Topologically and rationally slice knots

The smooth concordance group of topologically slice knots contains a $\mathbb{Z}^{\infty}\oplus(\mathbb{Z}/2\mathbb{Z})^{\infty}$ subgroup; the existence of a $\mathbb{Z}^{\infty}$ subgroup was first proved by Endo in 1995 using gauge theory [End95], while the existence of a $(\mathbb{Z}/2\mathbb{Z})^{\infty}$ subgroup was not proved until nearly 20 years later, using Heegaard Floer homology [HKL16]. Both of these proofs rely on Freedman’s landmark result [Fre82] that any knot with Alexander polynomial 1 is topologically slice.

Similarly, the smooth concordance group of rationally slice knots also contains a $\mathbb{Z}^{\infty}\oplus(\mathbb{Z}/2\mathbb{Z})^{\infty}$ subgroup. However, our understanding of rationally slice knots proceeded in the opposite direction: the proof of the $(\mathbb{Z}/2\mathbb{Z})^{\infty}$ subgroup [Cha07] preceded the proof of the $\mathbb{Z}^{\infty}$ subgroup [HKPS22] by 15 years. For their obstructive side, the former proof relies on J. Levine’s algebraic concordance group [Lev69] while the latter relies on involutive Heegaard Floer homology [HM17]. For their constructive side, they use the fact that the figure-eight knot is rationally slice and generalize its proof. This fact was observed by Cochran using work of Fintushel-Stern [FS84].

What about knots that are both topologically and rationally slice?

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  Rationally slice but not slice: The knots in [Cha07] are not topologically slice, as they are not even algebraically slice. Similarly, the generating set for the $\mathbb{Z}^{\infty}$ subgroup in [HKPS22] are not algebraically slice either.

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  Topologically slice but not slice: The knots representing the $(\mathbb{Z}/2\mathbb{Z})^{\infty}$ subgroup of [HKL16] can be shown to be rationally slice by a straightforward modification of the proof of [Cha07, Theorem 4.16]. The generating set for Endo’s family [End95] (the pretzel knots $P(-2k-1,4k+1,4k+3)$ for $k\geq 1$) are not rationally slice. There are more topologically slice knots in the literature and many of them are known to be not rationally slice (see e.g. [Hom15, OSS17, KP18, FPR19, DHST21]) and some of them are not known to be rationally slice or not (see e.g. [HK12, CHH13, CK21, Cha21, KKK22]).

Note that all previously known examples of knots that are both topologically and rationally slice have order two in the smooth concordance group. The goal of this paper is to show that the smooth concordance group of topologically and rationally slice knots does in fact contain a $\mathbb{Z}^{\infty}\oplus(\mathbb{Z}/2\mathbb{Z})^{\infty}$ subgroup. For any knot $K$, let $\mathrm{Wh}^{+}(K)$ denote the positive Whitehead double of $K$, and $K_{p,-1}$ denote the $(p,-1)$-cable of $K$.

As mentioned in Theorem 1.1, the knots $K_{n}$ are topologically and rationally slice. Indeed, up to topological concordance, we may ignore $\mathrm{Wh}^{+}(T)$, resulting in $E_{2n+1,-1}\#-E_{2n+1,-1}$, which is clearly slice. Similarly, up to rational concordance, we may ignore $E$, which also results in a slice knot. Our family of knots exploits the fact that cabling does not induce a homomorphism on the concordance group; see [HPC21] for further discussion of satellite operators and the concordance group.

We may naturally interpret our main theorem as follows. Let $\mathcal{T}$ be the smooth concordance group of topologically slice knots and $\mathcal{T}_{\mathbb{Q}}$ be the smooth rational concordance group of topologically slice knots. Since two concordant knots are rationally concordant, we have the following natural surjective homomorphism:

Note that the main theorem of [HKL16] implies that $\ker\psi_{\mathcal{T}}$ contains infinitely many order two elements. As a direct corollary of our main theorem, we have the following:

Furthermore, the knots in 1.1 can be used to prove the existence of a knot which is topologically slice and strongly rationally slice but not slice. Strongly rationally slice knots are knots $K$ which bound a smoothly embedded disk $\Delta$ in a rational homology ball $X$ such that the inclusion induces an isomorphism

The question of distinguishing strongly rationally slice knots from slice knots is very subtle; the first example of such a knot was found in [KP22]. By directly applying the proof of [KP22, Theorem 1.4] to the knots in 1.1, we immediately get the following corollary.

Recall that there exists a natural surjective homomorphism $\psi\colon\mathcal{C}\to\mathcal{C}_{\mathbb{Q}}$ defined from the smooth concordance group to the smooth rational concordance group. As its counterpart in the topological category, we can also consider the map

defined from the topological concordance group to the topological rational concordance group. Even though we know that there are subgroups $\mathbb{Z}^{\infty}\oplus(\mathbb{Z}/2\mathbb{Z})^{\infty}$ in $\ker\psi$ and $(\mathbb{Z}/2\mathbb{Z})^{\infty}$ in $\ker\psi^{\text{top}}$ [Cha07, HKPS22], the existence of an infinite order element in $\ker\psi^{\text{top}}$ is still not known. Hence, we ask:

Lastly, we make a remark that the positive Whitehead double of the figure-eight knot is also a good candidate that is likely to have infinite order in $\ker\psi_{\mathcal{T}}$. To the best of the authors’ knowledge it is not even known if this knot is slice or not. Here, we ask:

We assume that the reader is familiar with involutive knot Floer homology [HM17] and the $\iota_{K}$-local equivalence group [HMZ18]. See [HKPS22, Section 2] for an expository overview of these concepts. In this paper, we focus on the horizontal almost $\iota_{K}$-local equivalence group, which has the advantage that, modulo the image of the figure eight knot, it is totally ordered. What follows is a short summary of [KP22, Section 2].

A horizontal almost $\iota_{K}$-complex is a pair $(C,\iota)$ of a bigraded complex $C$ of finitely generated free modules over $\mathbb{F}_{2}[U]$ and a chain homotopy equivalence $\iota\colon\widehat{C}\rightarrow\widehat{C}$, where $\widehat{C}$ is the hat-flavored truncation of $C$, such that the following conditions are satisfied.

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  The formal variable $U$ has bigrading $(-2,0)$.

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  $U^{-1}C\simeq\mathbb{F}_{2}[U,U^{-1}]$

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  $\iota$ is skew-graded, i.e. maps a $(a,b)$-bigraded element to a $(b,a)$-bigraded one.

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  $\Phi\iota\Phi\iota\sim\iota\Phi\iota\Phi$.

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  $\iota^{2}\sim 1+\Phi\iota\Phi\iota$.

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  There exists a chain map $f\colon C\rightarrow C$ whose hat-flavored truncation is homotopic to $\iota\Phi\iota$.

Here, $\Phi$ denotes the formal derivative of the differential of $C$ with respect to the formal variable $U$, which is a chain map which is well-defined up to homotopy.

A degree-preserving chain map $f\colon C\rightarrow D$ between horizontal almost $\iota_{K}$-complexes $(C,\iota_{C})$ and $(D,\iota_{D})$ is almost $\iota_{K}$-local map if it satisfies the following conditions.

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  $f$ is local, i.e. the localized map $U^{-1}f\colon U^{-1}C\rightarrow U^{-1}D$ is a homotopy equivalence.

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  $\iota_{C}\widehat{f}\sim\widehat{f}\iota_{D}$, where $\widehat{f}$ is the hat-flavored truncation of $f$.

Furthermore, if there exist almost $\iota_{K}$-local maps $f\colon C\rightarrow D$ and $g:D\rightarrow C$, then we say that $(C,\iota_{C})$ and $(D,\iota_{D})$ are almost $\iota_{K}$-locally equivalent.

We denote the almost $\iota_{K}$-local equivalence classes of horizontal almost $\iota_{K}$-complexes by $\mathfrak{I}^{U}_{K}$. We endow this set with a tensor product operation $\otimes$ as $(C,\iota_{C})\otimes(D,\iota_{D})=(C\otimes D,\iota_{C\otimes D})$, where $\iota_{C\otimes D}$ is defined as $\iota_{C\otimes D}=\iota_{C}\otimes\iota_{D}+\Phi\iota_{C}\otimes\iota_{D}\Phi$. Although this operation is nonsymmetric, the group $\mathfrak{I}^{U}_{K}$ is indeed abelian [KP22, Proposition 2.6].

The involutive knot Floer homology package associates to a knot $K$ a well-defined element in $\mathfrak{I}^{U}_{K}$, given by the chain complex $CFK^{-}(S^{3},K)$, together with the $\iota_{K}$-action on $\widehat{CFK}(S^{3},K)$. For simplicity, we denote this element as $[K]$. It is clear that this defines a group homomorphism

where $\mathcal{C}$ is the smooth concordance group. The image of the unknot is denoted by $0\in\mathfrak{I}^{U}_{K}$.

Notice that the existence of almost $\iota_{K}$-local map gives a partial order on $\mathfrak{I}^{U}_{K}$. In particular, we say that $C\leq D$ if there exists an almost $\iota_{K}$-local map $C\rightarrow D$. It turns out that this is actually a total order $\mathfrak{I}^{U}_{K}$, modulo the figure-eight complex $[E]$, i.e. the 2-torsion element induced by the involutive knot Floer homology of the figure-eight knot $E$. In other words, two elements $C,D\in\mathfrak{I}^{U}_{K}$ are incomparable if and only if $C=D+[E]$ [KP22, Theorem 2.11]. We will exploit this ordering to prove the main theorem.

We assume that the reader is familiar with standard materials in bordered Heegaard Floer homology, in particular the materials in [LOT18].

For simplicity, from now on, we will denote the $\infty$-framed solid torus, as a bordered manifold, as $T_{\infty}$. Given a pattern $P\subset T_{\infty}$, we can define the minus-flavored type-A module $CFA^{-}(T_{\infty},P)$, which is a type-A structure over the torus algebra $\mathcal{A}(T^{2})$ with the coefficient ring $\mathbb{F}_{2}[U]$. We denote its truncation by $U=0$ as $\widehat{CFA}(T_{\infty},P)$. Furthermore, we can also remove a tubular neighborhood of $P$ from $T_{\infty}$ and endow the newly created torus boundary with the $0$-framing; this defines a bordered manifold with two torus boundaries, whose type-DA bordered Floer homology is denoted as $\widehat{CFDA}(T_{\infty}-P)$. Note that we have

via standard gluing formulae.

We will denote the longitudinal knot inside $T_{\infty}$ by $\nu$. In particular, we have a gluing formula [LOT18, Theorem 11.29]:

where $CFK^{-}$ denotes the truncation of $CFK_{UV}(S^{3},K)$ by $V=0$. Note that we can also apply the gluing formula to get an identification

Given a bordered manifold $M$ with one torus boundary, its type-D bordered involution takes the form

where $\mathbf{AZ}$ denotes the Auroux-Zarev piece, defined in [LOT11, Section 4]. Although $\iota_{M}$ is not well-defined up to homotopy due to the lack of naturality in bordered Floer homology, it is still a homotopy equivalence, and we will not need its uniquenss anyway. We denote the set of homotopy classes of all possible type-D bordered involutions of $M$ by $\mathbf{Inv}_{D}(M)$.

The following lemma follows from the proof of [KP22, Theorem 4.5]. Its proof is straightforward from the discussions preceding the proof of [Kan22, Theorem 1.2], so we omit it for the sake of simplicity.

We start with several explicit computations of bordered Floer homology modules. Given any knot $K$ in $S^{3}$, the type-D module $\widehat{CFD}(S^{3}-K)$ can be easily computed from the knot Floer chain complex $CFK_{UV}(S^{3},K)$. For example, for the right-handed trefoil $T$, the type-D module $\widehat{CFD}(S^{3}-T)$ for the 0-framed complement of $T$ can be described as follows:

Similarly, we can also describe $\widehat{CFD}(S^{3}-E)$ for the 0-framed complement of the figure-eight knot $E$, as shown below. We will call its summand generated by $a,b,c,e,y_{1},y_{2},y_{3},y_{4}$ as the square-module, and denote it by $S$.

We will also need the (hat-flavored) type-A module $\widehat{CFA}(T_{\infty},P_{2n+1,-1})$ of the $(2n+1,-1)$-cabling pattern in the $\infty$-framed solid torus $T_{\infty}$, which is shown below. Note that $CFA^{-}(T_{\infty},P_{2n+1,-1})$ was computed originally in [OSS17, Lemma 8.3]; we simply truncated their computation by taking $U=0$.

Based on these computations, we will now prove two computational lemmas, which play a crucial role in the proof of 1.1.

Recall that bordered Heegaard Floer homology comes with gradings by nonabelian groups; given a bordered 3-manifold $Y$ with boundary $\mathcal{Z}$, the associated type-D module $\widehat{CFD}(Y)$ is graded by a transitive $G(\mathcal{Z})$-set. Hence the notion of degree-preserving endomorphisms of $\widehat{CFD}(Y)$ is well-defined. See [LOT18, Chapter 10] for a detailed description of gradings in bordered Floer homology.

We need one more lemma regarding the $\iota_{K}$-local equivalence class of the involutive knot Floer homology of $\mathrm{Wh}^{+}(T)$, which we will denote as $D$ for simplicity.

We are now ready to prove 1.1; its proof will be divided in two parts. In the first part, we will prove that the knots

have infinite order in the smooth concordance group. Then, in the second part, we will prove that infinitely many $K_{n}$ form a linearly independent family of the smooth concordance group.