# Torsion in the knot concordance group and cabling

Sungkyung Kang  Center for Geometry and Physics, Institute for Basic Science
(IBS), Pohang 37673, Korea<EMAIL_ADDRESS>and  JungHwan Park
Department of Mathematical Sciences, Korea Advanced Institute for Science and
Technology<EMAIL_ADDRESS>
###### Abstract.

We define a nontrivial mod 2 valued additive concordance invariant defined on
the torsion subgroup of the knot concordance group using involutive knot Floer
package. For knots not contained in its kernel, we prove that their iterated
$(\text{odd},1)$-cables have infinite order in the concordance group and,
among them, infinitely many are linearly independent. Furthermore, by taking
$(2,1)$-cables of the aforementioned knots, we present an infinite family of
knots which are strongly rationally slice but not slice.

###### Key words and phrases:

Concordance knots, cabling, Horizontal almost $\iota_{K}$-complex

###### 2020 Mathematics Subject Classification:

57K10, 57K18

##  1\. Introduction

Given a knot $K$ and relatively prime integers $p$ and $q$, let $K_{p,q}$ be
the _$(p,q)$ -cable_ of $K$ where $p$ denotes the longitudinal winding.
Through out the paper we assume that $|p|>1$ since $K_{1,q}$ is isotopic to
$K$. Recall that a knot is called _slice_ if it bounds a smoothly embedded
disk in $B^{4}$ and the fact that $(p,1)$-cable and $(p,-1)$-cable of a slice
knot are both slice. Miyazaki [Miy94, Question 3] asked if the converse of
this fact also holds (see also [Mei17, Conjecture 1.3] and [CDR14, Section
1]). In fact, we go one step further and focus on the following stronger
conjecture. Let $\mathcal{C}$ denote the _smooth knot concordance group_.

######  Conjecture 1.

The $(p,q)$-cable of $K$ is torsion in $\mathcal{C}$ if and only if $K$ is
slice and $|q|=1$.

It can be easily seen that if $|q|>1$, then $K_{p,q}$ has infinite order in
$\mathcal{C}$.111One way to prove this is to use the cabling formulae for the
Ozsváth-Szabó concordance invariant $\tau$ [OS03] and the Ozsváth-Stipsicz-
Szabó concordance invariant $\Upsilon$ [OSS17]; see e.g. [Hom14a, Theorem 1]
for $\tau$ and [FPR19, Proposition 5.3] for $\Upsilon$. Therefore, for the
rest of the article we only consider the case when $K$ is not slice and
$|q|=1$. In fact, since $K_{p,q}$ is isotopic to $K_{-p,-q}$, we will further
assume that $q=1$.

There are many nonslice knots where their $(p,1)$-cables have infinite order
in $\mathcal{C}$. For example, if $K$ has nontrivial signature function
[Tri69, Lev69], then a formula of Litherland [Lit79] implies that $K_{p,1}$
has infinite order in $\mathcal{C}$. In fact, the same formula implies that
the set of cables $\\{K_{p,1}\\}_{p>1}$ is linearly independent in
$\mathcal{C}$. A similar conclusions obtained by various different techniques
can be found in [KP18, FPR19, Che21, DPR21]. These results crucially use the
fact that the starting knot $K$ has infinite order in $\mathcal{C}$ (e.g. a
knot with nontrivial signature function has infinite order in $\mathcal{C}$).

On the other hand, determining the order of $(p,1)$-cables of a _torsion_ knot
is much harder. The only known result of this kind is given recently in
[HKPS20]. They showed that when $K$ is the figure-eight knot, the set of
cables $\\{K_{2n+1,1}\\}_{n>0}$ is linearly independent in $\mathcal{C}$ using
involutive knot Floer homology [HM17]. (This also gave a first example of an
infinite order rationally slice knot.) In this article, we expand this result
to a much larger family. Roughly, we show that “half” of the torsion knots
have infinite order once cabled (compare it with [Miy94, Theorem 8.5] where
they consider cables of fibered knots). Let
$K_{p_{1},q_{1};p_{2},q_{2};\ldots;p_{m},q_{m}}$ denote the iterated cable of
$K$.

######  Theorem 1.1.

Denote by $\mathcal{C}_{T}$ the torsion subgroup of the knot concordance group
$\mathcal{C}$. Then there is a nontrivial group homomorphism

$\mathfrak{A}\colon\mathcal{C}_{T}\rightarrow\mathbb{Z}/2\mathbb{Z}.$

Moreover, if $K$ is torsion in $\mathcal{C}$ with $\mathfrak{A}(K)=1$, then
for any sequence of positive integers $n_{1},n_{2},\ldots,n_{m}$ the iterated
cable $K_{2n_{1}+1,1;2n_{2}+1,1;\ldots;2n_{m}+1,1}$ has infinite order in
$\mathcal{C}$. In particular, for any nonzero integer $n$ the cable
$K_{2n+1,1}$ has infinite order in $\mathcal{C}$.

We also show that knots obtained by certain cables on $K$ with
$\mathfrak{A}(K)=1$ are linearly independent.

######  Theorem 1.2.

If $K$ is torsion in $\mathcal{C}$ with $\mathfrak{A}(K)=1$, then the set of
cables $\\{K_{2n+1,1}\\}_{n>0}$ contains an infinite subset which is linearly
independent in $\mathcal{C}$.

For the nontrivial group homomorphism $\mathfrak{A}$, we first define the
almost local $\iota_{K}$-equivalence group $\mathfrak{I}^{U}_{K}$ of
horizontal almost $\iota_{K}$-complexes (see Section 2 for more details). Note
that the $\iota_{K}$-local equivalence group $\mathfrak{I}_{K}$ and the almost
$\iota_{K}$-local equivalence group $\widehat{\mathfrak{I}}_{K}$ were defined
previously in [Zem19] and [HKPS20], respectively. Moreover, we show that
$\mathfrak{A}$ is the composition of the following sequence of the natural
“forgetful maps” followed by the involutive knot Floer homology map
$\mathcal{C}\rightarrow\mathfrak{I}_{K}$;

$\mathfrak{I}_{K}\rightarrow\widehat{\mathfrak{I}}_{K}\rightarrow\mathfrak{I}^{U}_{K}.$

In other words, involutive knot Floer theory gives a well-defined homomorphism
$\mathfrak{A}\colon\mathcal{C}\rightarrow\mathfrak{I}^{U}_{K}$.

It turns out that $\mathfrak{I}^{U}_{K}$ is totally ordered modulo the figure-
eight knot $E$, so that its torsion subgroup is generated by $E$. This allows
us to consider its restriction
$\mathfrak{A}\colon\mathcal{C}_{T}\rightarrow\mathbb{Z}/2\mathbb{Z}$, where we
have that $\mathfrak{A}(E)=1$. Therefore, the above theorems can be thought of
it as a generalizations of [HKPS20]. Moreover, note that the “half” of the
torsion knots satisfy $\mathfrak{A}=1$; i.e. if $\mathfrak{A}(K)=0$, then
$\mathfrak{A}(K\\#E)=1$.

The phenomenon of total orderings of algebraic structures arising from
Heegaard Floer theory and its variants occurs in other settings as well. For
instance, the almost local equivalence group of $\iota$-complexes is totally
ordered [DHST18] and the almost local equivalence group of knot-like complexes
over $\mathbb{F}_{2}[U,V]/(UV)$ is also totally ordered [DHST21]. A notable
difference in our case is that our group $\mathfrak{I}^{U}_{K}$ contains a
torsion, and one has to take a quotient by that element to obtain a total
ordering.

Now, we restrict our attention to the case when $K$ allows a special symmetry
and $p=2$. Note that, so far, we have only considered $(\text{odd},1)$-cables,
but by considering $(2,1)$-cables we will obtain an interesting application to
rational knot concordance. We say a knot $K$ is _rationally slice_ if it
bounds a smoothly embedded disk $\Delta$ in a rational homology ball $X$.
Moreover, if the inclusion induces an isomorphism

$H_{1}(S^{3}\smallsetminus
K;\mathbb{Z})\xrightarrow{\cong}H_{1}(X\smallsetminus\Delta;\mathbb{Z})/\text{torsion},$

then we say $K$ is _strongly rationally slice_.222Strongly rationally slice
knots are rationally slice knots with _complexity_ one. See Section 6 for the
notion of complexity and [CO93, CK02, Cha07] for related works. Following the
proof in [Lev69], it can be easily verified that every strongly rationally
slice knot is algebraically slice. Moreover, slice obstructions such as
$\tau$-invariant [OS03], $\varepsilon$-invariant [Hom14b],
$\Upsilon$-invariant [OSS17], $\Upsilon^{2}$-invariant [KL18],
$\nu^{+}$-invariant [HW16], and $\varphi_{j}$-invariants [DHST21] all vanish
for strongly rationally slice knots. These facts make it difficult to find an
example of a strongly rationally slice knot that is not slice. In this
article, we present the first such examples:

######  Theorem 1.3.

There are infinitely many pairwise nonconcordant knots which are strongly
rationally slice but not slice.

The examples are constructed in the following way. Recall that a knot $K$ is
called _strongly negative-amphichiral_ if there is an orientation-preserving
involution $\phi\colon S^{3}\to S^{3}$ such that $\phi(K)=K$ and the fixed
point set of $\phi$ is a copy of $S^{0}$ contained in $K$. Kawauchi [Kaw09]
showed that every strongly negative-amphichiral knot $K$ is rationally slice;
that is, $K$ bounds a smoothly embedded disk $\Delta$ in a rational homology
ball $X$. Moreover, he showed the homomorphism induced by the inclusion sends
the generator of $H_{1}(S^{3}\smallsetminus K;\mathbb{Z})$ to twice the
generator of $H_{1}(X\smallsetminus\Delta;\mathbb{Z})/\text{torsion}$.
Combined with Mayer-Vietoris sequence (see 6.2), it implies that the iterated
cable $K_{p_{1},1;p_{2},1;\ldots;p_{m},1}$ of a strongly negative-amphichiral
knot $K$ is strongly rationally slice if $p_{i}$ is even for some $i$. Hence
the following theorem immediately implies Theorem 1.3.

######  Theorem 1.4.

If $K$ is torsion in $\mathcal{C}$ with $\mathfrak{A}(K)=1$, then for any
sequence of positive integers $n_{1},n_{2},\ldots,n_{m}$ the $(2,1)$-cable of
the iterated cable $K_{2n_{1}+1,1;2n_{2}+1,1;\ldots;2n_{m}+1,1}$ has infinite
order in $\mathcal{C}$.

### Organization and proof outline

We briefly sketch the ideas for the proofs of above theorems as follows. In
Section 2, we define the concept of horizontal almost $\iota_{K}$-complexes,
which provides an algebraic model for $CFK^{-}$ of knots in $S^{3}$, together
with the action of $\iota_{K}$ on $\widehat{CFK}$. This allows us to define
the almost $\iota_{K}$-local equivalence group $\mathfrak{I}^{U}_{K}$ of
horizontal almost $\iota_{K}$-complexes, which is equipped with a canonically
defined partial order. Then we prove that it is totally ordered modulo the
figure-eight knot, from which the definition of the invariant $\mathfrak{A}$
in 1.1 follows.

In Section 3, we show that the bordered involution on a 0-framed knot
complement naturally induces a horizontal almost $\iota_{K}$-complex, which is
defined uniquely up to almost $\iota_{K}$-local equivalence. Then, in Section
4, we prove a part of 1.1 which states that $K_{2n+1,1}$ has infinite order in
$\mathcal{C}$, by comparing the horizontal $\iota_{K}$-complex induced by
$(-K)_{2n+1,-1}$ with the complex induced by the $(2n+1,-1)$-cable of the
figure-eight knot. The comparison is done by passing to the bordered setting
and then applying involutive bordered Heegaard Floer theory of knot
complements, developed by the first author in [Kan22]. Then we develop the
arguments used in its proof to prove 1.2.

Finally, In Section 5, we present a full proof of 1.1 by finding an inductive
step which allows us to deal with iterated cables. We then use it in Section 6
to prove 1.4 by considering the 2-fold cyclic branched double covers of
$S^{3}$ along the given knots.

### Questions

We end this section with few questions. It can be easily verified that for
each alternating knot $K$ which is torsion in $\mathcal{C}$, we have that
$\mathfrak{A}(K)=\operatorname{Arf}(K)$. Naturally, we ask if the equality
holds in general.

######  Question 1.

If $K$ is torsion in $\mathcal{C}$, then is
$\mathfrak{A}(K)=\operatorname{Arf}(K)$?

Also, since the almost local equivalence group of $\iota$-complexes and the
almost local equivalence group of knot-like complexes over
$\mathbb{F}_{2}[U,V]/(UV)$ are known to be totally ordered and isomorphic to
$\mathbb{Z}^{\infty}$ [DHST18, DHST21], we ask the following question.

######  Question 2.

Is $\mathfrak{I}^{U}_{K}$ isomorphic to
$\mathbb{Z}^{\infty}\oplus\mathbb{Z}/2\mathbb{Z}$?

### Notation and conventions

Throughout the paper the unknot is denoted by $O$ and the figure-eight knot is
denoted by $E$. Given a knot $K$ and a positive integer $n$, the connected sum
of $n$ copies of $K$ is denoted by $nK$, the reverse of the mirror image of
$K$ is denoted by $-K$, and the reverse of $K$ is denoted $K^{r}$. Also, given
a closed 3-manifold $Y$, the connected sum of $n$ copies of $Y$ is denoted by
$nY$. Lastly, two morphisms (chain maps, type-$D$ morphisms, etc) $f$ and $g$
are homotopic, we write $f\sim g$. In contrast, when two objects $A$ and $B$
are homotopy equivalent, we write $A\simeq B$ instead.

### Acknowledgements

The authors wish to thank Matthew Stoffregen, Ian Zemke, and Jae Choon Cha for
helpful discussions. The first author is supported by the Institute for Basic
Science (IBS-R003-D1). The second author is partially supported by Samsung
Science and Technology Foundation (SSTF-BA2102-02) and the POSCO TJ Park
Science Fellowship.

##  2\. The horizontal almost $\iota_{K}$-local equivalence group

We briefly recall the theory of involutive knot Floer homology and
$\iota_{K}$-local equivalences. Given an oriented knot $K$ in $S^{3}$ (or in
general, a rationally nullhomologous knot in a rational homology sphere with a
prescribed self-conjugate $\textrm{Spin}^{c}$ structure), one starts by
representing it by a doubly-pointed Heegaard diagram
$H=(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},z,w)$. Note that one can
recover $K$ from $H$ in the following way. The given sets of $\alpha$-curves
and $\beta$-curves define handlebodies $H_{\alpha}$ and $H_{\beta}$ (in which
the given sets of curves bound disks, respectively) such that $H_{\alpha}\cup
H_{\beta}=S^{3}$. Choose directed paths $K_{\alpha}\subset H_{\alpha}$ and
$K_{\beta}\subset H_{\beta}$, such that $K_{\alpha}$ goes from $z$ to $w$ and
$K_{\beta}$ goes from $w$ to $z$. Then we have $K=K_{\alpha}\cup K_{\beta}$.

Then we consider the $2g$-dimensional smooth manifold
$\mathrm{Sym}^{g}(\Sigma)$, where $g$ denotes the genus of $\Sigma$. This
manifold contains two $g$-dimensional tori, $\mathbb{T}_{\alpha}$ and
$\mathbb{T}_{\beta}$, defined by the product of all $\alpha$-curves and
$\beta$-curves, respectively, and two 2-codimensional submanifolds $D_{z}$ and
$D_{w}$, defined by the images of the natural maps

$\begin{split}\mathrm{Sym}^{g-1}(\Sigma)\times\\{z\\}&\rightarrow\mathrm{Sym}^{g}(\Sigma),\\\
\mathrm{Sym}^{g-1}(\Sigma)\times\\{w\\}&\rightarrow\mathrm{Sym}^{g}(\Sigma).\end{split}$

Then a standard Floer-theoretic argument can be applied to construct the chain
complex

$CFK_{UV}(S^{3},K):=\left(\bigoplus_{\mathbf{x}\in\mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta}}\mathbb{F}_{2}[U,V]\cdot\mathbf{x},\partial\right)$

by counting pseudoholomorphic disks of Maslov index 0 with boundary conditions
given by $\mathbb{T}_{\alpha}$ and $\mathbb{T}_{\beta}$, while recording their
algebraic intersection numbers with $D_{z}$ and $D_{w}$ by formal variables
$U$ and $V$. This complex also comes equipped with a bigrading $(A,M)$ by a
pair of integers $A$ and $M$, called the _Alexander_ and _Maslov_ gradings. It
turns out that Heegaard moves induce homotopy equivalences between $CFK_{UV}$,
and thus the homotopy type of $CFK_{UV}(S^{3},K)$ depends only on the isotopy
class of $K$ [OS04, Zem17]. Furthermore, we have the following localization
formula:

$(U,V)^{-1}CFK_{UV}(S^{3},K)\simeq(U,V)^{-1}\mathbb{F}_{2}[U,V].$

In this paper, we will consider several truncations of the two-variable
complex $CFK_{UV}(S^{3},K)$. First of all, consider the ring
$\mathcal{R}:=\mathbb{F}_{2}[U,V]/(UV)$. The $\mathcal{R}$-coefficient chain
complex

$CFK_{\mathcal{R}}(S^{3},K):=CFK_{UV}(S^{3},K)\otimes_{\mathbb{F}_{2}[U,V]}\mathcal{R},$

which can be seen as ignoring all “diagonal” terms in the differential of
$CFK_{UV}(S^{3},K)$, will be used in this paper. We will also use the $V=0$
truncation, namely

$CFK^{-}(S^{3},K):=CFK_{UV}(S^{3},K)\otimes_{\mathbb{F}_{2}[U,V]}\mathbb{F}_{2}[U,V]/(V),$

and the hat-flavored truncation:

$\widehat{CFK}(S^{3},K):=CFK_{UV}(S^{3},K)\otimes_{\mathbb{F}_{2}[U,V]}\mathbb{F}_{2}[U,V]/(U,V).$

The above truncations, except for the hat-flavored version, admit localization
formulae as well:

$\begin{split}(U,V)^{-1}CFK_{\mathcal{R}}(S^{3},K)&\simeq(U,V)^{-1}\mathcal{R},\\\
U^{-1}CFK^{-}(S^{3},K)&\simeq\mathbb{F}_{2}[U,U^{-1}].\end{split}$

Now, given a doubly-pointed Heegaard diagram
$H=(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},z,w)$ representing
$(S^{3},K)$, we consider its conjugate diagram
$\overline{H}=(-\Sigma,\boldsymbol{\beta},\boldsymbol{\alpha},w,z)$, where
$-\Sigma$ denotes $\Sigma$ with the opposite orientation. The oriented (and
doubly-basepointed) knot represented by $\overline{H}$ is $(K,w,z)$, i.e. the
same oriented knot $K$ with the positions of basepoints exchanged. Since the
holomorphic disk counts in $H$ are the same as those in $\overline{H}$ while
the role of $U$ and $V$ are exchanged, the identity map

$\operatorname{conj}\colon CFK_{UV}(K,z,w)\rightarrow CFK_{UV}(K,w,z)$

is a chain homotopy _skew_ -equivalence. Choose a self-diffeomorphism $\phi$
of $S^{3}$ which is supported in a tubular neighborhood of $K$ and exchanges
$z$ and $w$. Then it induces a homotopy equivalene

$\phi_{\ast}\colon CFK_{UV}(K,w,z)\rightarrow CFK_{UV}(K,z,w).$

Then we define $\iota_{K}=\phi_{\ast}\circ\operatorname{conj}$, which is again
a homotopy skew-equivalence. Note that $\iota_{K}$ is defined up to
ambiguities given by monodromies along loops of Heegaard moves; since homotopy
equivalences induced by those loops are homotopic to identity due to the
first-order naturality of Heegaard Floer theory [JTZ21], $\iota_{K}$ is
defined uniquely up to homotopy. Since $\iota_{K}$ exchanges the actions of
$U$ and $V$, it descends to $CFK_{\mathcal{R}}$ and $\widehat{CFK}$, but not
$CFK^{-}$.

The involution $\iota_{K}$ satisfies the following properties, as shown in
[HM17] and [Zem19].

  * •

$\iota^{2}_{K}\sim 1+\Phi\Psi\sim 1+\Psi\Phi$, where $\Phi$ and $\Psi$ are the
formal derivatives of the differential of $CFK_{UV}(S^{3},K)$ with respect to
$U$ and $V$, respectively.

  * •

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

  * •

$\iota$ is skew-graded, i.e. it maps an element of bidegree $(A,M)$ to an
element of bidegree $(-A,M)$.

Using these properties, the concepts of $\iota_{K}$-complexes and
$\iota_{K}$-local maps are defined in [HKPS20] as follows. A free bigraded
chain complex $C$ over $\mathbb{F}_{2}[U,V]$, together with a chain homotopy
skew-equivalence $\iota:C\rightarrow C$, is an _$\iota_{K}$ -complex_ if it
satisfies the following conditions.

  * •

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

  * •

$\iota^{2}\sim 1+\Phi\Psi$, where $\Phi$ and $\Psi$ are the formal derivatives
of the differential of $C$ with respect to $U$ and $V$, respectively.

Furthermore, a chain map $f\colon C\rightarrow D$ between
$\iota_{K}$-complexes is an _$\iota_{K}$ -local map_ if the following
conditions are satisfied.

  * •

$(U,V)^{-1}f$ is a homotopy equivalence.

  * •

$f\iota\sim\iota f$.

If there exists an $\iota_{K}$-local map from $C$ to $D$ and also from $D$ to
$C$, then we say that $C$ and $D$ are _$\iota_{K}$ -locally equivalent_ and
$f$ is an _$\iota_{K}$ -local equivalence_.

######  Remark 2.1.

In this paper, we will sometimes use the notion of $\iota_{K}$-local maps for
chain maps between complexes over $\mathcal{R}$. The definition of
$\iota_{K}$-local maps in such a setting is identical to the original
definition.

The concept of almost local $\iota_{K}$-complexes is also defined in [HKPS20]
by considering instead the hat-flavored involution.

######  Definition 2.2.

A free bigraded chain complex $C$ over $\mathbb{F}_{2}[U,V]$, together with a
skew-graded homotopy equivalence
$\iota\colon\widehat{C}\rightarrow\widehat{C}$, where
$\widehat{C}=C\otimes_{\mathbb{F}_{2}[U,V]}\mathbb{F}_{2}[U,V]/(U,V)$, is an
_almost $\iota_{K}$-complex_ if the following conditions are satisfied.

  * •

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

  * •

$\iota^{2}\sim 1+\Phi\Psi$ as skew-graded chain endomorphisms of
$\widehat{C}$, where $\Phi$ and $\Psi$ are the formal derivatives of the
differential of $C$ with respect to $U$ and $V$, respectively.

Furthermore, a chain map $f\colon C\rightarrow D$ is an _almost
$\iota_{K}$-local map_ if the following conditions are satisfied.

  * •

$(U,V)^{-1}f$ is a homotopy equivalence.

  * •

The induced map $\widehat{f}:\widehat{C}\rightarrow\widehat{D}$, where
$\widehat{C}$ and $\widehat{D}$ are the truncations of $C$ and $D$ by $U=V=0$,
satisfies $\widehat{f}\iota_{K}\sim\iota_{K}\widehat{f}$.

If almost $\iota_{K}$-local maps exist between two almost
$\iota_{K}$-complexes in both directions, we say that they are _almost
$\iota_{K}$-locally equivalent_.

Note that, given a knot $K$, its knot Floer complex $CFK_{UV}(S^{3},K)$,
together with the action of $\iota_{K}$, defines an $\iota_{K}$-complex. If
one only considers the action of $\iota_{K}$ on $\widehat{CFK}(S^{3},K)$, one
then obtains an almost $\iota_{K}$-complex. It is shown in [Zem19] that the
(almost) $\iota_{K}$-local equivalence class of the resulting
$\iota_{K}$-complex (or almost $\iota_{K}$-complex) depends only on the
concordance class of $K$.

Motivated by the definitions of $\iota_{K}$-complexes and almost
$\iota_{K}$-complexes, we now develop a similar framework by mimicking the
structure of $CFK^{-}(S^{3},K)$ of knots $K$ in $S^{3}$, together with the
action of $\iota_{K}$ on $\widehat{CFK}(S^{3},K)$. For simplicity, we make the
following definitions. Given a chain complex $C$ over $\mathbb{F}_{2}[U]$, we
say that the chain complex
$\widehat{C}=C\otimes_{\mathbb{F}_{2}[U]}\mathbb{F}_{2}[U]/(U)$ induced by
taking $U=0$ is the _hat-flavored truncation_ of $C$. Also, given a chain map
$f\colon C\rightarrow D$ between chain complexes over $\mathbb{F}_{2}[U]$, the
induced map $\widehat{f}\colon\widehat{C}\rightarrow\widehat{D}$ between
$\widehat{C}$ and $\widehat{D}$ is called the _hat-flavored truncation_ of
$f$. Sometimes we will confuse $f$ and $\widehat{f}$ to reduce the amount of
symbols used.

######  Definition 2.3.

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.

  * •

The formal variable $U$ has bigrading $(-1,-2)$.

  * •

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

  * •

$\iota$ is skew-graded.

  * •

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

  * •

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

  * •

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.

######  Remark 2.4.

Since we do not have the formal variable $V$ anymore, we had to represent
$\Psi$ in terms of $\Phi$ and $\iota$. Luckily, we have
$\Phi\iota\sim\iota\Psi$, $\iota^{2}\sim 1+\Phi\Psi$, and
$\iota^{4}\sim\operatorname{id}$, so we replaced $\Psi$ using the formula

$\begin{split}\Psi\sim\iota^{-1}\Phi\iota&=\iota\circ(1+\Psi\Phi)\circ\Phi\iota\\\
&=\iota\circ(\Phi+\Psi\Phi^{2})\circ\iota\sim\iota\Phi\iota.\end{split}$

Note that, in the last line of the above equation, we used the fact that
$\Phi^{2}$ is nullhomotopic, which follows from the observation that the value
of $\Phi$ vanishes on cycles and $\ker(\partial)\subset\ker(\Phi)$. The last
condition in 2.3 requires that one can find $\Psi$ as an
$\mathbb{F}_{2}[U]$-linear chain map. This condition is crucial in the proof
that the tensor product operation between two horizontal almost
$\iota_{K}$-complexes is commutative.

######  Definition 2.5.

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.

  * •

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

  * •

$\iota_{C}\widehat{f}\sim\widehat{f}\iota_{D}$, 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

$\iota_{C\otimes
D}=\iota_{C}\otimes\iota_{D}+\Phi\iota_{C}\otimes\iota_{D}\Phi.$

Note that this operation is modeled after the connected sum formula for
involutive knot Floer homology [Zem19, Theorem 1.1], which reads

$\iota_{K_{1}\\#K_{2}}=\iota_{K_{1}}\otimes\iota_{K_{2}}+\Phi\iota_{K_{1}}\otimes\Psi\iota_{K_{2}}.$

######  Proposition 2.6.

$(\mathfrak{I}^{U}_{K},\otimes)$ is an abelian group.

###### Proof.

We first prove that the group operation is well-defined. Given
$C,D\in\mathfrak{I}^{U}_{K}$, formally differentiating both sides of

$\partial_{C\otimes D}=\partial_{C}\otimes 1+1\otimes\partial_{D}$

with respect to $U$ gives $\Phi_{C\otimes D}=1\otimes\Phi+\Phi\otimes 1$. From
now on we drop the subscripts when the meaning is clear from context.

Thus we have

$\begin{split}\Phi_{C\otimes D}\iota_{C\otimes
D}&\sim(1\otimes\Phi+\Phi\otimes
1)\circ(\iota\otimes\iota+\Phi\iota\otimes\iota\Phi)\\\
&\sim\Phi\iota\otimes\iota+\iota\otimes\Phi\iota+\Phi\iota\otimes\Phi\iota\Phi,\end{split}$

which squares to

$\begin{split}\Phi_{C\otimes D}\iota_{C\otimes D}\Phi_{C\otimes
D}\iota_{C\otimes
D}&\sim\Phi\iota\Phi\iota\otimes\iota^{2}+\Phi\iota^{2}\otimes\iota\Phi\iota+\Phi\iota\Phi\iota\otimes\iota\Phi\iota\Phi+\iota\Phi\iota\otimes\Phi\iota^{2}+\iota^{2}\otimes\Phi\iota\Phi\iota+\Phi\iota\Phi\iota\otimes\Phi\iota\Phi\iota\\\
&\sim 1\otimes\Phi\iota\Phi\iota+\Phi\iota\Phi\iota\otimes
1+\Phi\otimes\iota\Phi\iota+\iota\Phi\iota\otimes\Phi.\end{split}$

On the other hand, since

$\begin{split}\iota_{C\otimes D}\Phi_{C\otimes
D}&=(\iota\otimes\iota+\Phi\iota\otimes\iota\Phi)\circ(1\otimes\Phi+\Phi\otimes
1)\\\
&\sim\iota\otimes\iota\Phi+\iota\Phi\otimes\iota+\Phi\iota\Phi\otimes\iota\Phi,\end{split}$

we have

$\iota_{C\otimes D}\Phi_{C\otimes D}\iota_{C\otimes D}\Phi_{C\otimes D}\sim
1\otimes\Phi\iota\Phi\iota+\Phi\iota\Phi\iota\otimes
1+\Phi\otimes\iota\Phi\iota+\iota\Phi\iota\otimes\Phi$

as well, which implies that

$\Phi_{C\otimes D}\iota_{C\otimes D}\Phi_{C\otimes D}\iota_{C\otimes
D}\sim\iota_{C\otimes D}\Phi_{C\otimes D}\iota_{C\otimes D}\Phi_{C\otimes D}.$

Also, we have

$\begin{split}\iota_{C\otimes
D}^{2}&\sim\iota^{2}\otimes\iota^{2}+\iota\Phi\iota\otimes\iota^{2}\Phi+\Phi\iota^{2}\otimes\iota\Phi\iota+\Phi\iota\Phi\iota\otimes\iota\Phi\iota\Phi\\\
&\sim 1\otimes 1+\Phi\iota\Phi\iota\otimes
1+1\otimes\Phi\iota\Phi\iota+\Phi\otimes\iota\Phi\iota+\iota\Phi\iota\otimes\Phi\\\
&\sim 1+\Phi_{C\otimes D}\iota_{C\otimes D}\Phi_{C\otimes D}\iota_{C\otimes
D},\end{split}$

and furthermore, if $f\colon C\rightarrow C$ and $g:D\rightarrow D$ are chain
maps such that their hat-flavored truncations $\widehat{f}$ and $\widehat{g}$
are homotopic to $\iota\Phi\iota$, then we have

$\begin{split}\iota_{C\otimes D}\Phi_{C\otimes D}\iota_{C\otimes
D}&=\iota\Phi\iota\otimes\iota^{2}+\iota\Phi^{2}\iota\otimes\iota^{2}\Phi+\Phi\iota\Phi\iota\otimes\iota\Phi\iota+\Phi\iota\Phi^{2}\iota\otimes\iota\Phi\iota\Phi\\\
&\quad+\iota^{2}\otimes\iota\Phi\iota+\iota\Phi\iota\otimes\iota\Phi\iota\Phi+\Phi\iota^{2}\otimes\iota\Phi^{2}\iota+\Phi\iota\Phi\iota\otimes\iota\Phi^{2}\iota\Phi\\\
&\sim\iota\Phi\iota\otimes 1+1\otimes\iota\Phi\iota,\end{split}$

which is represented by the chain map $f\otimes 1+1\otimes g$. So we deduce
that $C\otimes D$ is also an element of $\mathfrak{I}^{U}_{K}$, and thus the
group operation is well-defined.

Next, we show that the group operation is associative. Choose any
$A,B,C\in\mathfrak{I}^{U}_{K}$. Then we have

$\begin{split}\iota_{A\otimes(B\otimes
C)}&\sim\iota\otimes(\iota\otimes\iota+\Phi\iota\otimes\iota\Phi)+\Phi\iota\otimes(\iota\Phi\otimes\iota+\iota\otimes\iota\Phi)\\\
&\sim\iota\otimes\iota\otimes\iota+\iota+\Phi\iota+\iota\Phi+\Phi\iota\otimes\iota\Phi\otimes\iota+\Phi\iota\otimes\iota\otimes\iota\Phi.\end{split}$

On the other hand, we have

$\begin{split}\iota_{(A\otimes B)\otimes
C}&\sim(\iota\otimes\iota+\Phi\iota\otimes\iota\Phi)\otimes\iota+(\Phi\iota\otimes\iota+\iota\otimes\Phi\iota)\otimes\iota\Phi\\\
&\sim\iota\otimes\iota\otimes\iota+\Phi\iota\otimes\iota\Phi\otimes\iota+\Phi\iota\otimes\iota\otimes\iota\Phi+\iota\otimes\Phi\iota\otimes\iota\Phi.\end{split}$

Hence $\iota_{A\otimes(B\otimes C)}\sim\iota_{(A\otimes B)\otimes C}$, and
thus the group operation is associative.

Now we show that every element of $\mathfrak{I}^{U}_{K}$ admits an inverse.
Given any $C\in\mathfrak{I}^{U}_{K}$, endow its dual complex $C^{\ast}$ with
the dual involution $\iota^{\ast}$, and note that
$\Phi_{C^{\ast}}=\Phi^{\ast}_{C}$. Consider the trace map
$\operatorname{tr}\colon C\otimes C^{\ast}\rightarrow\mathbb{F}[U]$ and its
dual $\operatorname{cotr}\colon\mathbb{F}[U]\rightarrow C\otimes C^{\ast}$.
Then, by the tensor-hom adjunction, we have

$\begin{split}\operatorname{tr}\circ\iota_{C\otimes
C^{\ast}}&\sim\operatorname{tr}\circ(\iota\otimes\iota^{\ast}+\Phi\iota\otimes\iota^{\ast}\Phi^{\ast})\\\
&\sim\operatorname{tr}\circ(\iota^{2}\otimes 1+\Phi\iota\Phi\iota\otimes
1)\sim\operatorname{tr}\end{split}$

and also

$\begin{split}\iota_{C\otimes
C^{\ast}}\circ\operatorname{cotr}&\sim(\iota\otimes\iota^{\ast}+\Phi\iota\otimes\iota^{\ast}\Phi^{\ast})\circ\operatorname{cotr}\\\
&\sim(\iota^{2}\otimes 1+\Phi\iota\Phi\iota\otimes
1)\circ\operatorname{cotr}\sim\operatorname{cotr}.\end{split}$

Hence $C\otimes C^{\ast}$ is almost $\iota_{K}$-localy trivial, and hence
$C^{\ast}$, endowed with $\iota^{\ast}$, is the inverse of $C$ in
$\mathfrak{I}^{U}_{K}$. Thus $\mathfrak{I}^{U}_{K}$ is a group.

It remains to show that the group operation is commutative. Given two elements
$C,D$ in $\mathfrak{I}^{U}_{K}$, consider two horizontal almost
$\iota_{K}$-complexes $(C\otimes D,\iota_{1})$ and $(C\otimes D,\iota_{2})$,
where $\iota_{1}$ and $\iota_{2}$ are defined as

$\iota_{1}=\iota\otimes\iota+\Phi\iota\otimes\iota\Phi\qquad\text{and}\qquad\iota_{2}=\iota\otimes\iota+\iota\Phi\otimes\Phi\iota,$

respectively. We claim that they are almost $\iota_{K}$-locally equivalent. To
prove this, define an $\mathbb{F}_{2}[U]$-linear chain map $f\colon C\otimes
D\rightarrow C\otimes D$ by

$F=\operatorname{id}\otimes\operatorname{id}+f\otimes\Phi,$

where $f\colon C\rightarrow C$ is a chain map whose hat-flavored truncation
$\widehat{f}$ is homotopic to $\iota\Phi\iota$. Then $F$ satisfies
$U^{-1}F\sim\operatorname{id}$ since

$F^{2}=\operatorname{id}\otimes\operatorname{id}+f^{2}\otimes\Phi^{2}\sim\operatorname{id}\otimes\operatorname{id},$

and we also have

$\iota_{1}F=\iota\otimes\iota+\Phi\iota\otimes\iota\Phi+\iota^{2}\Phi\iota\otimes\iota\Phi+\Phi\iota^{2}\Phi\iota\otimes\iota\Phi^{2}\sim\iota\otimes\iota$

since $\iota^{2}\sim 1+\Phi\iota\Phi\iota\sim 1+\iota\Phi\iota\Phi$ and
$\Phi^{2}\sim 0$. Similarly, we also have

$F\iota_{2}=\iota\otimes\iota+\iota\Phi\otimes\Phi\iota+\iota\Phi\iota^{2}\otimes\Phi\iota+\iota\Phi\iota^{2}\Phi\otimes\Phi^{2}\iota\sim\iota\otimes\iota.$

So $\iota_{1}F+F\iota_{2}\sim 0$, and thus $F$ is an almost $\iota_{K}$-local
map from $(C\otimes D,\iota_{1})$ to $(C\otimes D,\iota_{2})$. However the
same argument also shows that $F$ also works as an almost $\iota_{K}$-local
map from $(C\otimes D,\iota_{2})$ to $(C\otimes D,\iota_{1})$. Therefore we
deduce that $\mathfrak{I}^{U}_{K}$ is an abelian group. ∎

######  Remark 2.7.

As in the case of (almost) $\iota_{K}$-local maps, given a knot $K$, its
minus-flavored knot Floer complex $CFK^{-}(S^{3},K)$, together with the action
of $\iota_{K}$ on $\widehat{CFK}(S^{3},K)$, defines an element of
$\mathfrak{I}^{U}_{K}$, which depends only on the concordance class of $K$.
This gives a well-defined map

$\mathfrak{A}:\mathcal{C}\rightarrow\mathfrak{I}^{U}_{K}.$

It follows from [Zem19] that $\mathfrak{A}$ is a group homomorphism.

We define a partial order on the group $\mathfrak{I}^{U}_{K}$ in the following
way. Given two elements $C$ and $D$ of $\mathfrak{I}^{U}_{K}$, we define
$C\leq D$ if there exists an almost $\iota_{K}$-local map from $C$ to $D$. It
is clear that the relation $\leq$ on $\mathfrak{I}^{U}_{K}$ is indeed a
partial order, since $C_{1}\leq C_{2}$ implies $C_{1}\otimes D\leq
C_{2}\otimes D$ for any $D\in\mathfrak{I}^{U}_{K}$. If neither $C\leq D$ nor
$D\leq C$ holds, we say that $C$ and $D$ are _incomparable_.

######  Example 2.8.

The trivial complex $C_{O}$ is defined as the trivial complex
$\mathbb{F}_{2}[U]$, whose generator 1 has bidegree $(0,0)$, together with the
trivial involution on its hat-flavored truncation. It is clear that $C_{O}$
represents the identity element of $\mathfrak{I}^{U}_{K}$ and this complex is
represented by the involutive knot Floer homology of the unknot $O$.

######  Example 2.9.

Consider the chain complex $C_{E}$, generated over $\mathbb{F}_{2}[U]$ by
elements $a,b,c,d,x$, where $a$ and $x$ has bidegree $(0,0)$. The differential
is given by

$\partial a=Ub,\,\partial c=Ud,\,\partial b=\partial d=\partial x=0.$

Its hat-flavored truncation $\widehat{C}_{E}$ is still generated by
$a,b,c,d,x$, but has zero differential. We define an involution $\iota_{E}$ on
$\widehat{C}_{E}$ by

$\iota_{E}(a)=a+x,\,\iota_{E}(b)=c,\,\iota_{E}(c)=b,\,\iota_{E}(d)=d,\,\iota_{E}(x)=x+d.$

Then $C_{E}=(C_{E},\iota_{E})$ is an element of $\mathfrak{I}^{U}_{K}$. Note
that this complex is represented by the involutive knot Floer homology of the
figure-eight knot $E$.

To see that $C_{E}$ is incomparable to $C_{O}$ in $\mathfrak{I}^{U}_{K}$,
suppose on the contrary that it is. Since the figure-eight knot is negative-
amphichiral, we have $-[C_{E}]=[C_{E}]\in\mathfrak{I}^{U}_{K}$. Therefore, we
should have an almost $\iota_{K}$-local map $f\colon C_{E}\rightarrow C_{O}$.
Since $x$ generates the homology of $U^{-1}C_{E}$, we should have $f(x)=1$.
However, since $\widehat{C}_{E}$ has zero differential and $C_{O}$ has trivial
involution, we should also have

$f(x)=f(a+\iota_{E}(a))=f(a)+f(\iota_{E}(a))=f(a)+\iota_{O}(f(a))=0,$

a contradiction. Therefore $C_{E}$ and $C_{O}$ are incomparable.

2.9 already shows that $\mathfrak{I}^{U}_{K}$ is not totally ordered. However,
we will see that, after taking quotient by the order 2 subgroup generated by
$C_{E}$, the induced partial order on the quotient group is a total ordering.
In other words, $\mathfrak{I}^{U}_{K}$ is totally ordered modulo $C_{E}$.

######  Lemma 2.10.

Let $(C,\iota)$ be a horizontal almost $\iota_{K}$-complex. Suppose that there
exists an element $a^{\prime}$ and a cycle $x^{\prime}$ in $C$ such that the
homology class of $x^{\prime}$ generates the homology of $U^{-1}C$ and
$a^{\prime}+\iota(a^{\prime})=x^{\prime}$ in $\widehat{C}$. Then there exists
an almost $\iota_{K}$-local map from $C_{E}$ to $C$.

###### Proof.

We may assume without loss of generality that the differential $\partial$ of
$C$ is reduced, i.e. the induced differential on $\widehat{C}$ is zero. Then
there exists an element $b^{\prime}\in C$ such that $\partial
a^{\prime}=Ub^{\prime}$. Choose a lift $c^{\prime}\in C$ of
$\iota(b^{\prime})\in\widehat{C}$, and choose again an element $d^{\prime}\in
C$ such that $\partial c^{\prime}=Ud^{\prime}$. Then we have in $\widehat{C}$
that

$d^{\prime}=\Phi\iota\Phi\iota(a^{\prime})=(1+\iota^{2})(a^{\prime})=(1+\iota)^{2}(a^{\prime})=x^{\prime}+\iota(x^{\prime}).$

Therefore the map from $C_{E}$ to $C$ which maps $a,b,c,d,x$ to
$a^{\prime},b^{\prime},c^{\prime},d^{\prime},x^{\prime}$, respectively, is an
almost $\iota_{K}$-local equivalence. ∎

######  Theorem 2.11.

Two elements $C,D\in\mathfrak{I}^{U}_{K}$ are incomparable if and only if
$[C]=[D]+[C_{E}]$.

###### Proof.

Let $C,D\in\mathfrak{I}^{U}_{K}$. First, suppose that $[C]=[D]+[C_{E}]$, then
we have [$C\otimes D^{\ast}]=[C_{E}]$. Since $C_{O}$ and $C_{E}$ are
incomparable by 2.9, we conclude that $C$ and $D$ are also incomparable.

For the other direction, we will show that for each
$C\in\mathfrak{I}^{U}_{K}$, if there is no almost $\iota_{K}$-local map from
$C_{E}$ to $C$, then there is an almost $\iota_{K}$-local map from $C$ to
$C_{O}$. This is enough since if $C$ and $D$ are incomparable, then $C\otimes
D^{\ast}$ and $C_{O}$ are incomparable, which means that there exist no almost
$\iota_{K}$-local maps to $C_{O}$ from both $C\otimes D^{\ast}$ and
$C^{\ast}\otimes D$. Then it follows from the assertion that there exist
almost $\iota_{K}$-local maps from $C_{E}$ to them. Since $[C_{E}]=-[C_{E}]$,
we deduce that $C\otimes D^{\ast}$ is almost $\iota_{K}$-locally equivalent to
$C_{E}$, and thus $[C]=[D]+[C_{E}]$.

Possibly after replacing $C$ by a homotopy equivalent complex, we may assume
that there exists an $\mathbb{F}_{2}[U]$-linear basis $B=\\{x,y_{i},z_{i}\\}$
for $C$, so that for each $i$, we have $\partial y_{i}=U^{n_{i}}z_{i}$ for
some $n_{i}>0$. For simplicity, we will write $1+\iota$ as $\omega$. Consider
the $\mathbb{F}_{2}$-vector spaces

$Z=\operatorname{span}_{\mathbb{F}_{2}}\\{z_{i}\\}\qquad\text{and}\qquad
W=\operatorname{span}_{\mathbb{F}_{2}}\\{\omega(g)\mid g\in B\\}$

which are linear subspaces of $\widehat{C}$. We claim that $x$ is not
contained in $Z+W$. Indeed, suppose it were. Then $x=z+w$ where $z$ is a
torsion cycle and $w\in\operatorname{Im}(\omega)$, which implies that there
exists an element $b$ in the $\mathbb{F}_{2}$-linear span of $B$ such that
$\omega(b)=x+z$. But now the homology class of $x+z$ generates
$H_{\ast}(U^{-1}C)$, so it follows from 2.10 that there exists an almost
$\iota_{K}$-local map from $C_{E}$ to $C$, a contradiction. The claim follows.

Now choose a homogeneous $\mathbb{F}_{2}$-basis $\\{p_{1},\ldots,p_{r}\\}$ for
$Z+W$. Extend the set $\\{x,p_{1},\ldots,p_{r}\\}$ to a homogenous
$\mathbb{F}_{2}$-basis

$\\{x,p_{1},\ldots,p_{r},q_{1},\ldots,q_{s}\\}$

for all of $\operatorname{span}_{\mathbb{F}_{2}}(B)$. Define

$C^{\prime}=\operatorname{span}_{\mathbb{F}_{2}[U]}\\{p_{1},\ldots,p_{r},q_{1},\ldots,q_{s}\\}.$

Since the image of $\partial$ is contained in the $\mathbb{F}_{2}[U]$-linear
span of $Z$, it is clear that $C^{\prime}$ is a subcomplex of $C$. Then the
quotient map

$C\rightarrow C/C^{\prime}\simeq\mathbb{F}_{2}[U]$

is a local map from $C$ to $C_{O}$, completing the proof. ∎

2.11 then allows us to define an additive $\mathbb{Z}/2\mathbb{Z}$-valued
invariant on the torsion subgroup of $\mathcal{C}$ via involutive knot Floer
homology, as follows.

######  Corollary 2.12.

The group homomorphism
$\mathfrak{A}\colon\mathcal{C}\rightarrow\mathfrak{I}^{U}_{K}$ in 2.7
restricts to a homomorphism

$\mathfrak{A}\colon\mathcal{C}_{T}\rightarrow\mathbb{Z}/2\mathbb{Z},$

where $\mathcal{C}_{T}$ denotes the torsion subgroup of the knot concordance
group $\mathcal{C}$ and $\mathbb{Z}/2\mathbb{Z}$ is the subgroup of
$\mathfrak{I}^{U}_{K}$ generated by the chain complex $C_{E}$.

###### Proof.

We need to prove that if $K$ is torsion in $\mathcal{C}$, then either
$\mathfrak{A}(K)=[C_{O}]$ or $\mathfrak{A}(K)=[C_{E}]$ in
$\mathfrak{I}^{U}_{K}$. Suppose not; then it follows from 2.11 that
$\mathfrak{A}(K)$ is comparable to $C_{O}$. Assume without loss of generality
that $\mathfrak{A}(K)\geq C_{O}$. Then we get

$[C_{O}]\leq\mathfrak{A}(K)\leq 2\mathfrak{A}(K)\leq\cdots\leq
n\mathfrak{A}(K)=\mathfrak{A}(nK)=[C_{O}],$

which implies $\mathfrak{A}(K)=[C_{O}]$, a contradiction. ∎

Furthermore, the existence of a total ordering on $\mathfrak{I}^{U}_{K}$
modulo $C_{E}$ gives a new method to determine whether the given knot has
infinite order in the concordance group.

######  Corollary 2.13.

If $\mathfrak{A}(K)$ is comparable to both $C_{O}$ and $C_{E}$, then $K$ has
infinite order in $\mathcal{C}$.

###### Proof.

If $K$ is torsion in $\mathcal{C}$, then it follows from 2.12 that
$\mathfrak{A}(K)=[C_{O}]$ or $\mathfrak{A}(K)=[C_{E}]$ in
$\mathfrak{I}^{U}_{K}$. This is a contradiction, since we know from 2.9 that
$C_{O}$ and $C_{E}$ are not comparable to each other. ∎

######  Example 2.14.

Note that $\mathfrak{A}(O)=0$ and $\mathfrak{A}(E)=1$ by definition. In
general, it follows directly from [HM17, Proposition 8.3] that for any quasi-
alternating torsion knot $K$, we have that
$\mathfrak{A}(K)=\operatorname{Arf}(K)$.

Finally, we will see that the invariant $\mathfrak{A}$ is insensitive under
orientation reversal for torsion knots, although its definition involves
involutive knot Floer homology and thus takes the orientation on the given
oriented knot into account in an essential way. This fact will be used in
Section 5.

######  Proposition 2.15.

For any torsion knot $K$, we have $\mathfrak{A}(K^{r})=\mathfrak{A}(K)$.

###### Proof.

It suffices to show that $\mathfrak{A}(K)=0$ implies $\mathfrak{A}(K^{r})=0$.
Suppose that $\mathfrak{A}(K)=0$. As observed in [BKST21], the involution
$\iota_{K^{r}}$ on $CFK_{UV}(S^{3},K^{r})=CFK_{UV}(S^{3},K)$ is the homotopy
inverse of $\iota_{K}$. By assumption, there exists an $\iota_{K}$-local map

$f\colon C_{O}\rightarrow CFK^{-}(S^{3},K),$

i.e. $f$ is a local map whose hat-flavored truncation $\widehat{f}$ satisfies
$\widehat{f}\sim\iota_{K}\widehat{f}$. Composing both sides with
$\iota_{K^{r}}$ then gives

$\widehat{f}\sim\iota_{K^{r}}\iota_{K}\widehat{f}\sim\iota_{K^{r}}\widehat{f},$

and thus $f$ is actually an $\iota_{K}$-local map from $C_{O}$ to
$CFK^{-}(S^{3},K^{r})$. Hence $\mathfrak{A}(K^{r})$ is comparable to $C_{O}$.
Since $K$ is torsion in $\mathcal{C}$, we see that $K^{r}$ should also be
torsion. By 2.12 and 2.13, we should have $\mathfrak{A}(K^{r})=0$, as desired.
∎

##  3\. Horizontal almost $\iota_{K}$-complexes from bordered Floer homology

Bordered Heegaard Floer homology, defined by Lipshitz, Ozsvath, and Thurston
in [LOT18], associates to a bordered 3-manifold $M$ (with connected boundary
$\Sigma$) a type-$D$ module $\widehat{CFD}(M)$ and a type-$A$ module
$\widehat{CFA}(M)$ over the $\mathbb{F}_{2}$-algebra $\mathcal{A}(\Sigma)$,
which are defined via counting holomorphic disks on any choice of bordered
Heegaard diagram of $M$ with respect to certain boundary conditions. When we
have to specify the diagram we are working on, we will denote them as
$\widehat{CFD}(H)$ and $\widehat{CFA}(H)$ where $H$ is our choice of a
bordered Heegaard diagram representing $M$. The categories of type-$D$ (or
type-$A$) modules also have a notion for morphisms between them, which are the
$\mathcal{A}(\Sigma)$-linear maps which commute with the differential in the
type-$D$ case and the maps which are compatible with the $A_{\infty}$
relations in the type-$A$ case. Note that we are not giving precise
definitions here; in general, we expect the readers to be familiar with
bordered Heegaard Floer theoretic machinaries.

When $M$ has two boundary components, say $\Sigma_{1}$ and $\Sigma_{2}$, we
instead have a type-$DA$ module $\widehat{CFDA}(M)$, which is a
$\mathcal{A}(\Sigma_{1})$-$\mathcal{A}(\Sigma_{2})$-bimodule. This bimodule is
also defined using bordered Heegaard diagrams representing $M$, and we will
denote its type-$DA$ module as $\widehat{CFDA}(H)$ when we have to specify a
Heegaard diagram $H$ that we are working with. As in the case of type-$D$ and
type-$A$ modules, type-$DA$ modules also have a notion of morphisms between
them, which are called type-$DA$ morphisms.

A similar construction also works when we have a knot inside a bordered
3-manifold, in which case we use doubly-pointed bordered Heegaard diagram such
that one basepoint lies on the boundary. Such diagrams give minus-flavored
type-$D$ and type-$A$ modules over $\mathbb{F}_{2}[U]$, denoted $CFD^{-}$ and
$CFA^{-}$, in which algebraic intersection numbers of pseudoholomorphic disks
with the interior basepoint is recorded by a formal variable $U$. The
truncations of those modules by $U=0$ are denoted again by $\widehat{CFD}$ and
$\widehat{CFA}$. Note that the truncation by $U=1$ is equivalent to forgetting
the knot.

This framework is compatible with gluing constructions in the following way.
If $M$ and $N$ are bordered 3-manifolds with connected boundary and $\partial
M=\partial N$, then we have

$\widehat{CF}(M\cup N)\simeq\widehat{CFA}(M)\boxtimes\widehat{CFD}(N).$

Similar formulae also holds for gluing bordered manifolds with one or two
boundary components, and also when we have a knot inside either $M$ or $N$
(and then the formula involves $CFA^{-}$ or $CFD^{-}$). The general rule is
that a type-$D$ boundary should be glued to a type-$A$ boundary.

When bordered manifolds have boundary components diffeomorphic to torus, the
associated $\mathbb{F}_{2}$-algebra $\mathcal{A}(T^{2})$ admits a simple
description, as follows. As an $\mathbb{F}_{2}$-vector space, we have

$\mathcal{A}(T^{2})=\mathrm{Span}_{\mathbb{F}_{2}}(\iota_{0},\iota_{1},\rho_{1},\rho_{2},\rho_{3},\rho_{12},\rho_{23},\rho_{123}),$

where $\iota_{0}$ and $\iota_{1}$ are idempotents which sum up to 1. We will
mainly focus on the case of knot complements, i.e. $S^{3}\smallsetminus K$, in
which case its $\widehat{CFD}$ can be computed directly from the
$\mathcal{R}$-coefficient knot Floer chain complex of $K$. We will also make
use of bordered manifolds with two torus boundaries, which arise as
complements of pattern knots in the $\infty$-framed solid torus, denoted
$T_{\infty}$.

We now briefly recall the involutive theory for bordered Heegaard Floer
theory, as defined in [HL19]. Let $M$ be a bordered 3-manifold with a torus
boundary. Choose a bordered Heegaard diagram
$H=(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},z)$ representing $M$ and its
conjugate diagram
$\overline{H}=(-\Sigma,\boldsymbol{\beta},\boldsymbol{\alpha},z)$. Since the
holomorphic disk counts for $H$ and $\overline{H}$ are identical, the identity
maps give a canonical isomorphism

$\operatorname{conj}:\widehat{CFD}(\overline{H})\rightarrow\widehat{CFD}(H).$

Unlike involutive Heegaard Floer homology of closed 3-manifolds, the conjugate
diagram $\overline{H}$ does not represent $M$, but rather represents
$\mathsf{AZ}\cup M$, where $\mathsf{AZ}$ denotes the Auroux-Zarev dualizing
piece, which is represented by the bordered diagram in Figure 3.1. Choosing a
sequence of Heegaard moves from $\mathsf{AZ}\cup H$ to $\overline{H}$ gives a
homotopy equivalence between
$\widehat{CFDA}(\mathsf{AZ})\boxtimes\widehat{CFD}(H)$ and
$\widehat{CFD}(\overline{H})$, and composing it with $\operatorname{conj}$
gives a homotopy equivalence

$\iota_{M}\colon\widehat{CFDA}(\mathsf{AZ})\boxtimes\widehat{CFD}(M)\rightarrow\widehat{CFD}(M).$

Note that, due to the lack of naturality in bordered Heegaard Floer homology,
the homotopy equivalence $\iota_{M}$ might not be defined uniquely up to
homotopy, although this ambiguity has almost no effect on our arguments
involving $\iota_{M}$. We denote the set of all possible maps $\iota_{M}$ that
can possibly arise from its definition by $\operatorname{Inv}_{D}(M)$. The
elements of $\operatorname{Inv}_{D}(M)$ are called _bordered involutions_ of
$M$.

Figure 3.1. A $\alpha$-$\beta$-bordered Heegaard diagram representing the
Auroux-Zarev piece $\mathsf{AZ}$.

One can also define bordered involutions for bordered 3-manifolds with two
torus boundaries. Given such a bordered 3-manifold $N$ and an
$\alpha$-$\beta$\- bordered Heegaard diagram $H$ representing it, we can
similarly define the conjugate diagram $\overline{H}$ of $H$. In this case, it
is shown in [LOT11, Theorem 4.6] that $\overline{H}$ represents
$\mathsf{AZ}\cup N\cup\overline{\mathsf{AZ}}$, where $\overline{\mathsf{AZ}}$
denotes the mirror of $\mathsf{AZ}$. Proceeding as in the one torus boundary
case, one can define the set $\operatorname{Inv}(N)$ of _bordered involutions_
$\iota_{N}$ of $N$, which are homotopy equivalences of the form

$\iota_{N}\colon\widehat{CFDA}(\mathsf{AZ})\boxtimes\widehat{CFDA}(N)\boxtimes\widehat{CFDA}(\overline{\mathsf{AZ}})\rightarrow\widehat{CFDA}(N).$

Recall that, given a bordered 3-manifold $N$ and $M$, where $N$ has two torus
boundaries and $M$ has one torus boundary, we have a pairing formula

$\widehat{CFD}(N\cup M)\simeq\widehat{CFDA}(N)\boxtimes\widehat{CFD}(M).$

Given any choices of bordered involutions $\iota_{N}\in\operatorname{Inv}(N)$
and $\iota_{M}\in\operatorname{Inv}_{D}(M)$, we can describe a bordered
involution of the glued manifold $N\cup M$ as follows. Consider the homotopy
equivalence

$\begin{split}\iota_{\iota_{N},\iota_{M}}\colon&\widehat{CFDA}(\mathsf{AZ})\boxtimes\widehat{CFD}(N\cup
M)\\\
&\simeq\widehat{CFDA}(\mathsf{AZ})\boxtimes\widehat{CFDA}(N)\boxtimes\widehat{CFD}(M)\\\
&\simeq\widehat{CFDA}(\mathsf{AZ})\boxtimes\widehat{CFDA}(N)\boxtimes\widehat{CFDA}(\overline{\mathsf{AZ}})\boxtimes\widehat{CFDA}(\mathsf{AZ})\boxtimes\widehat{CFD}(M)\\\
&\xrightarrow{\iota_{N}\boxtimes\iota_{M}}\widehat{CFDA}(N)\boxtimes\widehat{CFD}(M)\simeq\widehat{CFD}(N\cup
M).\end{split}$

Note that, in the construction of $\iota_{\iota_{N},\iota_{M}}$, we used a
homotopy equivalence

$\Omega\colon\widehat{CFDA}(\mathbb{I})\xrightarrow{\simeq}\widehat{CFDA}(\overline{\mathsf{AZ}})\boxtimes\widehat{CFDA}(\mathsf{AZ}),$

where $\mathbb{I}$ denotes the trivial cylinder $T^{2}\times[0,1]$, induced
from the observation that $\overline{\mathsf{AZ}}\cup\mathsf{AZ}$ is
diffeomorphic to $\mathbb{I}$. It is shown in [HL19, Lemma 4.4] that
$\widehat{CFDA}(\mathbb{I})$ is homotopy-rigid, so that $\Omega$ is uniquely
defined up to homotopy, and hence the homotopy class of
$\iota_{\iota_{N},\iota_{M}}$ depends only on the homotopy classes of
$\iota_{N}$ and $\iota_{M}$. Following the proof of [HL19, Theorem 5.1], we
can immediately see that $\iota_{\iota_{N},\iota_{M}}$ is a bordered
involution for $N\cup M$, so that we have a well-defined map

$\begin{split}\operatorname{Inv}(N)\times\operatorname{Inv}_{D}(M)&\rightarrow\operatorname{Inv}_{D}(N\cup
M),\\\ (\iota_{N},\iota_{M})&\mapsto\iota_{\iota_{N},\iota_{M}}.\end{split}$

Now, given a torsion knot $K$, its horizontal almost $\iota_{K}$-complex
$\mathfrak{A}(K)$ can be algebraically recovered from the involutive bordered
Floer homology of $S^{3}\smallsetminus K$ as follows. Consider the
$\infty$-framed solid torus $T_{\infty}$ and the longitudinal knot $\nu\subset
T_{\infty}$. Then we have a pairing formula:

$CFK^{-}(S^{3},K)\simeq
CFA^{-}(T_{\infty},\nu)\boxtimes\widehat{CFD}(S^{3}\smallsetminus K),$

where $S^{3}\smallsetminus K$ is 0-framed. Its hat-flavored truncation is
given by

$\widehat{CFK}(S^{3},K)\simeq\widehat{CFA}(T_{\infty},\nu)\boxtimes\widehat{CFD}(S^{3},K).$

Choose any bordered involution $\iota_{S^{3}\smallsetminus
K}\in\operatorname{Inv}_{D}(S^{3}\smallsetminus K)$. Then, as shown (in a
type-$D$ version) in [Kan22, Theorem 1.3], there exists a type-$A$ morphism

$\mathfrak{f}^{A}_{dual}\colon\widehat{CFA}(T_{\infty},\nu)\rightarrow\widehat{CFA}(T_{\infty},\nu)\boxtimes\widehat{CFDA}(\mathsf{AZ})$

such that the map

$\begin{split}\widehat{CFA}(T_{\infty},\nu)\boxtimes\widehat{CFD}(S^{3},K)&\xrightarrow{\mathfrak{f}^{A}_{dual}\boxtimes\operatorname{id}}\widehat{CFA}(T_{\infty},\nu)\boxtimes\widehat{CFDA}(\mathsf{AZ})\boxtimes\widehat{CFD}(S^{3},K)\\\
&\xrightarrow{\operatorname{id}\boxtimes\iota_{S^{3}\smallsetminus
K}}\widehat{CFA}(T_{\infty},\nu)\boxtimes\widehat{CFD}(S^{3},K)\end{split}$

is homotopic to either $\iota_{K}$ and $\iota^{-1}_{K}$. Since it follows from
the proof of [Kan22, Theorem 1.3] that the above map is either always
homotopic to $\iota_{K}$ for all $K$ or always homotopic to $\iota^{-1}_{K}$
for all $K$, we will assume that it is actually always homotopic to
$\iota_{K}$ for all knots $K$. Note that, if it were homotopic to
$\iota^{-1}_{K}$ instead, all proofs work in the same way with a slight
modification.

Now we will prove two lemmas, 3.1 and 3.3, which describe how to relate
type-$D$ morphisms between bordered Floer homology of 0-framed knot
complements to $\iota_{K}$-local maps between horizontal almost
$\iota_{K}$-complexes. These lemmas are crucial in the proofs of the main
theorems.

######  Lemma 3.1.

Given two knots $K_{1},K_{2}$ and a type-$D$ morphism
$f\colon\widehat{CFD}(S^{3}\smallsetminus
K_{1})\rightarrow\widehat{CFD}(S^{3}\smallsetminus K_{2})$, denote by
$\operatorname{Hat}(f)$ the induced map

$\begin{split}\widehat{CFK}(S^{3},K_{1})&\simeq\widehat{CFA}(T_{\infty},\nu)\boxtimes\widehat{CFD}(S^{3}\smallsetminus
K_{1})\\\ &\xrightarrow{\operatorname{id}\boxtimes
f}\widehat{CFA}(T_{\infty},\nu)\boxtimes\widehat{CFD}(S^{3}\smallsetminus
K_{2})\simeq\widehat{CFK}(S^{3},K_{2}).\end{split}$

Then, for any choices of $\iota_{S^{3}\smallsetminus
K_{i}}\in\operatorname{Inv}_{D}(S^{3}\smallsetminus K_{i})$ for $i=1,2$, we
have that

$\operatorname{Hat}\left(\iota_{S^{3}\smallsetminus
K_{2}}\circ(\operatorname{id}\boxtimes f)\circ\iota^{-1}_{S^{3}\smallsetminus
K}\right)\sim\iota_{K_{2}}\circ\operatorname{Hat}(f)\circ\iota^{-1}_{K_{1}},$

where $\iota_{K_{i}}$ denotes the involutive actions on
$\widehat{CFK}(S^{3},K_{i})$ for $i=1,2$.

###### Proof.

Observe that

$\begin{split}\left(\operatorname{id}\boxtimes\left(\iota_{S^{3}\smallsetminus
K_{2}}\circ(\operatorname{id}\boxtimes f)\circ\iota^{-1}_{S^{3}\smallsetminus
K_{1}}\right)\right)\circ\left(\operatorname{id}\boxtimes\iota_{S^{3}\smallsetminus
K_{1}}\right)\circ\left(\mathfrak{f}^{A}_{dual}\boxtimes\operatorname{id}\right)&\sim\left(\operatorname{id}\boxtimes\left(\iota_{S^{3}\smallsetminus
K_{2}}\circ(\operatorname{id}\boxtimes
f)\right)\right)\circ\left(\mathfrak{f}^{A}_{dual}\boxtimes\operatorname{id}\right)\\\
&\sim\left(\operatorname{id}\boxtimes\iota_{S^{3}\smallsetminus
K_{2}}\right)\circ\left(\mathfrak{f}^{A}_{dual}\boxtimes\operatorname{id}\right)\circ(\operatorname{id}\boxtimes
f).\end{split}$

Therefore we deduce that

$\operatorname{Hat}\left(\iota_{S^{3}\smallsetminus
K_{2}}\circ(\operatorname{id}\boxtimes f)\circ\iota^{-1}_{S^{3}\smallsetminus
K_{1}}\right)\circ\iota_{K_{1}}\sim\iota_{K_{2}}\circ\operatorname{Hat}(f),$

which implies

$\operatorname{Hat}\left(\iota_{S^{3}\smallsetminus
K_{2}}\circ(\operatorname{id}\boxtimes f)\circ\iota^{-1}_{S^{3}\smallsetminus
K_{1}}\right)\sim\iota_{K_{2}}\circ\operatorname{Hat}(f)\circ\iota^{-1}_{K_{1}},$

as desired. ∎

######  Definition 3.2.

Given two knots $K_{1},K_{2}$, we say that a type-$D$ morphism
$f\colon\widehat{CFD}(S^{3}\smallsetminus
K_{1})\rightarrow\widehat{CFD}(S^{3}\smallsetminus K_{2})$ is _local_ if the
induced map

$\begin{split}\widehat{CF}(S^{3})&\simeq\widehat{CFA}(T_{\infty})\boxtimes\widehat{CFD}(S^{3}\smallsetminus
K_{1})\\\ &\xrightarrow{\operatorname{id}\boxtimes
f}\widehat{CFA}(T_{\infty})\boxtimes\widehat{CFD}(S^{3}\smallsetminus
K_{2})\simeq\widehat{CF}(S^{3})\end{split}$

is a homotopy equivalence. Furthermore, a local morphism
$f\colon\widehat{CFD}(S^{3}\smallsetminus
K_{1})\rightarrow\widehat{CFD}(S^{3}\smallsetminus K_{2})$ is said to be
_almost bordered- $\iota$-local_ if $f$ is local and

$\operatorname{Hat}\left(f+\iota_{S^{3}\smallsetminus
K_{2}}\circ(\operatorname{id}\boxtimes f)\circ\iota^{-1}_{S^{3}\smallsetminus
K_{1}}\right)\sim 0$

for some choices of $\iota_{S^{3}\smallsetminus
K_{i}}\in\operatorname{Inv}_{D}(S^{3}\smallsetminus K_{i})$ for $i=1,2$.

In fact, the definition of bordered-$\iota$-locality does not depend on the
choices of $\iota_{S^{3}\smallsetminus K_{i}}$ by the following lemma.

######  Lemma 3.3.

Given two knots $K_{1},K_{2}$ and a local morphism
$f\colon\widehat{CFD}(S^{3}\smallsetminus
K_{1})\rightarrow\widehat{CFD}(S^{3}\smallsetminus K_{2})$, denote by
$\operatorname{Minus}(f)$ the induced map

$\begin{split}CFK^{-}(S^{3},K_{1})&\simeq
CFA^{-}(T_{\infty},\nu)\boxtimes\widehat{CFD}(S^{3}\smallsetminus K_{1})\\\
&\xrightarrow{\operatorname{id}\boxtimes
f}CFA^{-}(T_{\infty},\nu)\boxtimes\widehat{CFD}(S^{3}\smallsetminus
K_{2})\simeq CFK^{-}(S^{3},K_{2}).\end{split}$

Then for any choices of $\iota_{S^{3}\smallsetminus
K_{i}}\in\operatorname{Inv}_{D}(S^{3}\smallsetminus K_{i})$ for $i=1,2$, the
map $\operatorname{Minus}(f)$ is an almost $\iota_{K}$-local map if and only
if $f$ is almost bordered-$\iota$-local.

###### Proof.

Since $f$ is local, we know that $\operatorname{Minus}(f)$ is also a local
map, since $\widehat{CFA}(T_{\infty})$ is the truncation of
$CFA^{-}(T_{\infty},\nu)$ by $U=1$. By 3.1, we have

$\begin{split}\operatorname{Hat}\left(f+\iota_{S^{3}\smallsetminus
K_{2}}\circ(\operatorname{id}\boxtimes f)\circ\iota^{-1}_{S^{3}\smallsetminus
K_{1}}\right)&\sim\operatorname{Hat}(f)+\iota_{K_{2}}\circ\operatorname{Hat}(f)\circ\iota_{K_{1}}^{-1}\\\
&\sim\left(\operatorname{Hat}(f)\circ\iota_{K_{1}}+\iota_{K_{2}}\circ\operatorname{Hat}(f)\right)\circ\iota_{K_{1}}^{-1}.\end{split}$

Since $\operatorname{Hat}(f)$ is the hat-flavored truncation of
$\operatorname{Minus}(f)$, we deduce that $\operatorname{Minus}(f)$ is an
almost $\iota_{K}$-local map if and only if

$\iota_{K_{2}}\circ\operatorname{Hat}(f)\sim\operatorname{Hat}(f)\circ\iota_{K_{1}},$

and if and only if

$\operatorname{Hat}\left(f+\iota_{S^{3}\smallsetminus
K_{2}}\circ(\operatorname{id}\boxtimes f)\circ\iota^{-1}_{S^{3}\smallsetminus
K_{1}}\right)\sim 0,$

which concludes the proof. ∎

##  4\. $(\text{odd},1)$-cables of torsion knots

In this section, we will prove a part of 1.1 which states that if $K$ is
torsion in $\mathcal{C}$ with $\mathfrak{A}(K)=1$, then $K_{2n+1,1}$ has
infinite order in $\mathcal{C}$, and then improve its proof a bit more to
prove 1.2. We start by observing that the $\mathcal{R}$-coefficient knot Floer
chain complex of a torsion knot has a $\mathcal{R}$ summand.

######  Lemma 4.1.

If $K$ is torsion in $\mathcal{C}$, then $CFK_{\mathcal{R}}(S^{3},K)$ admits a
direct summand isomorphic to $\mathcal{R}$, which is concentrated at bidegree
$(0,0)$.

###### Proof.

The local equivalence group $\mathfrak{K}$ of knot-like complexes, defined in
[DHST21], is endowed with a total ordering. In particular, $\mathfrak{K}$ is
torsion-free. Hence we deduce that $CFK_{\mathcal{R}}(S^{3},K)$ is locally
trivial, which implies that there exist almost local maps

$f\colon\mathcal{R}\rightarrow
CFK_{\mathcal{R}}(S^{3},K)\qquad\text{and}\qquad\,g:CFK_{\mathcal{R}}(S^{3},K)\rightarrow\mathcal{R}.$

Since $g\circ f$ is a local map from $\mathcal{R}$ to itself, it should be
homotopic to the identity map. Therefore $f$ and $g$ induce a splitting

$CFK_{\mathcal{R}}(S^{3},K)\simeq\mathcal{R}\oplus A$

for some $\mathcal{R}$-acyclic summand $A$. ∎

Recall that, for a torsion knot $K$, the value $\mathfrak{A}(K)$ is 0 if the
horizontal almost $\iota_{K}$-complex $CFK^{-}(K)$ is almost
$\iota_{K}$-locally equivalent to the $C_{O}$, and $1$ if it is almost
$\iota_{K}$-locally equivalent to $C_{E}$. Since the notion of almost
$\iota_{K}$-local equivalence involves chain maps which are not
$\mathcal{R}$-linear, but only $\mathbb{F}_{2}[U]$-linear, it is not quite
compatible with the bordered techniques. However, due to 4.1, we can say a bit
more than that, and as a result, we have the following lemma.

######  Lemma 4.2.

If $K$ is torsion in $\mathcal{C}$ with $\mathfrak{A}(K)=1$, then there exists
another knot $K^{\prime}$, concordant to $K$, such that
$\widehat{CFD}(S^{3}\smallsetminus K^{\prime})$ has a direct summand
isomorphic to $\widehat{CFD}(S^{3}\smallsetminus E)$. Furthermore, the
inclusion map $i\colon\widehat{CFD}(S^{3}\smallsetminus
E)\rightarrow\widehat{CFD}(S^{3}\smallsetminus K^{\prime})$ is almost
bordered-$\iota$-local.

###### Proof.

Let $n$ be a positive integer such that $nK$ is slice. Then
$2n(K\\#E)=2nK\\#2nE$ is slice, so $K\\#E$ also is torsion in $\mathcal{C}$.
By 4.1, we see that $CFK_{\mathcal{R}}(S^{3},K\\#E)$ admits a direct summand
isomorphic to $\mathcal{R}$, concentrated at bidegree $(0,0)$; denote its
generator as $x$. Since
$\mathfrak{A}(K\\#E)=\mathfrak{A}(K)+\mathfrak{A}(E)=0$, the knot Floer
complex of $K\\#E$ is almost $\iota_{K}$-locally equivalent to $C_{O}$, so
there exists an almost $\iota_{K}$-local map $f\colon C_{O}\rightarrow
CFK^{-}(S^{3},K\\#E)$. Write $f(1)=x+c$ for some cycle $c$ which represents a
$U$-torsion class in homology. Then we have

$\partial
y=(\widehat{f}\circ\iota_{C_{O}}+\iota_{K\\#E}\circ\widehat{f})(1)=x+c+\iota_{K\\#E}(x+c)$

for some $y\in\widehat{CFK}(S^{3},K\\#E)$. Choose any lifts to
$CFK_{\mathcal{R}}(S^{3},K)$ of $c$ and $y$, and denote them again by $c$ and
$y$. Then we get

$\iota_{K\\#E}(x)=\iota_{K\\#E}(x+c)+\iota_{K\\#E}(c)=x+c+\iota_{K\\#E}(c)+\partial
y\pmod{U,V},$

Assume without loss of generality that the differential on
$CFK_{\mathcal{R}}(S^{3},K\\#E)$ is reduced, i.e. its truncation by $U=V=0$ is
zero. Then, for some $\alpha,\beta\in CFK_{\mathcal{R}}(S^{3},K\\#E)$, we can
write

$\iota_{K\\#E}(x)=x+c+\iota_{K\\#E}(c)+U\alpha+V\beta.$

Also, we can decompose the differential $\partial$ of
$CFK_{\mathcal{R}}(S^{3},K\\#E)$ as

$\partial=\partial_{U}+\partial_{V},$

where $\partial_{U}$ and $\partial_{V}$ consists of terms containing powers of
$U$ and $V$, respectively.

Recall that $c$ is a lift of a cycle in $CFK^{-}(S^{3},K\\#E)$. Since $UV=0$
in $\mathcal{R}$, we should have $\partial_{U}(c)=0$. Since $\iota_{K\\#E}$ is
a chain skew-map, we have
$\partial_{U}\circ\iota_{K\\#E}=\iota_{K\\#E}\circ\partial_{V}$. Thus we get

$\begin{split}\partial_{V}(c)&=\partial_{V}(x+\iota_{K\\#E}(x))+\partial_{V}(\iota_{K\\#E}(c))+U\partial_{V}(\alpha)+V\partial_{V}(\beta)\\\
&=\iota_{K\\#E}(\partial_{U}(c))+V\partial_{V}(\beta)=V\partial_{V}(\beta).\end{split}$

Hence, if we define $c^{\prime}=c+V\beta$, then we have

$\begin{split}\partial_{U}(c^{\prime})&=V\partial_{U}(\beta)=0,\\\
\partial_{V}(c^{\prime})&=\partial_{V}(c)+V\partial_{V}(\beta)=0,\end{split}$

so $c^{\prime}$ is a cycle. Since $x$ generates a free $\mathcal{R}$ summand
of $CFK_{\mathcal{R}}(S^{3},K\\#E)$ and the homology class of $c^{\prime}$ is
$\mathcal{R}$-torsion, we deduce that $x+c^{\prime}$ also generates a free
summand. Furthermore, since $c^{\prime}=c\pmod{U,V}$, we have

$\iota_{K\\#E}(x+c^{\prime})=\iota_{K\\#E}(x+c)=x+c\pmod{U,V}.$

Now take $K^{\prime}:=K\\#2E$. Since $x+c^{\prime}$ generates a free summand
of $CFK_{\mathcal{R}}(S^{3},K\\#E)$, we see that $(x+c^{\prime})\otimes
CFK_{\mathcal{R}}(S^{3},E)$ is a direct summand of
$CFK_{\mathcal{R}}(S^{3},K^{\prime})$. It follows from [LOT11, Theorem 11.26]
that direct summands of $CFK_{\mathcal{R}}$ of a knot induce direct summands
of $\widehat{CFD}$ of its 0-framed complement, so there exists a direct
summand of $\widehat{CFD}(S^{3}\smallsetminus K^{\prime})$ isomorphic to
$\widehat{CFD}(S^{3}\smallsetminus E)$ such that the inclusion map
$i\colon\widehat{CFD}(S^{3}\smallsetminus
E)\rightarrow\widehat{CFD}(S^{3}\smallsetminus K^{\prime})$ satisfies the
property that $\mathrm{Minus}(i)$ is an almost $\iota_{K}$-local map. ∎

For $p>1$, denote the $(2n+1,-1)$-cabling pattern by $C_{p}$. The type-$A$
module $\widehat{CFA}(T_{\infty},C_{p})$ was computed in [OSS17, Lemma 8.3].
To summarize, it admits a model generated by an $\iota_{0}$-element $z$ and
$\iota_{1}$-elements $b_{1},\cdots,b_{2n+1},a_{1},\cdots,a_{2n+1}$. The
$A_{\infty}$-operations are given as follows.

$\textstyle{a_{2n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{2}}$$\scriptstyle{U^{2n+1}}$$\textstyle{a_{2p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{2},\rho_{1}}$$\scriptstyle{U^{2p}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{2},\rho_{1}}$$\textstyle{a_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{2},\rho_{1}}$$\scriptstyle{U^{2}}$$\textstyle{a_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{2},\rho_{1}}$$\scriptstyle{U}$$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{3}}$$\textstyle{b_{2n+1}}$$\textstyle{b_{2p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U\cdot(\rho_{2},\rho_{1})}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U\cdot(\rho_{2},\rho_{1})}$$\textstyle{b_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U\cdot(\rho_{2},\rho_{1})}$$\textstyle{b_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U\cdot(\rho_{2},\rho_{1})}$

Recall from [HKPS20, Proposition 3.1] that $CFK_{UV}(S^{3},E_{2n+1,-1})$ has
generators

$a,b,c,d,e,f,g,b_{0,i},\cdots,g_{0,i},b_{1,i},\cdots,g_{1,i},b_{0,p},\cdots,f_{0,p},b_{1,p},\cdots,g_{1,p}.$

for $1\leq i\leq p-1$. Among them, $a$ generates a direct summand and
$b,c,d,e,f,g$ generate the “central” summand, satisfying
$\iota_{E_{2n+1,-1}}(b)=b+a\pmod{U,V}$ and
$\iota_{E_{2n+1,-1}}(a)=a\pmod{U,V}$.

It follows from [LOT18, Theorem 11.26] that the type-$D$ module
$\widehat{CFD}(S^{3}\smallsetminus E)$ is generated by 9 elements

$e_{0},f_{0},g_{0},h_{0},e_{1},f_{1},g_{1},h_{1},w,$

where the differential is given as follows.

$\textstyle{f_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{1}}$$\textstyle{e_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{2}}$$\textstyle{e_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{3}}$$\scriptstyle{\rho_{1}}$$\textstyle{f_{1}}$$\textstyle{h_{1}}$$\textstyle{\oplus}$$\textstyle{w\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{12}}$$\textstyle{g_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{123}}$$\textstyle{g_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{2}}$$\textstyle{h_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{3}}$$\scriptstyle{\rho_{123}}$

We consider the elements of $CFK^{-}(S^{3},E_{2n+1,-1})$ which correspond to
$z\otimes w$, $z\otimes e_{0}$, and $b_{2p+i}\otimes h_{1}$ under the pairing
formula

$CFK^{-}(S^{3},E_{2n+1,-1})\simeq
CFA^{-}(T_{\infty},\nu)\boxtimes\widehat{CFD}(S^{3}\smallsetminus E).$

######  Lemma 4.3.

The element $b$ can be written mod $U$ as an $\mathbb{F}_{2}$-linear
combination of $a_{1}\otimes h_{1},\cdots,a_{2n+1}\otimes h_{1}$.

###### Proof.

It follows from the $A_{\infty}$ operations of $CFA^{-}(T_{\infty},\nu)$ that
$\partial(a_{i}\otimes h_{1})=U^{i}b_{i}\otimes h_{1}$. Also, $b_{i}\otimes
h_{1}$ are cycles in $HFK^{-}(S^{3},E_{2n+1,-1})$, which are not boundaries
since $\rho_{3}h_{1}$ is not a cycle. Furthermore, it is clear that $z\otimes
w$ and $a_{i}\otimes h_{1}$ have bidegree $(0,0)$. Comparing this with
computations in [HKPS20], we deduce that the $\mathbb{F}_{2}$-linear span of
those elements should contain either $b+a+U\gamma$ or $b+U\gamma$ for some
cycle $\gamma$. After a basis change by $b\mapsto b+a$, we may assume without
loss of generality that the span actually contains $b+U\gamma$. ∎

######  Lemma 4.4.

Let $K$ be a knot such that $CFK_{\mathcal{R}}(S^{3},K)$ admits a direct
summand isomorphic to $\mathcal{R}$, which is concentrated at bidegree
$(0,0)$. If $f\colon\widehat{CFD}(S^{3}\smallsetminus
E)\rightarrow\widehat{CFD}(S^{3}\smallsetminus K)$ is a type-$D$ morphism with
$\operatorname{Hat}(f)\sim 0$, then the map

$\begin{split}f_{2n+1,-1}\colon\widehat{CFD}(S^{3}\smallsetminus
E_{2n+1,-1})&\simeq\widehat{CFDA}(T_{\infty}\smallsetminus
C_{p})\boxtimes\widehat{CFD}(S^{3}\smallsetminus E)\\\
&\xrightarrow{\operatorname{id}\boxtimes
f}\widehat{CFDA}(T_{\infty}\smallsetminus
C_{p})\boxtimes\widehat{CFD}(S^{3}\smallsetminus
K)\simeq\widehat{CFD}(S^{3}\smallsetminus K_{2n+1,-1})\end{split}$

satisfies $\operatorname{Hat}(f_{2n+1,-1})(a_{i}\otimes h_{1})=0$ for each
$1\leq i\leq 2n+1$.

###### Proof.

We may assume without loss of generality that the tensored differential
obtained by box-tensoring the type-$D$ modules of 0-framed knot complements
with $\widehat{CFA}(T_{\infty},\nu)$ is zero. Then the condition
$\operatorname{Hat}(f)\sim 0$ becomes $\operatorname{Hat}(f)=0$. Since
$\widehat{CFA}(T_{\infty},\nu)$ has one $\iota_{0}$-generator, no
$\iota_{1}$-generators, and no nontrivial $A_{\infty}$ operations, it follows
that $f(w)=\sum_{b\in B}\rho_{b}b$, where $B$ is a basis of
$\widehat{CFD}(S^{3}\smallsetminus K)$ and
$\rho_{b}\in\\{\rho_{1},\rho_{2},\rho_{3},\rho_{12},\rho_{23},\rho_{123}\\}$.
Suppose that $\operatorname{Hat}(f_{2n+1,-1})(a_{i}\otimes h_{1})\neq 0$ for
some $1\leq i\leq 2n+1$. Recall from [LOT18, Theorem 11.26] that
$\widehat{CFD}$ of a knot complement consists of stable chains and unstable
chains, together with possible linear changes of basis along the endpoints of
stable chains. Due to the assumption that $CFK_{\mathcal{R}}(S^{3},K)$ admits
an $\mathcal{R}$ summand, we should have a splitting

$\widehat{CFD}(S^{3}\smallsetminus K)\simeq\widehat{CFD}(S^{3}\smallsetminus
O)\oplus N$

and $N$ consists only of stable chains. Denote by $t$ the generator of
$\widehat{CFD}(S^{3}\smallsetminus O)$, so that $\partial t=\rho_{12}t$.
Suppose that $f(h_{1})$, which is an element of
$\widehat{CFD}(S^{3}\smallsetminus K)$, is not contained in the $N$ summand.
Denote by $p_{u}$ the projection

$p_{u}\colon\widehat{CFD}(S^{3}\smallsetminus
K)\rightarrow\widehat{CFD}(S^{3}\smallsetminus O).$

Then we should have $p_{u}(f(h_{1}))=\rho_{2}t$. But then we get

$\rho_{12}p_{u}(f(e_{0}))=\partial p_{u}(f(e_{0}))=p_{u}(f(\partial
e_{0}))=\rho_{3}p_{u}(f(e_{1}))+\rho_{12}w.$

Since $\rho_{12}w$ is not a multiple of $\rho_{3}$, it follows that
$p_{u}(f(e_{0}))=w$, which would imply that $\operatorname{Hat}(f)$ cannot be
nullhomotopic, a contradiction. Hence we should have $f(h_{1})\in N$.

Recall again that $N$ consists of stable chains. Denote the sets of its
$\iota_{0}$-generators and $\iota_{1}$-generators by $B_{0}$ and $B_{1}$,
respectively. Since $h_{1}$ is a cycle and all nontrivial
$A_{\infty}$-operations of $\widehat{CFA}(T_{\infty},C_{p})$ starting from
$a_{i}$ begins with $\rho_{2}$, in order to have
$\operatorname{Hat}(f_{2n+1,-1})(a_{i}\otimes h_{1})\neq 0$, we should have

$f(h_{1})=\rho_{2}\cdot\sum_{\alpha\in S_{0}}\alpha+(\text{terms involving
}\iota_{1}\text{-generators})$

for some nonempty subset $S_{0}\subset B_{0}$. Indeed, assume that
$S_{0}=\emptyset$. Then we can write

$f(h_{1})=\sum_{\beta\in S_{1}}\beta+\rho_{23}\cdot\sum_{\beta\in T_{1}}\beta$

for some $S_{1},T_{1}\subset B_{1}$ with $S_{1}\cap T_{1}=\emptyset$. Since
$h_{1}$ is a cycle and no nontrivial $A_{\infty}$-operations of
$\widehat{CFA}(T_{\infty},C_{p})$ starts with $\rho_{23}$, we should have
$S_{1}\neq\emptyset$. Since $\partial e_{0}=\rho_{3}e_{1}+\rho_{1}h_{1}$, we
have

$\partial f(e_{0})=\rho_{3}f(e_{1})+\rho_{1}\sum_{\beta\in
S_{1}}\beta+\rho_{123}\sum_{\beta\in T_{1}}\beta.$

Since $N$ consists only of stable chains, the above equation implies that each
$\beta\in S_{1}$ must be contained in a stable chain of the form

$a_{\beta}\xrightarrow{\rho_{1}}\beta\xleftarrow{\rho_{23}}\cdots\xleftarrow{\rho_{123}}a^{\prime}_{\beta}$

and $f(e_{0})$ contains $a_{\beta}$. But then, if $s$ denotes the generator of
$\widehat{CFA}(T_{\infty},\nu)$, the image under $\operatorname{Hat}(f)$ of
$e_{0}\otimes s$ must contain $a_{\beta}\otimes s$, which implies that
$\operatorname{Hat}(f)(e_{0}\otimes s)$ is a cycle in $\widehat{CFK}(S^{3},K)$
whose homology class is nontrivial. Since $e_{0}\otimes s$ is a cycle of
$\widehat{CFK}(S^{3},E)$, this is a contradiction. Hence
$\operatorname{Hat}(f)$ is not nullhomotopic, a contradiction.

Since each $\alpha\in S_{0}$ is either the startpoint or the endpoint of a
stable chain, we should have either
$\partial(\rho_{2}\alpha)=\rho_{23}b_{\alpha}$ for some $b_{\alpha}\in B_{1}$
or $\rho_{2}\alpha=\partial b_{\alpha}$ for some $b_{\alpha}\in B_{1}$.
However, if $\rho_{2}\alpha=\partial b_{\alpha}$, then since $h_{1}$ is a
cycle, one can eliminate $\rho_{2}\alpha$ from $f(h_{1})$ by homotoping $f$.
Hence we may assume without loss of generality that we do not have such basis
elements in $S_{0}$. Then we must have

$0=f(\partial h_{1})=\partial f(h_{1})=\rho_{23}\cdot\sum_{\alpha\in
S_{0}}b_{\alpha}+\partial(\text{terms involving
}\iota_{1}\text{-generators}).$

Furthermore, for any $\iota_{1}$-generator $\beta\in B_{1}$, its differential
$\partial\beta$ is always of the form $\rho_{2}\gamma$ or $\rho_{23}\gamma$,
and the latter terms should cancel out the $\rho_{23}b_{\alpha}$ terms in the
above equation. However, since each $b_{\alpha}$ is in a stable chain of the
form

$\alpha\xrightarrow{\rho_{3}}b_{\alpha}\xrightarrow{\rho_{23}}\cdots\xrightarrow{\rho_{2}}c_{\alpha},$

there is no $\iota_{1}$-generator whose differential can cancel out
$\rho_{23}b_{\alpha}$. Hence no cancellation can happen, which means that we
should have $S_{0}=\emptyset$, a contradiction. Therefore
$\operatorname{Hat}(f_{2n+1,-1})(a_{i}\otimes h_{1})=0$. ∎

We are finally ready to prove that $K_{2n+1,1}$ has infinite order in
$\mathcal{C}$.

######  Theorem 4.5.

If $K$ is torsion in $\mathcal{C}$ with $\mathfrak{A}(K)=1$, then for any
nonzero integer $n$ the cable $K_{2n+1,1}$ has infinite order in
$\mathcal{C}$.

###### Proof.

Note that $K_{2n+1,1}$ has infinite order in $\mathcal{C}$ if and only if
$-K_{2n+1,1}=(-K)_{2n+1,-1}$ has infinite order in $\mathcal{C}$. Moreover,
since $\mathfrak{A}(K)=\mathfrak{A}(-K)$, it is enough to show that
$K_{2n+1,-1}$ has infinite order in $\mathcal{C}$.

By 3.3 and 4.2, we may assume that there exists a type-$D$ morphism
$f\colon\widehat{CFD}(S^{3}\smallsetminus
E)\rightarrow\widehat{CFD}(S^{3}\smallsetminus K)$ such that, for any choice
of bordered involutions $\iota_{S^{3}\smallsetminus
E}\in\operatorname{Inv}_{D}(S^{3}\smallsetminus E)$ and
$\iota_{S^{3}\smallsetminus K}\in\operatorname{Inv}_{D}(S^{3}\smallsetminus
K)$, we have

$\operatorname{Hat}\left(f+\iota_{S^{3}\smallsetminus K}\circ
f\circ\iota^{-1}_{S^{3}\smallsetminus E}\right)\sim 0.$

Then, by 4.3 and 4.4, we have

$\operatorname{Hat}\left(f_{2n+1,-1}+\left(\iota_{S^{3}\smallsetminus
K}\circ\left(\operatorname{id}\boxtimes
f\right)\circ\iota^{-1}_{S^{3}\smallsetminus
E}\right)_{2n+1,-1}\right)(b)=\operatorname{Hat}\left(\left(f+\iota_{S^{3}\smallsetminus
K}\circ\left(\operatorname{id}\boxtimes
f\right)\circ\iota^{-1}_{S^{3}\smallsetminus E}\right)_{2n+1,-1}\right)(b)=0.$

Since it follows from the description of bordered involutions of glued
manifolds that

$\left(\iota_{S^{3}\smallsetminus K}\circ(\operatorname{id}\boxtimes
f)\circ\iota^{-1}_{S^{3}\smallsetminus
E}\right)_{2n+1,-1}\sim\iota_{S^{3}\smallsetminus
K_{2n+1,-1}}\circ(\operatorname{id}\boxtimes
f_{2n+1,-1})\boxtimes\iota^{-1}_{S^{3}\smallsetminus E_{2n+1,-1}},$

we deduce from 3.1 that

$\begin{split}&\operatorname{Hat}\left(f_{2n+1,-1}\right)(b)+\left(\iota_{K_{2n+1,-1}}\circ\operatorname{Hat}(f_{2n+1,-1})\circ\iota^{-1}_{E_{2n+1,-1}}\right)(b)\\\
&=\operatorname{Hat}\left(f_{2n+1,-1}+\iota_{S^{3}\smallsetminus
K_{2n+1,-1}}\circ(\operatorname{id}\boxtimes
f_{2n+1,-1})\boxtimes\iota^{-1}_{S^{3}\smallsetminus
E_{2n+1,-1}}\right)(b)=0.\end{split}$

Since $\iota^{-1}_{E_{2n+1,-1}}(b)=b+a\pmod{U,V}$ on
$CFK_{\mathcal{R}}(S^{3},E_{2n+1,-1})$, we get

$\iota^{-1}_{K_{2n+1,-1}}\circ\operatorname{Hat}(f_{2n+1,-1})(b)=\operatorname{Hat}(f_{2n+1,-1})(b+a).$

Denote the differentials of minus-flavored knot Floer complexes by
$\partial^{-}$. Then $\partial^{-}b$ is a multiple of $U^{n}$, so
$\partial^{-}\operatorname{Minus}(f_{2n+1,-1})(b)$ should also be a multiple
of $U^{n}$. Since $n>1$, we have
$\Phi\circ\operatorname{Minus}(f_{2n+1,-1})(b)=0\pmod{U}$, which implies that
$\Phi\circ\operatorname{Hat}(f_{2n+1,-1})(b)=0$. Hence we have

$\begin{split}\iota_{K_{2n+1,-1}}\circ\operatorname{Hat}(f_{2n+1,-1})(b)&=\iota^{-1}_{K_{2n+1,-1}}\circ\left(1+\iota_{K_{2n+1,-1}}\Phi\iota_{K_{2n+1,-1}}\Phi\right)\circ\mathrm{Hat}(f_{2n+1,-1})(b)\\\
&=\iota^{-1}_{K_{2n+1,-1}}\circ\operatorname{Hat}(f_{2n+1,-1})(b)\\\
&=\operatorname{Hat}(f_{2n+1,-1})(b+a).\end{split}$

Furthermore, since $a$ is a cycle which generates the homology of
$U^{-1}CFK^{-}(S^{3},E_{2n+1,-1})$, its image
$\operatorname{Minus}(f_{2n+1,-1})(a)$ should also be a cycle which generates
the homology of $U^{-1}CFK^{-}(S^{3},K_{2n+1,-1})$. It then follows from 2.10
that $C_{E}\leq\mathfrak{A}(K_{2n+1,-1})$ in $\mathfrak{I}^{U}_{K}$.

Now, we have

$\begin{split}\iota_{K_{2n+1,-1}}\circ\operatorname{Hat}(f_{2n+1,-1})(a)&=\iota_{K_{2n+1,-1}}\circ\left(\operatorname{Hat}(f_{2n+1,-1})(b)+\iota_{K_{2n+1,-1}}\circ\operatorname{Hat}(f_{2n+1,-1})(b)\right)\\\
&=\operatorname{Hat}(f_{2n+1,-1})(a)+\iota_{K_{2n+1,-1}}\Phi\iota_{K_{2n+1,-1}}\Phi\circ\operatorname{Hat}(f_{2n+1,-1})(b)\\\
&=\operatorname{Hat}(f_{2n+1,-1})(a).\end{split}$

Hence the inclusion of the subcomplex of $CFK^{-}(S^{3},K_{2n+1,-1})$
generated by $\operatorname{Minus}(f_{2n+1,-1})(a)$ is an almost
$\iota_{K}$-local map by 3.3. This implies that
$C_{O}\geq\mathfrak{A}(K_{2n+1,-1})$. Therefore it follows from 2.13 that
$C_{O}<\mathfrak{A}(K_{2n+1,-1})$ and hence $K_{2n+1,-1}$ has infinite order
in $\mathcal{C}$. ∎

To prove 1.2, we have to approximate $CFK^{-}(S^{3},K_{2n+1,-1})$ by simpler
horizontal almost $\iota_{K}$-complexes. To do so, for each $n>1$, we consider
an element $C_{n}$ of $\mathfrak{I}^{U}_{K}$, generated by elements
$a_{n},b_{n},c_{n},d_{n},x_{n}$, where $a_{n}$ and $x_{n}$ have bidegree
$(0,0)$. The differential is given by

$\partial a_{n}=U^{n}b_{n},\,\partial c_{n}=U^{n}d_{n},\,\partial
b_{n}=\partial d_{n}=\partial x_{n}=0$

and the involution (on its hat-flavored truncation $\widehat{C}_{n}$) is given
by

$\iota_{C_{n}}(a_{n})=a_{n}+x_{n},\,\iota_{C_{n}}(b_{n})=c_{n},\,\iota_{C_{n}}\text{
fixes }d_{n},x_{n}.$

Since $\Phi=0$ on $\widehat{C}_{n}$, it is clear that $C_{n}$ is a well-
defined horizontal almost $\iota_{K}$-complex. Furthermore, since the chain
map $f_{n}:C_{n}\rightarrow C_{n+1}$ defined by

$f_{n}(a_{n})=a_{n+1},\,f_{n}(b_{n})=Ub_{n+1},\,f_{n}(c_{n})=f_{n}(d_{n})=0,\,f_{n}(x_{n})=x_{n+1}$

satisfies $f_{n}\circ\iota=\iota\circ f_{n}$ on $\widehat{C}_{n}$, and it is
clear that there exists an almost $\iota_{K}$-local map from $C_{O}$ to
$C_{n}$ but not from $C_{n}$ to $C_{O}$, it follows from 2.11 that

$0<C_{2}\leq C_{3}\leq C_{4}\leq\cdots.$

We will see that the complexes $C_{n}$ provide both lower and upper bounds for
$CFK^{-}(S^{3},K_{2n+1,-1})$.

######  Lemma 4.6.

Let $K$ be a knot such that $CFK^{-}(S^{3},K)$ admits an $\mathbb{F}_{2}[U]$
direct summand, generated by an element $x$ of bidegree $(0,0)$. If $n$ is a
positive integer such that $U^{n}$ annihilates the torsion submodule of
$HFK^{-}(S^{3},K)$, then we have that $\mathfrak{A}(K)<C_{n+1}$.

###### Proof.

Suppose that there exists an almost $\iota_{K}$-local map $f\colon
C_{n+1}\rightarrow CFK^{-}(S^{3},K)$. By 4.1, $CFK_{\mathcal{R}}(S^{3},K)$
admits a splitting

$CFK_{\mathcal{R}}(S^{3},K)\simeq A\oplus\mathbb{F}_{2}[U]\cdot x$

for an acyclic summand $A$. Choose a basis
$\\{y_{1},\cdots,y_{k},z_{1},\cdots,z_{k}\\}$ of $A$ which is _horizontally
simplified_ , i.e. $\partial y_{i}=U^{n_{i}}z_{i}$ for some $n_{i}\leq n$ and
$\partial z_{i}=0$ for all $i=1,\cdots,k$. Since $\partial
a_{n+1}=U^{n+1}b_{n+1}$ in $C_{n+1}$, we get

$\partial f(a_{n+1})=f(\partial a_{n+1})=U^{n+1}f(b_{n+1}).$

Since $f(b_{n+1})$ is a cycle whose homology class is torsion, we have

$f(b_{n+1})=\sum_{i\in S_{z}}z_{i}$

for some $S_{z}\subset\\{1,\cdots,k\\}$. Write

$f(a_{n+1})=z+\sum_{i\in S_{y}}U^{c_{i}}y_{i}$

for some $S_{y}\subset\\{1,\cdots,k\\}$, $c_{i}>0$, and
$z\in\operatorname{span}_{\mathbb{F}_{2}[U]}(x,z_{1},\cdots,z_{k})$. Then we
get

$U^{n+1}\sum_{i\in S_{z}}z_{i}=U^{n+1}f(b_{n+1})=\partial
f(a_{n+1})=\sum_{i\in S_{y}}U^{n_{i}+c_{i}}z_{i},$

which implies that $S_{y}=S_{z}$ and $n_{i}+c_{i}=n+1$ for all $i\in S_{y}$.
But this means $c_{i}=n+1-n_{i}\geq 1$ for all $i$, so we deduce that
$f(a_{n+1})=z\pmod{U}$. Since $\iota(x_{n+1})=x_{n+1}$, we have

$0=(\iota
f+f\iota)(a_{n+1})=f(a_{n+1})+\iota(f(a_{n+1}))+f(x_{n+1})=f(x_{n+1})+z+\iota(z).$

Furthermore, since $U^{-1}f$ is a homotopy equivalence, the homology class of
$f(x_{n+1})$ generates $U^{-1}H_{\ast}(C)$. Hence $x$ should appear when we
write $\iota(z)$ as an $\mathbb{F}_{2}$-linear combination of the given basis
elements.

Recall that the involution $\iota$ on $\widehat{CFK}(S^{3},K)$ is a truncation
of an involution on $\widehat{CFK}(S^{3},K)$ which is a homotopy skew-
equivalence. Assume without loss of generality that the differential on
$CFK_{\mathcal{R}}(S^{3},K)$ is reduced, and write

$\partial=\partial_{U}+\partial_{V}$

where $\partial_{U}$ and $\partial_{V}$ are the sums of terms in $\partial$
involving $U$-powers and $V$-powers, respectively. Since $z$ is a
$\partial_{U}$-cycle whose homology class is $U$-torsion, $\iota(z)$ should be
the truncation from $CFK_{\mathcal{R}}(S^{3},K)$ of a $\partial_{V}$-cycle
whose homology class is $V$-torsion. In particular, $\iota(z)$ cannot contain
$x$, as $x$ comes from a cycle whose homology class is not annihilated by any
powers of $U$ and $V$, a contradiction. So we conclude that there does not
exist an almost $\iota_{K}$-local map from $C_{n+1}$ to $CFK^{-}(S^{3},K)$.

By 2.11, we are now left with two possibilities: $\mathfrak{A}(K)<[C_{n+1}]$
or $\mathfrak{A}(K)+[C_{E}]=[C_{n+1}]$. However, $CFK^{-}(S^{3},K)\otimes
E\simeq CFK^{-}(S^{3},K\\#E)$ also satisfies the assumptions of the lemma, so
there is no almost $\iota_{K}$-local map from $C_{n+1}$ to
$CFK^{-}(S^{3},K)\otimes C_{E}$. Therefore we conclude that
$\mathfrak{A}(K)<[C_{n+1}]$. ∎

We are now ready to prove 1.2, whose statement we recall.

######  Theorem 4.7.

If $K$ is torsion in $\mathcal{C}$ with $\mathfrak{A}(K)=1$, then the set of
cables $\\{K_{2n+1,1}\\}_{n>0}$ contains an infinite subset which is linearly
independent in $\mathcal{C}$.

###### Proof.

As in the proof of 4.5, it is enough to show that $\\{K_{2n+1,-1}\\}_{n>0}$
contains an infinite subset which is linearly independent in $\mathcal{C}$.
Also, recall that, in the proof of 4.5, we constructed a local chain map

$\operatorname{Minus}(f_{2n+1,-1})\colon CFK^{-}(S^{3},E_{2n+1,-1})\rightarrow
CFK^{-}(S^{3},K_{2n+1,-1})$

for each $n>1$, such that its hat-flavored truncation
$\operatorname{Hat}(f_{2n+1,-1})$ satisfies

$\begin{split}\iota_{K_{2n+1,-1}}\circ\operatorname{Hat}(f_{2n+1,-1})(b)&=\operatorname{Hat}(f_{2n+1,-1})(b+a),\\\
\iota_{K_{2n+1,-1}}\circ\operatorname{Hat}(f_{2n+1,-1})(a)&=\operatorname{Hat}(f_{2n+1,-1})(a).\end{split}$

Since $\partial b$ is a multiple of $U^{n+1}$, we see that
$\partial(\operatorname{Minus}(f_{2n+1,-1})(b))$ is also a multiple of
$U^{n+1}$. Therefore, we can write

$\partial(\operatorname{Minus}(f_{2n+1,-1})(b))=U^{n+1}c.$

Recall that $C_{n}$ is a horizontal almost $\iota_{K}$-chain complex generated
by $a_{n},b_{n},c_{n},d_{n},x_{n}$. Assume that $n>2$, and define a chain map
$g\colon C_{n}\rightarrow CFK^{-}(S^{3},K_{2n+1,-1})$ as

$g(a_{n-1})=\operatorname{Minus}(f_{2n+1,-1})(b),\,g(b_{n-1})=Uc,\,g(c_{n-1})=g(d_{n-1})=0,\,g(x_{n-1})=\operatorname{Minus}(f_{2n+1,-1})(a).$

Then $\iota g+g\iota=0$ as a map from $\widehat{C}_{n}$ to
$\operatorname{Minus}(K_{2n+1,-1})$, and $g$ is obviously local. Thus $g$ is
an almost $\iota_{K}$-local map, which implies
$[C_{n}]\leq\mathfrak{A}(K_{2n+1,-1})$.

Now we recursively construct a sequence $n_{1}<n_{2}<\cdots$ of integers as
follows. We start with the initial value $n_{1}=2$. Assuming that we have
already determined $n_{1},\cdots,n_{i}$, we take $N_{i}$ to be an integer such
that $N_{i}\geq n_{i}$ and $U^{N_{i}}$ annihilates the torsion submodule
$HFK_{tor}(S^{3},K_{2n_{i}+1,-1})$ of $HFK^{-}(S^{3},K_{2n_{i}+1,-1})$. By 4.1
and 4.6, we have $C\mathfrak{A}(K_{2n_{i}+1,-1})<[C_{N_{i}+1}]$ for any
integer $C$; this is because $HFK_{tor}(S^{3},CK_{2n_{i}+1,-1})$ is also
annihilated by $U^{N_{i}}$. Then we take $n_{i+1}=N_{i}+1$ and continue the
recursion.

Suppose that the family $\\{K_{2n_{i}+1,-1}\\}_{i>0}$ is not linearly
independent in $\mathcal{C}$. Then there exist integers $0<i_{1}<\cdots<i_{k}$
and $C_{1},\ldots,C_{k}$ such that $C_{k}>0$ and

$C_{k}\mathfrak{A}(K_{2n_{k}+1,-1})=\sum_{j=1}^{k-1}C_{j}\mathfrak{A}(K_{2n_{i_{j}}+1,-1}).$

However, since $[C_{n_{i}}]\leq\mathfrak{A}(K_{2n_{i}+1,-1})$ and
$C\mathfrak{A}(K_{2n_{i}+1,-1})<[C_{n_{i+1}}]$ for any integer $C$, we should
have

$\begin{split}\sum_{j=1}^{k-1}C_{j}\mathfrak{A}(K_{2n_{i_{j}}+1,-1})&=C_{1}\mathfrak{A}(K_{2n_{i_{1}}+1,-1})+\sum_{j=2}^{k-1}C_{j}\mathfrak{A}(K_{2n_{i_{j}}+1,-1})\\\
&<(C_{2}+1)\mathfrak{A}(K_{2n_{i_{2}}+1,-1})+\sum_{j=3}^{k-1}C_{j}\mathfrak{A}(K_{2n_{i_{j}}+1,-1})\\\
&<(C_{k-1}+1)\mathfrak{A}(K_{2n_{i_{k-1}}+1,-1})\\\
&<[C_{n_{i_{k-1}+1}}]\leq[C_{n_{i_{k}}}]\leq\mathfrak{A}(K_{2n_{i_{k}}+1,-1})\leq
C_{k}\mathfrak{A}(K_{2n_{i_{k}}+1,-1}),\end{split}$

a contradiction. Therefore the given family is linearly independent. ∎

##  5\. Generalization to iterated cables

Recall from 3.3 that a type-$D$ morphism
$f\colon\widehat{CFD}(S^{3}\smallsetminus
K_{1})\rightarrow\widehat{CFD}(S^{3}\smallsetminus K_{2})$ is _almost
bordered- $\iota$-local_ if and only if $\operatorname{Minus}(f)$ is an almost
$\iota_{K}$-local map. The arguments used in the previous section to prove
that $K_{2n+1,-1}$ has infinite order in $\mathcal{C}$ can be rephrased as
follows. In particular, we do not require $K$ is torsion in $\mathcal{C}$;
having a type-$D$ morphism $f$ satisfying the given conditions is enough.

######  Proposition 5.1.

Let $K$ be a knot. If $\widehat{CFD}(S^{3}\smallsetminus K)$ admits a direct
summand isomorphic to $\widehat{CFD}(S^{3}\smallsetminus O)$ and there exists
an almost bordered-$\iota$-local map from $\widehat{CFD}(S^{3}\smallsetminus
E)$ to $\widehat{CFD}(S^{3}\smallsetminus K)$, then for any integer $n>0$, we
have $\mathfrak{A}(K_{2n+1,-1})\geq C_{O}$ and $\mathfrak{A}(K_{2n+1,-1})\geq
C_{E}$ in $\mathfrak{I}^{U}_{K}$, from which it follows that
$\mathfrak{A}(K_{2n+1,-1})>C_{O}$, and that $K_{2n+1,-1}$ has infinite order
in $\mathcal{C}$.

Recall that $CFK_{UV}(S^{3},E_{2n+1,-1})$ admits a direct summand $C$,
generated by elements $a,b,c,d,e,f,g$, where the differential acts as

$\partial b=U^{n}c+UVd+V^{n}e,\,\partial c=Vf,\,\partial e=Ug,\,\partial
a=\partial d=\partial f=\partial g=0.$

Furthermore, we have $\iota_{E_{2n+1,-1}}(b)=b+a$ or $b+a+U^{n-1}f$; so we may
assume that $\iota_{E_{2n+1,-1}}(b)=b+a$ possibly after a basis change given
by $a\mapsto a+a+U^{n-1}f$. Then we can adjust $\iota_{E_{2n+1,-1}}$ by a
homotopy to get

$\iota_{E_{2n+1,-1}}(a)=\iota_{E_{2n+1,-1}}(b+\iota_{E_{2n+1,-1}}(b))=b+a+\iota^{2}_{E_{2n+1,-1}}(b)=a+\Phi\Psi(b)=a+U^{n-1}f.$

We claim that the summand $C$ can be made $\iota_{E_{2n+1,-1}}$-stable.

######  Lemma 5.2.

There exists a chain map, homotopic to $\iota_{E_{2n+1,-1}}$, such that its
action on the basis elements $a,b,c,d,e,f,g$ is described as

$a\mapsto a+U^{n-1}f,\,b\mapsto b+a,\,c\mapsto e,\,e\mapsto c,\,f\mapsto
g,\,g\mapsto f,\,d\mapsto d.$

###### Proof.

We have already achieved $a\mapsto a+U^{n-1}f$ and $b\mapsto b+a$, so it
remains to achieve the rest. Since $\iota_{E_{2n+1,-1}}$ is a chain map, we
have

$U^{n}c+V^{n}e=\partial
b=\partial\iota_{E_{2n+1,-1}}(b)=\iota_{E_{2n+1,-1}}\partial(b)=V^{n}\iota_{E_{2n+1,-1}}(c)+U^{n}\iota_{E_{2n+1,-1}}(e).$

Hence $c+\iota_{E_{2n+1,-1}}(e)$ is a multiple of $V$ and
$e+\iota_{E_{2n+1,-1}}(c)$ is a multiple of $U$. However, by considering the
bidegrees of $c$ and $C_{E}$, it is easy to see that the only possibility is
$c+\iota_{E_{2n+1,-1}}(e)=e+\iota_{E_{2n+1,-1}}(c)=0$. Therefore we get
$\iota_{E_{2n+1,-1}}(c)=e$ and $\iota_{E_{2n+1,-1}}(e)=c$.

We then proceed to remaining basis elements. Since $\partial c=Vf$ and
$\partial e=Ug$, we have

$\begin{split}U\iota_{E_{2n+1,-1}}(f)&=\iota_{E_{2n+1,-1}}\partial
c=\partial\iota_{E_{2n+1,-1}}(c)=\partial e=Ug,\\\
V\iota_{E_{2n+1,-1}}(g)&=\iota_{E_{2n+1,-1}}\partial
e=\partial\iota_{E_{2n+1,-1}}(e)=\partial c=Vf,\end{split}$

which implies $\iota_{E_{2n+1,-1}}(f)=g$ and $\iota_{E_{2n+1,-1}}(g)=f$.
Furthermore, since $\partial d=U^{n-1}f+V^{n-1}g$, we have

$\partial(d+\iota_{E_{2n+1,-1}}(d))=\partial
d+\partial\iota_{E_{2n+1,-1}}(d)=(1+\iota_{E_{2n+1,-1}})(U^{n-1}f+V^{n-1}g)=0,$

so $\iota_{E_{2n+1,-1}}(d)$ differs from $d$ by a cycle. Since $d$ has
bidegree $(1,1)$ and every other basis elements of the complex
$CFK_{UV}(S^{3},E_{2n+1,-1})$ lies in a bidegree in which at least one of the
two degrees is nonpositive, we should have

$\iota_{E_{2n+1,-1}}(d)=d+\sum(\alpha_{i}d_{0,i}+\beta_{i}d_{1,i})$

for some $\alpha_{i},\beta_{i}\in\\{0,1\\}$. However, it is clear that
$\sum(\alpha_{i}d_{0,i}+\beta_{i}d_{1,i})$ is cycle if and only if all
$\alpha_{i}$ and $\beta_{i}$ vanish. Therefore we get
$\sum(\alpha_{i}d_{0,i}+\beta_{i}d_{1,i})(d)=d$. ∎

Using 5.2, we can construct a bordered-$\iota$-local map from the figure-eight
knot complement to $S^{3}\smallsetminus E_{2n+1,-1}$ as follows.

######  Lemma 5.3.

For any postive integer $n$, there exists an almost bordered-$\iota$-local map
$f$ from $\widehat{CFD}(S^{3}\smallsetminus E)$ to
$\widehat{CFD}(S^{3}\smallsetminus E_{2n+1,-1})$ such that the image of
$\operatorname{Hat}(f)$ is contained in the subcomplex generated by
$a,b,c,d,e,f,g$.

###### Proof.

It follows from the proof of [LOT18, Proposition 11.38] that for any two knots
$K_{1},K_{2}$ and any chain map $g:CFK_{UV}(S^{3},K_{1})\rightarrow
CFK_{UV}(S^{3},K_{2})$, there exists a type-D morphism
$G:\widehat{CF}(S^{3}\smallsetminus
K_{1})\rightarrow\widehat{CFD}(S^{3}\smallsetminus K_{2})$ such that
$\operatorname{Hat}(G)\sim\widehat{g}$. Hence the existence of the desired
almost bordered-$\iota$-local map $F$ can be shown by proving that there
exists an $\iota_{K}$-local map $f_{n}$ from $CFK_{UV}(S^{3},E)$ to
$CFK_{UV}(S^{3},E_{2n+1,-1})$, whose image is contained in the subcomplex
generated by $a,b,c,d,e,f,g$.

Recall that the knot Floer chain complex for the figure-eight knot $E$ is
generated by five elements, $x,h,s,t,z$, where the differential acts by

$\partial h=Us+Vt,\,\partial s=Vz,\,\partial t=Uz,\,\partial z=\partial x=0,$

and the involution acts by

$\iota_{K}(h)=h+x,\,\iota_{K}(x)=x+z,\,\iota_{K}(s)=t,\,\iota_{K}(t)=s,\,\iota_{K}(z)=z.$

For any $n>0$, consider the map $f_{n}\colon CFK_{UV}(S^{3},E)\rightarrow
CFK_{UV}(S^{3},E_{2n+1,-1})$ defined as

$f_{n}(x)=a,\,f_{n}(h)=b,\,f_{n}(s)=U^{n}c+Vd,\,f_{n}(t)=V^{n}e,\,f_{n}(z)=V^{n}d.$

It is straightforward to check that $\partial f_{n}+f_{n}\partial=0$, so that
$f_{n}$ is a local map. Furthermore, by 5.2, the map
$\iota_{E_{2n+1,-1}}f_{n}+f_{n}\iota_{E}$ is given by

$h\mapsto 0,\,s\mapsto Ud,\,t\mapsto Vd,\,z,x\mapsto U^{n}f+V^{n}g.$

Consider the linear map $H\colon CFK_{UV}(S^{3},E)\rightarrow
CFK_{UV}(S^{3},E_{2n+1,-1})$ defined by

$H(h)=H(s)=H(t)=0,\,H(z)=H(x)=d.$

Then we get $\iota_{E_{2n+1,-1}}f_{n}+f_{n}\iota_{E}=\partial H+H\partial$, so
we deduce that $f_{n}$ is an $\iota_{K}$-local map. Furthermore, the image of
$f_{n}$ consists of $\mathbb{F}_{2}[U]$-linear combinations of
$a,b,c,d,e,f,g$. Therefore $f_{n}$ satisfies the desired properties. ∎

Using the almost bordered-$\iota$-local map we got from 5.3, one can now prove
the following lemma, which will be used as an inductive step in the proof of
1.1.

######  Lemma 5.4.

Let $K$ be a knot and $n$ be a positive integer. If there exists an almost
bordered-$\iota$-local map $f\colon\widehat{CFD}(S^{3}\smallsetminus
E)\rightarrow\widehat{CFD}(S^{3}\smallsetminus K)$, then there also exists an
almost bordered-$\iota$-local map from $\widehat{CFD}(S^{3}\smallsetminus E)$
to $\widehat{CFD}(S^{3}\smallsetminus K_{2n+1,-1})$.

###### Proof.

Recall from Section 4 that the type-$D$ morphism

$f_{2n+1,-1}\colon\widehat{CFD}(S^{3}\smallsetminus
E_{2n+1,-1})\rightarrow\widehat{CFD}(S^{3}\smallsetminus K_{2n+1,-1})$

satisfies
$\iota_{K_{2n+1,-1}}(\operatorname{Hat}(f_{2n+1,-1})(b))=\operatorname{Hat}(f_{2n+1,-1})(b+a)$
and
$\iota_{K_{2n+1,-1}}(\operatorname{Hat}(f_{2n+1,-1})(a))=\operatorname{Hat}(f_{2n+1,-1})(a)$;
note that we do not know whether $f_{2n+1,-1}$ is an almost
bordered-$\iota$-local map. Consider the bordered-$\iota$-local map

$F\colon\widehat{CFD}(S^{3}\smallsetminus
E)\rightarrow\widehat{CFD}(S^{3}\smallsetminus E_{2n+1,-1})$

given in 5.3. Using the notation from the proof of 5.3 that
$CFK_{UV}(S^{3},E)$ is generated by $x,h,s,t,z$, since the image of
$\operatorname{Hat}(F)$ is contained in the $\mathbb{F}_{2}$-linear span of
$a,b,c,d,e,f,g$, it is clear that $\operatorname{Hat}(F)$ maps $x$ and $h$ to
$a$ and $b$, respectively, and maps $s,t,z$ to 0. Hence the map

$\operatorname{Hat}(f_{2n+1,-1}\circ
F)=\operatorname{Hat}(f_{2n+1,-1})\circ\operatorname{Hat}(F)$

satisfies $\iota_{K_{2n+1,-1}}\circ\operatorname{Hat}(f_{2n+1,-1}\circ
F)\sim\operatorname{Hat}(f_{2n+1,-1}\circ F)\circ\iota_{E}$, and therefore
$f_{2n+1,-1}\circ F$ is almost bordered-$\iota$-local. ∎

######  Lemma 5.5.

Let $K$ be a knot such that $\widehat{CFD}(S^{3}\smallsetminus K)$ admits a
direct summand isomorphic to $\widehat{CFD}(S^{3}\smallsetminus O)$. Then for
any pattern $P\subset T_{\infty}$ such that $P(O)$ is slice,
$CFK_{\mathcal{R}}(S^{3},P(K))$ also admits a direct summand isomorphic to
$\widehat{CFD}(S^{3}\smallsetminus O)$.

###### Proof.

By [LOT18, Theorem 11.26], we know that $\widehat{CFD}(S^{3},K)$ admits a
summand isomorphic to $\widehat{CFD}(S^{3},O)$. Then, by taking a box tensor
product with the bimodule $\widehat{CFDA}(T_{\infty}\smallsetminus P)$, we see
that $\widehat{CFD}(S^{3}\smallsetminus P(K))$ admits a summand isomorphic to
$\widehat{CFD}(S^{3}\smallsetminus P(O))$. Also, since $P(O)$ is slice, it
follows from [Hom17, Theorem 1] that $CFK_{UV}(S^{3},P(O))$ has a direct
summand isomorphic to $\mathbb{F}_{2}[U,V]$ concentrated in bidegree $(0,0)$,
from which it follows from [LOT18, Theorem 11.26] that
$\widehat{CFD}(S^{3}\smallsetminus P(O))$ admits a direct summand isomorphic
to $\widehat{CFD}(S^{3}\smallsetminus O)$. ∎

We are now ready to prove 1.1, whose statement we recall.

######  Theorem 5.6.

Denote by $\mathcal{C}_{T}$ the torsion subgroup of the knot concordance group
$\mathcal{C}$. Then there is a nontrivial group homomorphism
$\mathfrak{A}\colon\mathcal{C}_{T}\rightarrow\mathbb{Z}/2\mathbb{Z}$.
Moreover, if $K$ is torsion in $\mathcal{C}$ with $\mathfrak{A}(K)=1$, then
for any sequence of positive integers $n_{1},n_{2},\ldots,n_{m}$ the iterated
cable $K_{2n_{1}+1,1;2n_{2}+1,1;\ldots;2n_{m}+1,1}$ has infinite order in
$\mathcal{C}$.

###### Proof.

The existence of the nontrivial group homomorphism follows from 2.12. Also, as
in the proof of 4.5, it is enough to prove that
$K_{2n_{1}+1,-1;2n_{2}+1,-1;\ldots;2n_{m}+1,-1}$ has infinite order in
$\mathcal{C}$.

In order to do so, we show that there exists an almost bordered-$\iota$-local
map

$\widehat{CFD}(S^{3}\smallsetminus
E)\rightarrow\widehat{CFD}(S^{3}\smallsetminus
K_{2n_{1}+1,-1;2n_{2}+1,-1;\ldots,2n_{m-1}+1,-1}),$

from which the theorem follows by 5.5 and 5.1. We already know, possibly after
replacing $K$ by another knot concordant to it, that the result holds when
$m=1$, as shown in 4.2. Moreover, 5.4 implies that the claim holds for any
positive integer $m$.∎

######  Remark 5.7.

In particular, the proof of 5.6 implies that
$\mathfrak{A}(K_{2n_{1}+1,-1;2n_{2}+1,-1;\ldots,2n_{m-1}+1,-1})>0$.

##  6\. Strongly rationally slice knots that are not slice

In this section, we prove 1.4 using 5.6 and the following two lemmas. For
simplicity, we first define the _complexity_ of rationally slice knots as
follows.

######  Definition 6.1.

Let $K$ be a rationally slice knot that bounds a smoothly embedded disk
$\Delta$ in a rational homology ball $X$. The inclusion induces a homomorphism

$H_{1}(S^{3}\smallsetminus K;\mathbb{Z})\cong\mathbb{Z}\to
H_{1}(X\smallsetminus\Delta;\mathbb{Z})/\text{torsion}\cong\mathbb{Z},$

which is given by multiplication by a nonzero integer $c$, and we may assume
further that $c$ is positive by changing orientations if necessary. We define
the _complexity_ of $\Delta$ to be the positive integer $c$ and the
_complexity_ of $K$ to be the minimum complexity of all possible rational
slice disks for $K$.

Note that a rationally slice knot has complexity one if and only if it is
strongly rationally slice. The following lemma allows us to construct many
strongly rationally slice knots. We use $\nu Y$ to denote a open tubular
neighborhood of a submanifold $Y$.

######  Lemma 6.2.

Let $K$ be a complexity $c$ rationally slice knot and $P\subset T_{\infty}$ be
a winding number $w$ pattern. If $P(O)$ is slice, then $P(K)$ is rationally
slice knot of complexity at most $c/\gcd(c,w)$.

###### Proof.

Through out the proof all homology groups have integral coefficients. Let
$\Delta$ be a complexity $c$ smoothly embedded disk in a rational homology
ball $X$. Then the disk $\Delta$ induces a rational concordance $C\subset
X\smallsetminus B^{4}$ from $K$ to the unknot $O$. Using the Mayer-Vietoris
sequence, it can be easily verified that the homomorphism induced by the
inclusion

$H_{1}(S^{3}\smallsetminus K)\cong\mathbb{Z}\to
H_{1}\left(X\smallsetminus\left(B^{4}\cup
C\right)\right)/\text{torsion}\cong\mathbb{Z}$

is given by multipication by $c$. Moreover, using the pattern $P$, we get a
rational concordance $C_{P}\subset X\smallsetminus B^{4}$ from $P(K)$ to the
slice knot $P(O)$. Combined with a slice disk $\Delta_{P(O)}$ for $P(O)$, we
get a rational slice disk

$\Delta_{P(K)}:=C_{P}\cup\Delta_{P(O)}\subset\left(X\smallsetminus
B^{4}\right)\cup B^{4}=X$

for $P(K)$. Then as before we also have that

$H_{1}(S^{3}\smallsetminus P(K))\cong\mathbb{Z}\to
H_{1}\left(X\smallsetminus\left(B^{4}\cup
C_{P}\right)\right)/\text{torsion}\cong\mathbb{Z}$

is given by multiplication by the complexity of $\Delta_{P(K)}$. Hence, it is
enough to analize this homomorphism induced by the inclusion (see $v_{3}$
below).

Note that from the following Mayer-Vietoris sequence

$0\to H_{1}(P\times S^{1})\rightarrow H_{1}(T_{\infty}\smallsetminus\nu
P)\oplus H_{1}(\nu P)\to H_{1}(T_{\infty})\to 0,$

it can be easily verified that $H_{1}(T_{\infty}\smallsetminus\nu
P)\cong\mathbb{Z}\oplus\mathbb{Z}\cong\langle\lambda_{T},\mu_{P}\rangle,$
where

$\lambda_{T}:=S^{1}\times\\{\operatorname{pt}\\}\subset\partial
T_{\infty}\subset T_{\infty}=S^{1}\times
D^{2}\qquad\text{and}\qquad\mu_{P}:=\\{\operatorname{pt}\\}\times\nu P=P\times
D^{2}\subset T_{\infty}.$

Now, we consider the following commutative diagram of Mayer-Vietoris
sequences,

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}(K\times
S^{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}}$$\scriptstyle{v_{1}}$$\textstyle{H_{1}(T_{\infty}\smallsetminus
P)\oplus H_{1}(S^{3}\smallsetminus\nu
K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{2}}$$\scriptstyle{v_{2}}$$\textstyle{H_{1}(S^{3}\smallsetminus
P(K))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{3}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}(C\times
S^{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{3}}$$\textstyle{H_{1}(\nu
C\smallsetminus C_{P})\oplus H_{1}\left(X\smallsetminus(B^{4}\cup\nu
C)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{4}}$$\textstyle{H_{1}\left(X\smallsetminus\left(B^{4}\cup
C_{P}\right)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where the vertical homomorphisms are induced by the inclusions. Let $\mu$ be
the meridian of $P(K)$. We make the following observations:

  * •

$h_{1}([K\times\\{\operatorname{pt}\\}])=([\lambda_{T}],0)\in
H_{1}(T_{\infty}\smallsetminus P)\oplus H_{1}(S^{3}\smallsetminus\nu K)$.

  * •

$h_{1}([\\{\operatorname{pt}\\}\times S^{1}])=(w[\mu_{P}],1)\in
H_{1}(T_{\infty}\smallsetminus P)\oplus H_{1}(S^{3}\smallsetminus\nu K)$.

  * •

$h_{2}\left(([\mu_{P}],0)\right)=[\mu]\in H_{1}(S^{3}\smallsetminus P(K))$.

  * •

$v_{1}$ is an isomorphism.

  * •

$v_{2}{\big{|}}_{H_{1}(T_{\infty}\smallsetminus P)}\colon
H_{1}(T_{\infty}\smallsetminus P)\to H_{1}(\nu C\smallsetminus C_{P})$ is an
isomorphism.

  * •

$v_{2}{\big{|}}_{H_{1}(S^{3}\smallsetminus\nu K)}\colon
H_{1}(T_{\infty}\smallsetminus P)\to H_{1}\left(X\smallsetminus(B^{4}\cup\nu
C)\right)/\text{torsion}$ is given by multiplication by $c$.

Hence, we see that

$\operatorname{Im}(h_{3})\cong\langle([\lambda_{T}],0),(w[\lambda_{P}],c))\rangle\subset
H_{1}(\nu C\smallsetminus C_{P})\oplus H_{1}\left(X\smallsetminus(B^{4}\cup\nu
C)\right)/\text{torsion}\cong\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}.$

This implies that

$H_{1}\left(X\smallsetminus(B^{4}\cup
C_{P})\right)/\text{torsion}\cong\mathbb{Z}\oplus\mathbb{Z}/\langle(w,c)\rangle,$

and $v_{3}([\mu])=h_{4}\circ v_{2}(([\mu_{P}],0))$ is represented by
$(1,0)\in\mathbb{Z}\oplus\mathbb{Z}/\langle(w,c)\rangle,$ which completes the
proof.∎

######  Lemma 6.3.

Let $K$ be a knot. If the connected sum of some copies of $S^{3}_{+1}(K)$
bounds a spin rational homology ball, then there exists an almost
$\iota_{K}$-local map from $\mathbb{F}_{2}[U,V]$ to $CFK_{UV}(S^{3},K)$, and
thus $\mathfrak{A}(K)\geq[C_{O}]$.

###### Proof.

The assumption, together with the fact that the almost $\iota$-local
equivalence group is totally ordered, implies that the $CF^{-}(S^{3}_{+1}(K))$
is almost $\iota$-locally trivial; we refer to [DHST18] for the definitions of
almost $\iota$-complexes and almost $\iota$-local equivalence classes and
[DHST18, Theorem 3.25] for the fact that the almost $\iota$-local equivalence
group is totally ordered. We know from [HHSZ20, Theorem 1.6] that the almost
$\iota$-local equivalence class of $CF^{-}(S^{3}_{+1}(K))$ is given by
$(A_{0}(K),\iota)$, where $A_{0}(K)\subset CFK_{UV}(S^{3},K)$ is the subset of
elements of Alexander grading 0 and $\iota$ is the restriction of the action
of $\iota_{K}$ on $CFK_{UV}(S^{3},K)$ to $A_{0}(K)$. Hence $(A_{0}(K),\iota)$
is almost $\iota$-locally trivial.

Choose an almost $\iota$-local map $f\colon\mathbb{F}_{2}[U]\rightarrow
A_{0}(K)$. Recall that the $U$-action on $A_{0}(K)$ corresponds to the
$UV$-action on $CFK_{UV}(S^{3},K)$. Thus we can extend $f$ by tensoring with
$\mathbb{F}_{2}[U,V]$, along the ring inclusion
$\mathbb{F}_{2}[UV]\rightarrow\mathbb{F}_{2}[U,V]$, to get a local map

$\tilde{f}\colon\mathbb{F}_{2}[U,V]\rightarrow
A_{0}(K)\otimes_{\mathbb{F}_{2}[UV]}\mathbb{F}_{2}[U,V]\xrightarrow{\text{multiplication
map}}CFK_{UV}(S^{3},K).$

Since $\iota\circ f\sim f$ mod $U$ and the action of $\iota$ on $A_{0}(K)$ is
the restriction of the action of $\iota_{K}$ on $CFK_{UV}(S^{3},K)$, it is
clear that $\iota_{K}\circ\tilde{f}\sim\tilde{f}$ mod $UV$, so $\tilde{f}$ is
an almost $\iota_{K}$-local map. ∎

We are finally able to prove 1.4, whose statement we recall.

######  Theorem 6.4.

If $K$ is torsion in $\mathcal{C}$ with $\mathfrak{A}(K)=1$, then for any
sequence of positive integers $n_{1},n_{2},\ldots,n_{m}$ the $(2,1)$-cable of
the iterated cable $K_{2n_{1}+1,1;2n_{2}+1,1;\ldots;2n_{m}+1,1}$ has infinite
order in $\mathcal{C}$.

###### Proof.

Let $J$ be the iterated cable $K_{2n_{1}+1,1;2n_{2}+1,1;\ldots;2n_{m}+1,1}$
and $\Sigma(J_{2,1})$ be the 2-fold cyclic branched cover of $S^{3}$ branched
along $J_{2,1}$. By applying the Akbulut-Kirby [AK80] technique to $J_{2,1}$,
we have that

$\Sigma\left(J_{2,1}\right)\cong S^{3}_{+1}\left(J\\#J^{r}\right).$

Suppose that $nJ_{2,1}$ is slice for some positive integer $n$ and let $D$ be
its slice disk. Then the 2-fold cyclic branched cover of $B^{4}$ branched
along $D$ is a rational homology ball with a spin structure that restricts to
the unique spin structure on its boundary $\Sigma\left(nJ_{2,1}\right)\cong
n\Sigma\left(J_{2,1}\right)$. By 6.3, this implies

$\mathfrak{A}\left(K_{2n_{1}+1,1;2n_{2}+1,1;\ldots;2n_{m}+1,1}\\#K^{r}_{2n_{1}+1,1;2n_{2}+1,1;\ldots;2n_{m}+1,1}\right)\geq[C_{O}].$

However, we have

$\mathfrak{A}\left((-K)_{2n_{1}+1,-1;2n_{2}+1,-1;\ldots;2n_{m}+1,-1}\right)>[C_{O}]\qquad\text{and}\qquad\mathfrak{A}\left((-K)^{r}_{2n_{1}+1,1;2n_{2}+1,-1;\ldots;2n_{m}+1,-1}\right)>[C_{O}]$

by 2.15 and 5.7, which gives

$\mathfrak{A}\left(K_{2n_{1}+1,1;2n_{2}+1,1;\ldots;2n_{m}+1,1}\\#K^{r}_{2n_{1}+1,1;2n_{2}+1,1;\ldots;2n_{m}+1,1}\right)<[C_{O}];$

a contradiction. Hence $J_{2,1}$ has infinite order in $\mathcal{C}$. ∎

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