (κ′, β′) of every banded loop system (κ, β) in R³ with surgery link k₀ = k + o such that (k′, β′) does not meet the interior of d and has the surgery knot $k_0^{\text{'}}=k+o$ .

Proof of (2.4). By (2.2), a based disk system δ of κ transversely meets the extra disk system d with interior simple arcs or normal proper arcs in d except for the complementary arc system ${\alpha }_1^c$ . Let A be the interior arc system which is made disjoint from β by isotopic deformations of β respecting the arc system α₁ and the loop system κ. By taking a splitting of (κ, β) along A, it is considered that the based disk system δ transversely meets d only with normal proper arcs in d except for ${\alpha }_1^c$ . Then κ does not meet the interior of the extra disk system d. By (2.3), the proof of (2.4) is completed.

Let (κ, β) be a banded loop system a banded loop system with surgery link k₀ = k + o such that $P\left(F_0^1\right)$ is equivalent to F. By (2.4), there is a renewal embedding (κ′, β′) such that (κ′, β′) does not meet the interior of the extra disk system d, and has the surgery link k + o. Let b′ be the band system dual to the band system β′, and ${b^{\prime }}_1$ the band sub-system of b′ such that ${b^{\prime }}_1$ connects to o with just one band for every component of o. Let ${b^{\prime }}_2=b^{\prime }\backslash {b^{\prime }}_1$ . Since ${b^{\prime }}_1$ does not meet the interior of d, the surgery link of the banded link $(k+o,b^{\prime })$ is equivalent to the link k and the upper-closed realizing surface ucl $\left(F_0^1\right)\text{'}$ of the banded link $(k,{b^{\prime }}_2)$ is equivalent to the proper realizing surface $P\left(F_0^1\right)\text{'}$ of (κ′, β′) which is a ribbon surface in $R_{+}^4$ and is a renewal embedding of the proper realizing surface $P\left(F_0^1\right)$ of the banded loop system (κ, β) with the surgery link k + o. Since $P\left(F_0^1\right)$ is equivalent to F in $R_{+}^4$ and ucl ${\left(F_0^1\right)}^{\prime }$ is a ribbon surface with $\partial ucl\left(F_0^1\right)\text{'}=\partial F=k$ , there is a boundary-relative renewal embedding from ucl ${\left(F_0^1\right)}^{\prime }$ to F. This completes the proof of the classical ribbon theorem.

The author thanks to a referee for suggesting the content for making easier content. This paper is motivated by T. Shibuya’s comments pointing out insufficient explanation on Lemma 2.3 in [2] (Corollary 2 in this paper). This work was partly supported by JSPS KAKENHI Grant Number JP21H00978 and MEXT Promotion of Distinctive Joint Research Center Program JPMXP0723833165.

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