Knot Floer homology and strongly homotopy-ribbon concordances

TL;DR

This paper proves that strongly homotopy-ribbon concordances induce injective maps on knot Floer homology, revealing new structural properties of these concordances in knot theory.

Contribution

It establishes the injectivity of the map on knot Floer homology induced by strongly homotopy-ribbon concordances, a novel result in the study of knot invariants.

Findings

Injective maps on knot Floer homology from strongly homotopy-ribbon concordances

New insights into the structure of knot concordances

Advancement in understanding knot invariants and their relations

Abstract

We prove that the map on knot Floer homology induced by a strongly homotopy-ribbon concordance is injective.

Figures

Equations

$${\pi }_1(K_0)\to {\pi }_1(C)\text{is an injection,}\text{and}{\pi }_1(K_1)\to {\pi }_1(C)\text{is a surjection.}$$

$$\{\text{ribbon concordances}\}\subseteq \{\text{strongly homotopy-ribbon concordances}\}$$

$$\subseteq \{\text{homotopy-ribbon concordances}\}⊊\{\text{concordances}\}.$$

$$\widehat{HFK}(K)=\underset{i,j\in \mathbb{Z}}{⨁}{\widehat{HFK}}_i(K,j).$$

$${\hat{F}}_C:\widehat{HFK}(K_0)\to \widehat{HFK}(K_1),$$

$${CFK}^{\infty }(K),$$

$$F_C:{CFK}^{\infty }(K_0)\to {CFK}^{\infty }(K_1).$$

$$F_C:{CFK}^{\infty }(K_0)\to {CFK}^{\infty }(K_1)$$

$${\hat{F}}_C:\widehat{HFK}(K_0)\to \widehat{HFK}(K_1)$$

$$F_{\overline{C}}\circ F_C\simeq {\mathrm{id}}_{{CFK}^{\infty }(K_0)}.$$

$$g_3(K)=\max \left\{j:\widehat{HFK}(K,j)\ne \{0\}\right\}.$$

$$g_3(K_0)\le g_3(K_1).$$

$$k^{\prime }=\{0\}\times S^1,$$

$$\partial \mathbf{x}=\sum _{\mathbf{y}\in {\mathbb{T}}_{\alpha }\cap {\mathbb{T}}_{\beta }}\sum _{\begin{array}{c}\varphi \in {\pi }_2(\mathbf{x},\mathbf{y})\\ \mu (\varphi )=1\end{array}}\#\left(\mathcal{M}(\varphi )/\mathbb{R}\right)\cdot u^{n_{\mathbf{w}}(\varphi )}{v}^{n_{\mathbf{z}}(\varphi )}\cdot \mathbf{y}.$$

$$F_{W,\mathcal{F},\mathfrak{s}}:{\mathcal{C}\mathcal{F}\mathcal{K}}^{-}(Y_1,{\mathbb{K}}_1,\mathfrak{s}{|}_{Y_1})\to {\mathcal{C}\mathcal{F}\mathcal{K}}^{-}(Y_2,{\mathbb{K}}_2,\mathfrak{s}{|}_{Y_2}).$$

$$F_{\mathcal{C}}\simeq F_{{\mathcal{C}}^{\prime }},$$

$$F_{N(S),\mathcal{D},{\mathfrak{t}}_0},F_{N(S),{\mathcal{D}}^{\prime },{\mathfrak{t}}_0}:{\mathcal{C}\mathcal{F}\mathcal{K}}^{-}(\varnothing )\to {\mathcal{C}\mathcal{F}\mathcal{K}}^{-}(S^1\times S^2,\mathbb{O},{\mathfrak{s}}_0),$$

$${\mathcal{C}\mathcal{F}\mathcal{K}}^{-}(S^1\times S^2,\mathbb{O},{\mathfrak{s}}_0)\cong \left(({\mathbb{F}}_2{)}_{(-\frac{1}{2},-\frac{1}{2})}\oplus ({\mathbb{F}}_2{)}_{(\frac{1}{2},\frac{1}{2})}\right){\otimes }_{{\mathbb{F}}_2}{\mathbb{F}}_2[u,v],$$

$$F_{\overline{\mathcal{C}}}\circ F_{\mathcal{C}}\simeq F_{\overline{\mathcal{C}}\circ \mathcal{C}},$$

Full text

MM was supported by NSF grant DGE-1656466. IZ was supported by NSF grant DMS-1703685.

1. Introduction

A concordance $C$ from $K_{0}$ to $K_{1}$ is a smoothly embedded annulus in $[0,1]\times S^{3}$ such that $\partial C=-K_{0}\cup K_{1}$. A ribbon concordance from $K_{0}$ to $K_{1}$ is a concordance $C$ such that the projection of $C$ to $[0,1]$ is Morse and has only index 0 and 1 critical points.

If $C$ is a concordance from $K_{0}$ to $K_{1}$, write $\pi_{1}(C)$, $\pi_{1}(K_{0})$ and $\pi_{1}(K_{1})$ for the fundamental groups of the complements. A homotopy-ribbon concordance is a concordance such that

[TABLE]

The definition is justified by a result of Gordon [Gordon]*Lemma 3.1 which implies that ribbon concordances are homotopy-ribbon.

In fact, Gordon showed something stronger: if $C$ is a ribbon concordance, then its complement can be built by attaching only 1-handles and 2-handles to $S^{3}\setminus K_{0}$.

Justified by Gordon’s observation, one can make the following definition:

Definition 1.1**.**

A strongly homotopy-ribbon concordance $C$ from $K_{0}$ to $K_{1}$ is a concordance such that the complement of $C$ can be constructed by attaching 4-dimensional 1-handles and 2-handles to $S^{3}\setminus K_{0}$.

Strongly homotopy-ribbon concordance has been studied by, e.g., Cochran [CochranRibbon] and Larson-Meier [LarsonFiberedRibbon].

By definition, we have

[TABLE]

[TABLE]

It is unknown whether the first two inclusions are strict.

In this paper, we show that knot Floer homology obstructs strongly homotopy-ribbon concordance. Our argument is based on a result of the second author [ZemRibbon] for ribbon concordance.

1.1. Knot Floer homology and strongly homotopy-ribbon concordances

Knot Floer homology is an invariant of knots in 3-manifolds discovered independently by Ozsváth and Szabó [OSKnots] and Rasmussen [RasmussenKnots].

If $K$ is a knot in $S^{3}$, the simplest version of knot Floer homology is the hat version, which is a bigraded $\mathbb{F}_{2}$ vector space:

[TABLE]

The index $i$ denotes the Maslov grading, and $j$ the Alexander grading.

Associated to a concordance $C$ from $K_{0}$ to $K_{1}$, Juhász and Marengon [JMConcordance] described a bigraded homomorphism

[TABLE]

which is well defined up to bigraded automorphisms of $\widehat{\mathit{HFK}}(K_{0})$ and $\widehat{\mathit{HFK}}(K_{1})$. The ambiguity can be eliminated by picking a decoration of the concordance $C$ consisting of a pair of disjoint arcs on $C$ running from $K_{0}$ to $K_{1}$. Their construction used a more general construction of link cobordism maps described by Juhász  [JCob] for link Floer homology.

There is a more general version of knot Floer homology, the infinity version, denoted

[TABLE]

which is a $\mathbb{Z}\oplus\mathbb{Z}$-filtered, bigraded chain complex over the ring $\mathbb{F}_{2}[U,U^{-1}]$. We refer the reader to [Man-Intro-HFK] for a nice introduction to the numerous versions of knot Floer homology.

The second author [ZemCFLTQFT] constructed a bigraded, filtered, $\mathbb{F}_{2}[U,U^{-1}]$-equivariant cobordism map

[TABLE]

The second author showed that the cobordism map induced by a ribbon concordance is left invertible [ZemRibbon]*Theorem 1.1. Our main result is that the same is true for strongly homotopy-ribbon concordances:

Theorem 1.2**.**

If $C$ is a strongly homotopy-ribbon concordance from $K_{0}$ to $K_{1}$, then the induced map

[TABLE]

admits a left inverse. In particular, the induced map

[TABLE]

is an injection.

Our proof uses a similar doubling trick as [ZemRibbon]: if $\overline{C}$ denotes the concordance from $K_{1}$ to $K_{0}$ obtained by turning around and reversing the orientation of $C$, then we will show that

[TABLE]

Remark 1.3*.*

In fact, the proof of Theorem 1.2 works for strongly homotopy-ribbon concordances in an arbitrary $h$-cobordism of $S^{3}$. See Remark 4.3. This is potentially interesting because one may allow a homotopy-ribbon disk to live in any homotopy $B^{4}$. See, e.g.,  [CassonLoop].

Ozsváth and Szabó [OSgenusbounds] proved that knot Floer homology detects the Seifert genus:

[TABLE]

Combined with our Theorem 1.2, we obtain the following generalization of [ZemRibbon]*Theorem 1.5:

Corollary 1.4**.**

If there is a strongly homotopy-ribbon concordance from $K_{0}$ to $K_{1}$, then

[TABLE]

By Remark 1.3, Corollary 1.4 applies whenever there is a strongly homotopy-ribbon concordance from $K_{0}$ to $K_{1}$ in any $h$-cobordism of $S^{3}$.

2. Strongly homotopy-ribbon concordances

In this section, we demonstrate that ribbon concordances are strongly homotopy-ribbon concordances. This is described in the proof of [Gordon]*Lemma 3.1, however we give a proof for the reader’s convenience.

Lemma 2.1**.**

If $C$ is a ribbon concordance from $K_{0}$ to $K_{1}$, then $C$ is a strongly homotopy-ribbon concordance.

Proof.

Suppose $C$ is a ribbon concordance. For notational simplicity, we restrict to the case when the concordance has a single 0-handle and 1-handle. The concordance $C$ has a movie consisting of a single birth followed by a saddle, which is shown in Figure 2.1. The birth adds an unknot $U$, and the saddle adds a band $B$.

In Figure 2.2, another movie of a concordance is shown, featuring a 4-dimensional 1-handle and 2-handle, but no births or saddles. Write $S^{0}$ for the attaching sphere of the 1-handle, and $k$ for the attaching sphere of the 2-handle. In Figure 2.2, we use the dotted unknot notation for 1-handle attachment. We note that the transition through frames 3, 4 and 5 in Figure 2.1 is achieved by a sequence of isotopies (we remind the reader that handlesliding the knot $K_{0}$ over the knot $k$ or the dotted unknot corresponding to $S^{0}$ corresponds to an isotopy of the knot $K_{0}$ in the 3-manifold obtained by surgering $S^{3}$ on $S^{0}$ and $k$).

We claim that the movies shown in Figure 2.1 and Figure 2.2 can be related by a set of 4-dimensional movie moves. This is achieved by taking the movie in Figure 2.1 and adjoining a canceling pair of 4-dimensional 1- and 2-handles. Write $k$ for the 2-handle, which we assume is a 0-framed unknot. After sliding $U$ over the dotted unknot forming the 1-handle, and sliding the band $B$ over $k$ (repeatedly), the band $B$ now cancels the birth which added $U$, and we are left with the movie in Figure 2.2. ∎

3. A handle decomposition of the doubled concordance

We now describe a handle decomposition of the complement of the doubled concordance $\overline{C}\circ C$, when $C$ is strongly homotopy-ribbon.

Lemma 3.1**.**

Suppose that $C$ is a strongly homotopy-ribbon concordance from $K_{0}$ to $K_{1}$, and $C$ has a handle decomposition consisting of $n$ 1-handles, attached along 0-spheres $S^{0}_{1},\dots,S^{0}_{n}$, and $n$ 2-handles, attached along framed knots $k_{1},\dots,k_{n}$. A handle decomposition for the complement of $\overline{C}\circ C\subseteq[0,1]\times S^{3}$ is as follows:

1. (1)

$n$* 1-handles, attached along $S^{0}_{1},\dots,S_{n}^{0}$.* 2. (2)

$n$* 2-handles, attached along $k_{1},\dots,k_{n}$.* 3. (3)

$n$* 2-handles, attached along 0-framed meridians $k_{1}^{\prime},\dots,k_{n}^{\prime}$ of $k_{1},\dots,k_{n}$.* 4. (4)

$n$* 3-handles, attached along 2-spheres obtained by taking the belt spheres of the 1-handles, removing disks where they intersect $k_{1},\dots,k_{n}$, and adding a collection of pairwise disjoint annuli which connect the punctures to the 0-framed meridians $k_{1}^{\prime},\dots,k_{n}^{\prime}$, and are contained in a neighborhood of the knots $k_{1},\dots,k_{n}$. See Figure 3.1 for an example.*

Proof.

The proof is standard, though it is usually presented in the context of closed $4$-manifolds, where the $3$-handles need not be described explicitly. See [GompfStipsicz]*Example 4.6.3. For notational simplicity, we focus on the case when the complement of $C$ has a handle decomposition with one 1-handle and one 2-handle. The general case is a straightforward modification.

Let $S^{0}$ denote the attaching 0-sphere of the 1-handle. Write $S^{3}(S^{0})$ for the 3-manifold obtained from $S^{3}$ by surgery along $S^{0}$. (In general, we write $Y(j)$ for the manifold obtained by surgery on a framed sphere $j$ in a 3-manifold $Y$). Note that $S^{3}(S^{0})\cong S^{1}\times S^{2}$. Let $k\subseteq S^{3}(S^{0})$ be the knot along which the 2-handle is attached. The framing of $k$ induces an identification of a neighborhood of $k$ with $S^{1}\times D^{2}$. Write $N$ for this neighborhood. The surgered manifold $S^{3}(S^{0})(k)$ is obtained from $S^{3}(S^{0})$ by removing $N=S^{1}\times D^{2}$ and gluing in $N^{\prime}=D^{2}\times S^{1}$. The belt sphere of the 2-handle is the knot

[TABLE]

which is framed using the same identification of $N^{\prime}$ as $D^{2}\times S^{1}$.

The surgered manifold $S^{3}(S^{0})(k)(k^{\prime})$ is canonically diffeomorphic to $S^{3}(S^{0})$. The attaching sphere of the 3-handle in $S^{3}(S^{0})(k)(k^{\prime})$ is easy to see with this description: it coincides with the belt sphere of 1-handle, under the identification of $S^{3}(S^{0})(k)(k^{\prime})$ with $S^{3}(S^{0})$.

We consider now the image of the attaching 2-sphere of the 3-handle in $S^{3}(S^{0})(k)$. Outside of $N$ (the neighborhood of $k$), the attaching sphere of the 3-handle coincides with the belt sphere of the 1-handle attached along $S^{0}$. Inside of $N^{\prime}$, it is equal to a collection of annuli of the form $a\times S^{1}$, where $a\subseteq D^{2}$ is a radial arc extending from $\{0\}$ to $\partial D^{2}$. There is one annulus for each intersection point of $k$ with the belt sphere of the 1-handle.

Next, we push $k^{\prime}$ out of the neighborhood $N^{\prime}$, so that we obtain a more standard Kirby calculus picture. When we do this, it is straightforward to check that $k^{\prime}$ becomes a 0-framed meridian, and that the annuli are as described. ∎

4. Proof of the main theorem

Our proof follows from a “sphere tubing” property of the link Floer TQFT from [ZemCFLTQFT], which we state below in Lemma 4.2. We first review some background about the link Floer TQFT.

The link Floer TQFT uses the following decorated link cobordism category, originally described by Juhász [JCob]:

Definition 4.1**.**

1. (1)

A multi-based link $\mathbb{L}=(L,\bm{\mathbf{w}},\bm{\mathbf{z}})$ in a 3-manifold $Y$ is an orientated link $L$, containing two finite collections of basepoints, $\bm{\mathbf{w}}$ and $\bm{\mathbf{z}}$, such that each component of $L$ contains at least one $\bm{\mathbf{w}}$ basepoint and one $\bm{\mathbf{z}}$ basepoint. Furthermore, as one traverses $L$, the basepoints alternate between $\bm{\mathbf{w}}$ and $\bm{\mathbf{z}}$. 2. (2)

A decorated link cobordism $(W,\mathcal{F})$ from $(Y_{1},\mathbb{L}_{1})$ to $(Y_{2},\mathbb{L}_{2})$ consists of a cobordism $W$ from $Y_{1}$ to $Y_{2}$, as well as a decorated surface $\mathcal{F}=(\Sigma,\mathcal{A})$, as follows. The surface $\Sigma$ is a properly and smoothly embedded surface in $W$, such that $\partial\Sigma=-L_{1}\cup L_{2}$. Furthermore, $\mathcal{A}\subseteq\Sigma$ is a properly embedded 1-manifold which divides $\Sigma$ into two subsurfaces, $\Sigma_{\bm{\mathbf{w}}}$ and $\Sigma_{\bm{\mathbf{z}}}$, which satisfy $\bm{\mathbf{w}}_{1},\bm{\mathbf{w}}_{2}\subseteq\Sigma_{\bm{\mathbf{w}}}$ and $\bm{\mathbf{z}}_{1},\bm{\mathbf{z}}_{2}\subseteq\Sigma_{\bm{\mathbf{z}}}$.

The link Floer TQFT from [ZemCFLTQFT] assigns cobordism maps to a slightly different version of knot Floer homology than we described in the introduction. If $(Y,\mathbb{K})$ is a multi-pointed knot and $\mathfrak{s}\in\operatorname{{Spin}}^{c}(Y)$, we consider a version of knot Floer homology which we denote by $\mathcal{C\!F\!K}^{-}(Y,\mathbb{K},\mathfrak{s})$. Explicitly, if $(\Sigma,\bm{\mathbf{\alpha}},\bm{\mathbf{\beta}},\bm{\mathbf{w}},\bm{\mathbf{z}})$ is a Heegaard diagram, then $\mathcal{C\!F\!K}^{-}(Y,\mathbb{K},\mathfrak{s})$ is freely generated over the 2-variable polynomial ring $\mathbb{F}_{2}[u,v]$ by intersection points $\bm{\mathbf{x}}\in\mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta}$ with $\mathfrak{s}_{\bm{\mathbf{w}}}(\bm{\mathbf{x}})=\mathfrak{s}$. The differential counts Maslov index 1 holomorphic disks in the symmetric product $\operatorname{{Sym}}^{n}(\Sigma)$ (where $n=g(\Sigma)+|\bm{\mathbf{w}}|-1$) via the formula

[TABLE]

In the above expression, $n_{\bm{\mathbf{w}}}(\phi)$ denotes the total multiplicity of a class $\phi$ over the $\bm{\mathbf{w}}$ basepoints, and $n_{\bm{\mathbf{z}}}(\phi)$ is defined similarly.

The chain complex $\mathcal{C\!F\!K}^{-}(Y,\mathbb{K},\mathfrak{s})$ has a $\mathbb{Z}\oplus\mathbb{Z}$ filtration given by powers of $u$ and $v$. If $c_{1}(\mathfrak{s})$ is torsion and $K$ is null-homologous, there are also three gradings: two Maslov gradings, $\operatorname{{gr}}_{\bm{\mathbf{w}}}$ and $\operatorname{{gr}}_{\bm{\mathbf{z}}}$, as well as an Alexander grading $A$. The gradings are related by the equation $A=\tfrac{1}{2}(\operatorname{{gr}}_{\bm{\mathbf{w}}}-\operatorname{{gr}}_{\bm{\mathbf{z}}})$. The actions of $u$ and $v$ have $(\operatorname{{gr}}_{\bm{\mathbf{w}}},\operatorname{{gr}}_{\bm{\mathbf{z}}})$-bigradings of $(-2,0)$ and $(0,-2)$, respectively.

For a knot $K$ in $S^{3}$, the previously described knot Floer complex $\mathit{CFK}^{\infty}(K)$ is obtained by decorating $K$ with two basepoints to form $\mathbb{K}$, inverting $u$ and $v$ in $\mathcal{C\!F\!K}^{-}(S^{3},\mathbb{K})$, and taking the subcomplex in Alexander grading zero. The action of $U$ on $\mathit{CFK}^{\infty}$ corresponds to the action of the product $uv$ on $\mathcal{C\!F\!K}^{-}$. The differential drops both Maslov gradings by 1, and preserves the Alexander grading.

If $(W,\mathcal{F})$ is a decorated link cobordism from $(Y_{1},\mathbb{K}_{1})$ to $(Y_{2},\mathbb{K}_{2})$, and $\mathfrak{s}\in\operatorname{{Spin}}^{c}(W)$, then the construction from [ZemCFLTQFT] gives a map

[TABLE]

When $W$ and $\mathfrak{s}$ are understood from context, we will simply write $F_{\mathcal{F}}$ for the cobordism map.

If $(W,\mathcal{F})$ is a concordance in $[0,1]\times S^{3}$, decorated with a pair of arcs, the link cobordism maps preserve the Maslov and Alexander gradings by [ZemAbsoluteGradings]*Theorem 1.4. Correspondingly, the cobordism maps restrict to give maps on $\mathit{CFK}^{\infty}$ (the subcomplex in Alexander grading 0), as described in the introduction.

Before we prove Theorem 1.2, we recall the key lemma proven by the second author [ZemRibbon]*Lemma 3.1:

Lemma 4.2**.**

Suppose that $\mathcal{C}=(C,\mathcal{A})$ is a decorated concordance in $[0,1]\times S^{3}$, and $S\subseteq[0,1]\times S^{3}$ is a smoothly embedded 2-sphere in the complement of $C$. Let $\mathcal{C}^{\prime}=(C^{\prime},\mathcal{A}^{\prime})$ be a decorated concordance obtained by connecting $C$ and $S$ with a tube, away from the decorations. Then

[TABLE]

as maps from $\mathit{CFK}^{\infty}(K_{0})$ to $\mathit{CFK}^{\infty}(K_{1})$.

Proof.

Since $\mathit{CFK}^{\infty}$ is obtained by inverting $u$ and $v$ in $\mathcal{C\!F\!K}^{-}$, and then taking the subcomplex in Alexander grading zero, it suffices to prove the analogous claim on $\mathcal{C\!F\!K}^{-}$.

We factor the cobordism map $F_{\mathcal{C}}$ through a closed neighborhood $N(S)$ of the sphere $S$. Here we use the fact that the tube meets $\partial N(S)\cong S^{1}\times S^{2}$ in an unknot. Write $\mathcal{D}$ and $\mathcal{D}^{\prime}$ for intersections of $(C,\mathcal{A})$ and $(C^{\prime},\mathcal{A}^{\prime})$ with $N(S)$. After an isotopy, we may assume that both $\mathcal{D}$ and $\mathcal{D}^{\prime}$ are disks which intersect the dividing set in a single arc. We obtain cobordism maps

[TABLE]

where $\mathbb{O}$ is a doubly pointed unknot, $\mathfrak{t}_{0}$ is the $\operatorname{{Spin}}^{c}$ structure on $N(S)$ which evaluates trivially on $S$, and $\mathfrak{s}_{0}$ is its restriction to $\partial N(S)$. (Note that by definition, $\mathcal{C\!F\!K}^{-}(\emptyset):=\mathbb{F}_{2}[u,v]$, with vanishing differential). Since $\mathcal{C}$ and $\mathcal{C}^{\prime}$ coincide outside of $N(S)$, using the composition law, it suffices to show that $F_{N(S),\mathcal{D},\mathfrak{t}_{0}}\simeq F_{N(S),\mathcal{D}^{\prime},\mathfrak{t}_{0}}$.

There is a chain isomorphism

[TABLE]

where we view the latter complex as having vanishing differential, and $(\mathbb{F}_{2})_{(p,q)}$ denotes the rank 1 vector space over $\mathbb{F}_{2}$, concentrated in $(\operatorname{{gr}}_{\bm{\mathbf{w}}},\operatorname{{gr}}_{\bm{\mathbf{z}}})$-bigrading $(p,q)$.

The grading change formula from [ZemAbsoluteGradings]*Theorem 1.4 implies that $F_{N(S),\mathcal{D},\mathfrak{t}_{0}}$ and $F_{N(S),\mathcal{D}^{\prime},\mathfrak{t}_{0}}$ both have $(\operatorname{{gr}}_{\bm{\mathbf{w}}},\operatorname{{gr}}_{\bm{\mathbf{z}}})$-bigrading $\left(-\tfrac{1}{2},-\tfrac{1}{2}\right)$. Noting that $u$ and $v$ have $(\operatorname{{gr}}_{\bm{\mathbf{w}}},\operatorname{{gr}}_{\bm{\mathbf{z}}})$ bigrading $(-2,0)$ and $(0,-2)$, from Equation (1) we find that $\mathcal{C\!F\!K}^{-}(S^{1}\times S^{2},\mathbb{O},\mathfrak{s}_{0})$ has rank 1 in $(\operatorname{{gr}}_{\bm{\mathbf{w}}},\operatorname{{gr}}_{\bm{\mathbf{z}}})$-bigrading $\left(-\tfrac{1}{2},-\tfrac{1}{2}\right)$. Hence, if $F_{N(S),\mathcal{D},\mathfrak{t}_{0}}$ and $F_{N(S),\mathcal{D}^{\prime},\mathfrak{t}_{0}}$ are both non-zero, then they must be equal. By capping off with a 3-handle and a 4-handle, we obtain a 2-knot in $S^{4}$ decorated with a single dividing curve. The corresponding cobordism map is non-zero by [ZemRibbon]*Lemma 4.1, so we conclude $F_{N(S),\mathcal{D},\mathfrak{t}_{0}}$ and $F_{N(S),\mathcal{D}^{\prime},\mathfrak{t}_{0}}$ are both non-zero, and hence equal. ∎

Proof of Theorem 1.2.

See Figure 4.1 for a schematic example of this proof.

Let $C$ be a strongly homotopy-ribbon concordance from $K_{0}$ to $K_{1}$, and let $\overline{C}$ be the concordance from $K_{1}$ to $K_{0}$ obtained by turning around and reversing the orientation of $C$. Lemma 3.1 gives a handle decomposition of the complement of $\overline{C}\circ C$. Let $\mathcal{C}$ be $C$, decorated with a pair of dividing arcs, and let $\overline{\mathcal{C}}$ denote $\overline{C}$, decorated with the mirrored decoration.

By the composition law for link cobordisms,

[TABLE]

so it suffices to show that $F_{\overline{\mathcal{C}}\circ\mathcal{C}}\simeq\operatorname{{id}}_{\mathit{CFK}^{\infty}(K_{0})}.$

We use the handle decomposition described in Lemma 3.1. Note that if we ignore the knot $K_{0}$, then the handle decomposition describes $[0,1]\times S^{3}$. The knot $K_{0}$ may of course be tangled with the 2-handle attaching spheres $k_{1},\dots,k_{n}$.

The idea is as follows. We “pull” the knot $K_{0}$ away from the attaching spheres for the 4-dimensional handles. Of course to achieve this, we must pull $K_{0}$ through the framed knots $k_{1},\dots,k_{n}$ (i.e. we must perform a sequence of crossing changes of $K_{0}$ with the knots $k_{1},\dots,k_{n}$). To achieve a crossing change of $K_{0}$ with a framed knot $k_{i}$, we can isotope a segment of $K_{0}$ parallel to $k_{i}$ until we reach the 0-framed meridian $k_{i}^{\prime}$, and then “handleslide” $K_{0}$ across $k_{i}$. Given the explicit description of the attaching spheres of the 3-handles stated in Lemma 3.1, this isotopy and handleslide can be performed in the complement of the attaching 2-spheres of the 3-handles.

Note that handlesliding $K_{0}$ across $k_{i}^{\prime}$ is not actually a handleslide of the described $4$-manifold, as $K_{0}$ is not a handle attaching circle. Rather, this move changes the concordance $\overline{C}\circ C$. Indeed the two concordances are related by tubing to $\overline{C}\circ C$ the 2-sphere $S_{i}$, which is isotopic to the union of a small Seifert disk of unknot $k_{i}^{\prime}$ in $S^{3}$ together with a core of the 2-handle attached along $k_{i}^{\prime}$. (That is, $S_{i}$ consists of the cocore of the $2$-handle attached along $k_{i}$ and the core of the $2$-handle attaching along $k_{i}^{\prime}$, glued along their common boundary.) This move does not change the cobordism map by Lemma 4.2.

After untangling $K_{0}$ from all of the attaching spheres of the 4-dimensional handles, we are left with the identity knot cobordism $([0,1]\times S^{3},[0,1]\times K_{0})$, completing the proof. ∎

Remark 4.3*.*

The above proof works for strongly homotopy-ribbon concordances in an $h$-cobordism $W$ from $S^{3}$ to itself. Upon disentangling $K_{0}$ from the attaching spheres for the complement of $\overline{C}\circ C$, we are left with the complement of the concordance $[0,1]\times K_{0}$ inside of $([0,1]\times S^{3})\#\hat{W}$, where $\hat{W}$ is the homotopy 4-sphere obtained by filling in $\partial W$ with two 4-balls. By factoring the cobordism map through the connected sum sphere, the induced map is easily seen to be the identity.

Acknowledgments

We would like to thank András Juhász for helpful conversations, in particular for pointing out that the tubing operation from [ZemRibbon] can be interpreted in terms of sliding a surface over a 2-sphere. We would also like to thank the anonymous referee for some helpful comments.