# Homology concordance and An Infinite rank free subgroup

Hugo Zhou  School of Mathematics, Georgia Institute of Technology, Atlanta, GA
30332<EMAIL_ADDRESS>
###### Abstract.

Two knots are homology concordant if they are smoothly concordant in a
homology cobordism. The group $\widehat{\mathcal{C}}_{\mathbb{Z}}$ (resp.
${\mathcal{C}}_{\mathbb{Z}}$) was previously defined as the set of knots in
homology spheres that bounds homology balls (resp. in $S^{3}$), modulo
homology concordance. We prove
$\widehat{\mathcal{C}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$ contains a
$\mathbb{Z}^{\infty}$ subgroup. We construct our family of examples by
applying the filtered mapping cone formula to L-space knots, and prove linear
independence with the help of the connected knot complex.

00footnotetext: The author was partially supported by NSF grant DMS-1552285.

##  1\. Introduction

Two knots in $S^{3}$ are smoothly concordant if they cobound a smoothly
embedded annulus in the product $S^{3}\times I$. One can consider the set of
knots in $S^{3}$ modulo smooth concordance. This set becomes an abelian group
called the knot concordance group, under the operation induced by connected
sum. It is a well-known result that the knot concordance group contains a
$\mathbb{Z}^{\infty}$ summand (see, for example, [Lev69]).

As a generalization of smooth concordance, the notion homology concordance is
studied in [DR16] and [HLL18]. This paper will build on the established
results in [HLL18] regarding the homology concordance classes of knots in
integer homology spheres.

Given two knots $K\subset Y$, $K^{\prime}\subset Y^{\prime}$, where both $Y$
and $Y^{\prime}$ are integer homology spheres that are homology cobordant to
$S^{3}$, we call $K$ and $K^{\prime}$ homology concordant if they cobound a
smoothly embedded annulus in some homology cobordism between $Y$ and
$Y^{\prime}$, denoted $(Y,K)\sim(Y^{\prime},K^{\prime})$.

Let $\widehat{\mathcal{C}}_{\mathbb{Z}}$ denote the group consists of pairs
$(Y,K)$, where $Y$ is a homology null-cobordant homology $3$-sphere and $K$ a
knot in $Y$, modulo homology concordance. The operation, still induced by
connected sum now changes both the knot and the $3$-manifold:
$(Y,K)\mathbin{\\#}(Y^{\prime},K^{\prime})$ is given by
$(Y\mathbin{\\#}Y^{\prime},K\mathbin{\\#}K^{\prime})$. The inverse of $(Y,K)$
is $(-Y,-K)$.

Similarly, let ${\mathcal{C}}_{\mathbb{Z}}$ denote the group consists of pairs
$(S^{3},K)$ modulo homology concordance, which is a natural subgroup of
$\widehat{\mathcal{C}}_{\mathbb{Z}}$.

One wants to study the group structure of
$\widehat{\mathcal{C}}_{\mathbb{Z}}$, since this is the group analogous to the
knot concordance group for knots in homology spheres. Note that
$\widehat{\mathcal{C}}_{\mathbb{Z}}$ naturally inherits the complicated group
structure from the knot concordance group. In fact, there is a canonical map
from the knot concordance group to ${\mathcal{C}}_{\mathbb{Z}}$. Though
whether this map is injective remains unknown, J. Levine’s examples pass
through and constitute a $\mathbb{Z}^{\infty}$ summand in
${\mathcal{C}}_{\mathbb{Z}}$ (For a construction of Levine’s examples, see
[Lev69] or [Liv05]). Due to this observation, we consider instead
$\widehat{\mathcal{C}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$. The quotient
group $\widehat{\mathcal{C}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$ measures
the “difference” between knots in $S^{3}$ and knots in arbitrary homology
spheres that bound homology balls. Adam Simon Levine showed in [Lev16] that
$\widehat{\mathcal{C}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$ is not
trivial. Inspired by the study of the knot concordance group, we ask: Is
$\widehat{\mathcal{C}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$ infinitely
generated? Does it contain a $\mathbb{Z}^{\infty}$ subgroup?

The first question has been given an affirmative answer in [HLL18]. Moreover,
two approaches to the question have been demonstrated. The first approach
relies on an invariant $\theta(Y,K)$, defined as the maximal difference of
$d$-invariants of $1/n$-surgeries on $K$ (For a definition of $d$-invariant
see [OS03]). A family of pairs $(Y_{j},K_{j})$ was constructed whose
$\theta(Y_{j},K_{j})\rightarrow\infty$ as $j$ approaches infinity. With some
more argument, this shows the infinite generation of
$\widehat{\mathcal{C}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$. However, it
is unknown to the author at this point whether the pairs constructed in the
proof have infinite order. Another approach is to generalize the knot Floer
invariants including $\tau,\nu,\varepsilon,\Upsilon$ which originally were
only defined for knots in $S^{3}$, to homology concordance invariants. An
advantage of this approach is that many of these invariants still have the
same additivity properties under connected sums of the knots in an arbitrary
integer homology sphere as they do for knots in $S^{3}$. We will adopt this
approach in the paper, and answer the second question affirmatively, as
follows.

######  Theorem 1.1.

$\widehat{\mathcal{C}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$ contains a
$\mathbb{Z}^{\infty}$ subgroup.

One of the main tools we use in the proof is the following concordance
invariant. The invariant is more or less known to the experts. Since its
definition has not yet appeared in the literature, we write it down for
completeness.

######  Theorem 1.2.

Given a knot $K$ in an integer homology sphere $Y$, suppose
$C=CFK^{\infty}(Y,K)$ is the knot Floer complex. There is a homology
concordance invariant $C_{\mathrm{conn}}$, taking value in isomorphism classes
of absolutely-graded, $(i,j)$-filtered $\mathbb{F}[U,U^{-1}]$ chain complexes.
Moreover, $C$ is filtered chain homotopy equivalent to
$C_{\mathrm{conn}}\oplus A$ as a filtered chain complex, where $A$ is an
acyclic complex.

The invariant is defined in the same way as the connected Heegaard Floer
homology in [HHL18], ignoring the involution in the original definition. We
describe the construction in Section $6$ over the ring
$\mathbb{F}[\mathcal{U},\mathcal{V}]$. The same construction works over the
ring $\mathbb{F}[U,U^{-1}]$ as well.

Regarding $\tau$ and $\varepsilon$ invariants, we review the definitions and
properties in section 2. The key observation is:

######  Proposition 1.3 (Proposition 3.6(2) in [Hom14]).

Suppose $K$ is a knot in $S^{3}$. If $\varepsilon(S^{3},K)=0$, then
$\tau(S^{3},K)=0$.

We construct the $\mathbb{Z}^{\infty}$ subgroup by explicitly finding
infinitely many generating pairs. For $n>1$, let $M_{n}$ denote $+1$-surgery
on $T_{2,4n-1}$, the $(2,4n-1)$-torus knot, let
$Y_{n}=M_{n}\mathbin{\\#}-M_{n}$, and let $K_{n}\subset Y_{n}$ denote the
connected sum of the core of the surgery in $M_{n}$ with the unknot in
$-M_{n}$.

######  Proposition 1.4.

For $n>1$ and $(Y_{n},K_{n})$ as above, we have the following,

  1. (1)

$\tau(Y_{n},K_{n})=-1$;

  2. (2)

$\varepsilon(Y_{n},K_{n})=0$;

  3. (3)

$(Y_{n},K_{n})$ are linearly independent in
$\widehat{\mathcal{C}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$.

Theorem 1.1 directly follows from Proposition 1.4 (3). The construction of
$(Y_{n},K_{n})$ and the evaluation of their $\tau$ and $\varepsilon$
invariants are due to a computational result, Theorem 3.1.

######  Remark 1.5.

The quotient group
$\widehat{\mathcal{C}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$ can also be
interpreted in PL(piecewise-linear) terms. Recall that every knot in $S^{3}$
bounds a PL disk in $B^{4}$ by taking the cone. Such a disk is smooth outside
the cone point. In [Lev16], Adam Simon Levine gives a pair $(Y,K)$ such that
$Y$ bounds a contractible manifold but $K$ does not bound a PL disk in any
homology ball whose boundary is $Y$. Note that a knot $K$ in $Y$ bounds a PL
disk if and only if $(Y,K)$ is trivial in
$\widehat{\mathcal{C}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$. One can see
this by adding or deleting open balls near the cone points. Likewise, two
pairs $(Y_{0},K_{0}),(Y_{1},K_{1})\in\widehat{\mathcal{C}}_{\mathbb{Z}}$
differ by an element of ${\mathcal{C}}_{\mathbb{Z}}$ if and only if $K_{0}$
and $K_{1}$ cobound a PL annulus in some homology cobordism from $Y_{0}$ to
$Y_{1}$. Thus, the quotient
$\widehat{\mathcal{C}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$ can be
interpreted as the group of knots in homology null-cobordant homology spheres
modulo PL concordance in homology cobordisms.

######  Remark 1.6.

In the paper [HLL18] (Remark 1.13.), a variation of
$\widehat{\mathcal{C}}_{\mathbb{Z}}$ was brought into discussion. Let
$\widehat{\mathcal{C}}_{\mathbb{Z}}^{\prime}$ denote the subgroup of
$\widehat{\mathcal{C}}_{\mathbb{Z}}$ that consists of all pairs $(Y,K)$ where
$Y$ bounds a homology ball $X$ in which $K$ is freely nulhomotopic (meaning
$K$ is trivial in $\pi_{1}(X)$, or equivalently $K$ bounds a immersed disk in
$X$). From the $4$-manifold perspective,
$\widehat{\mathcal{C}}_{\mathbb{Z}}^{\prime}$ is arguably a more appropriate
generalization of the knot concordance group, as it measures the failure of
replacing immersed disks by embedded ones. It has been shown in [Dae18] that
$\widehat{\mathcal{C}}_{\mathbb{Z}}^{\prime}$ is a proper subgroup of
$\widehat{\mathcal{C}}_{\mathbb{Z}}$, answering a question raised up in
[HLL18]. Moreover, according to Corollary $1$ of [Dae18], each pair
$(Y_{n},K_{n})$ in our construction lies in
$\widehat{\mathcal{C}}_{\mathbb{Z}}\backslash\widehat{\mathcal{C}}_{\mathbb{Z}}^{\prime}$
since the core of the surgery solid torus normally generates the foundamental
group of $+1$-surgery on $T_{2,4n-1}$. We do not know whether their linear
combinations lie in $\widehat{\mathcal{C}}_{\mathbb{Z}}^{\prime}$ or not.

One can also consider the analogues of ${\mathcal{C}}_{\mathbb{Z}}$ and
$\widehat{\mathcal{C}}_{\mathbb{Z}}$ in the topological category. Namely, we
say knots $(Y_{0},K_{0})$ and $(Y_{1},K_{1})$ are _topologically homology
concordant_ if they cobound a locally flat embedded annulus in a topological
homology cobordism between $Y_{0}$ and $Y_{1}$ (which need not carry any
smooth structure), and we let $\mathcal{C}_{\mathbb{Z},\mathrm{top}}$ and
$\widehat{\mathcal{C}}_{\mathbb{Z},\mathrm{top}}$ denote the corresponding
concordance groups, as in [DR16]. Note that every homology $3$-sphere bounds a
contractible topological $4$-manifold, so there is no need to impose
restriction on which pairs $(Y,K)$ are represented in
$\widehat{\mathcal{C}}_{\mathbb{Z},\mathrm{top}}$ as we did in smooth case
(Recall that we required the homology spheres to bound homology balls in the
smooth version). There is strong evidence that the natural inclusion
$\varphi\colon\mathcal{C}_{\mathbb{Z},\mathrm{top}}\to\widehat{\mathcal{C}}_{\mathbb{Z},\mathrm{top}}$
is an isomorphism (see [Dav18]).

Define the natural map
$\psi\colon\widehat{\mathcal{C}}_{\mathbb{Z}}\rightarrow\widehat{\mathcal{C}}_{\mathbb{Z},\mathrm{top}}$
by simply forgetting the smooth structure. The kernel of the map $\psi$
consists of the pairs that are topologically homology concordant to
$(S^{3},O)$, where $O$ is the unknot in $S^{3}$. We can refine our
construction of the infinite generating pairs such that they are in
$\operatorname{ker}\psi$. The following result is suggested by JungHwan Park.

######  Theorem 1.7.

For $n>1$, there exists
$(Y^{\prime}_{n},J_{n})\in\widehat{\mathcal{C}}_{\mathbb{Z}}$ such that

  1. (1)

$(Y^{\prime}_{n},J_{n})\in\operatorname{ker}\psi$;

  2. (2)

$(Y^{\prime}_{n},J_{n})$ are linearly independent in
$\widehat{\mathcal{C}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$.

The proof of Theorem 1.7 is in Section 3.

### Organization

In Section $2$ we review invariants $\tau$ and $\varepsilon$, generalized as
concordance invariants. In Section $3,4$ and $5$ we present a computational
result of core of surgery on L-space knots using the filtered mapping cone
formula by Hedden and Levine [HL19], which will in turn give the proof of
Proposition 1.4 (1),(2). In Section $6$ we give the definition of the
connected knot complex over the ring $\mathbb{F}[\mathcal{U},\mathcal{V}]$. In
Section $7$ we prove the linear independence of the family $(Y_{n},K_{n})$.

### Acknowledgements

The author wants to thank his advisor Jennifer Hom for her patient guidance
throughout the project. The author is grateful to JungHwan Park for explaining
Theorem 1.7 and would like to thank Tye Lidman, Adam Simon Levine and Chuck
Livingston for helpful comments. The author also thanks the referee for
helpful comments.

##  2\. Concordance invariants from knot floer homology

In this section, we review the concordance invariants that will be useful for
the construction, defined in [HLL18] using tools from [Zem19b]. We assume that
the reader is familiar with the knot Floer invariant package in $S^{3}$, see
[Hom17] for a survey on the topic.

Recall that for a knot $K$ in an integer homology sphere $Y$, over the ring
$\mathbb{F}[U]$ knot floer complex $C=\operatorname{CFK}^{\infty}(Y,K)$
decomposes as a vector space $C=\bigoplus_{i,j\in\mathbb{Z}}C(i,j)$. Up to
chain homotopy we can choose the filtered basis such that it is reduced,
namely no differential preserves the $(i,j)$ coordinate. Let
$X\subset\mathbb{Z}\oplus\mathbb{Z}$ be a set such that if $a,c\in X$ and
$a\prec b\prec c$, then $b\in X$, where $\prec$ is the natural partial order
on $\mathbb{Z}\oplus\mathbb{Z}$. Let $CX=\bigoplus_{(i,j)\in X}C(i,j)$. $CX$
is naturally a subquotient complex of $C$.

Consider the maps

$\displaystyle\iota_{s}$ $\displaystyle\colon C\\{i=0,j\leq s\\}\rightarrow
C\\{i=0\\},$ $\displaystyle v_{s}$ $\displaystyle\colon
C\\{\max(i,j-s)=0\\}\rightarrow C\\{i=0\\},$ $\displaystyle v^{\prime}_{s}$
$\displaystyle\colon C\\{i=0\\}\rightarrow C\\{\min(i,j-s)=0\\},$

where $\iota_{s}$ is inclusion, $v_{s}$ consists of quotienting by
$C\\{i<0,j=s\\}$ followed by inclusion and $v^{\prime}_{s}$ consists of
quotienting by $C\\{i=0,j<s\\}$ followed by inclusion. Recall that
$C\\{i=0\\}\simeq\widehat{\operatorname{CF}}(Y)$ and $C\\{i\geq
0\\}\simeq\operatorname{CF}^{+}(Y)$. Also, let
$\rho\colon\widehat{\operatorname{CF}}(Y)\rightarrow\operatorname{CF}^{+}(Y)$
denote inclusion.

######  Definition 2.1 (Definition 4.2 in [HLL18]).

Let $K$ be a knot in an integer homology sphere $Y$. Define

$\displaystyle\tau(Y,K)$
$\displaystyle=\min\\{s\mid\operatorname{im}(\rho_{*}\circ\iota_{s*})\cap
U^{N}\operatorname{HF}^{+}(Y)\neq 0\ \forall N\gg 0\\}$
$\displaystyle\nu(Y,K)$
$\displaystyle=\min\\{s\mid\operatorname{im}(\rho_{*}\circ v_{s*})\cap
U^{N}\operatorname{HF}^{+}(Y)\neq 0\ \forall N\gg 0\\}$
$\displaystyle\nu^{\prime}(Y,K)$ $\displaystyle=\max\\{s\mid
v^{\prime}_{s*}(x)\neq 0\ \forall x\in\widehat{\operatorname{HF}}(Y)\textup{
s.t. }\rho_{*}(x)\neq 0\textup{ and }\rho_{*}(x)\in
U^{N}\operatorname{HF}^{+}(Y)\ \forall N\gg 0\\}.$

The definition of $\varepsilon$ relies on the relation between $\tau,\nu$ and
$\nu^{\prime}$; we have the following lemma:

######  Lemma 2.2 (Lemma 4.5 in [HLL18]).

Let $K$ be a knot in an integer homology sphere $Y$. The following three cases
are exhaustive and mutually exclusive:

  * •

$\nu(Y,K)=\tau(Y,K)+1$ and $\nu^{\prime}(Y,K)=\tau(Y,K)$,

  * •

$\nu(Y,K)=\tau(Y,K)$ and $\nu^{\prime}(Y,K)=\tau(Y,K)-1$,

  * •

$\nu(Y,K)=\tau(Y,K)$ and $\nu^{\prime}(Y,K)=\tau(Y,K)$.

The three cases in the above lemma correspond to the three different values of
$\varepsilon$.

######  Definition 2.3 (Definition 4.6 in [HLL18]).

Let $K$ be a knot in an integer homology sphere $Y$. Define

$\varepsilon(Y,K)=\begin{cases}-1\quad&\text{if }\nu(Y,K)=\tau(Y,K)+1,\\\
1\quad&\text{if }\nu^{\prime}(Y,K)=\tau(Y,K)-1,\\\
0\quad&\text{otherwise}.\end{cases}$

The following properties will be especially useful.

######  Proposition 2.4 (Propositions 4.7, 4.10, 4.11 in [HLL18]).

Both $\tau,\varepsilon$ are homology concordance invariants. Moreover, given
pairs $(Y,K)$ and $(Y^{\prime},K^{\prime})$ as group elements in
$\widehat{\mathcal{C}}_{\mathbb{Z}}$,

  1. (1)

$\tau(-Y,K)=-\tau(Y,K)$.

  2. (2)

$\tau(Y\mathbin{\\#}Y^{\prime},K\mathbin{\\#}K^{\prime})=\tau(Y,K)+\tau(Y^{\prime},K^{\prime})$.

  3. (3)

$\varepsilon(-Y,K)=-\varepsilon(Y,K)$.

  4. (4)

If $\varepsilon(Y,K)=\varepsilon(Y^{\prime},K^{\prime})$, then
$\varepsilon(Y\mathbin{\\#}Y^{\prime},K\mathbin{\\#}K^{\prime})=\varepsilon(Y,K)$.

  5. (5)

If $\varepsilon(Y,K)=0$, then
$\varepsilon(Y\mathbin{\\#}Y^{\prime},K\mathbin{\\#}K^{\prime})=\varepsilon(Y^{\prime},K^{\prime})$.

Lastly, we recall the following obstruction to knots in
${\mathcal{C}}_{\mathbb{Z}}$ observed in [HLL18]:

######  Corollary 2.5.

Let $(Y,K)$ be an element in $\widehat{\mathcal{C}}_{\mathbb{Z}}$. If
$\varepsilon(Y,K)=0$, but $\tau(Y,K)\neq 0$, then $(Y,K)$ generates a
$\mathbb{Z}$-subgroup in
$\widehat{\mathcal{C}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$.

###### Proof.

According to Proposition 1.3, $(Y,K)$ is not homology concordant to any knot
in $S^{3}$. In other words, $(Y,K)$ represents a non-trivial element in
$\widehat{\mathcal{C}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$. Suppose
$\tau(Y,K)=a$, where $a\neq 0$. Now by Proposition 2.4,
$\varepsilon(\mathbin{\\#}_{n}(Y,K))=0$ while
$\tau(\mathbin{\\#}_{n}(Y,K))=na$, and this completes the proof. ∎

##  3\. The core of surgery on L-space knots

Starting from this section, we will present the computational result that
leads to our construction of the $\mathbb{Z}^{\infty}$-subgroup. It relies on
the knot Floer homology of the core of a Dehn surgery on L-space knots. We now
explain the context and give a more precise description of the result.

A knot $K\subset S^{3}$ is called an L-space knot if the surgery
$S^{3}_{n}(K)$ is an L-space for $n$ large enough. These knots have relatively
simple knot Floer complexes that are determined by the Alexander polynomial.
See Section 7 of [HM17] for a more detailed review regarding L-space knots.
Since their notations give a certain convenience in representing the symmetry
relation of knot Floer complex of L-space knots, we will borrow them in this
section and throughout the computation in the next two sections.

In [OS05], Ozsváth and Szabó proved that if $K$ is an L-space knot, then the
Alexander polynomial of $K$ is of the form

$\Delta_{K}(t)=(-1)^{m}+\sum_{i=1}^{m}(-1)^{m-i}(t^{n_{i}}+t^{-n_{i}})$

for a sequence of positive integers $0<n_{1}<n_{2}<\dots<n_{m}.$ Here,
$n_{m}=g(K)$ is the genus of $K$. Let $n(K)\geqslant 0$ be the quantity

$n(K):=n_{m}-n_{m-1}+\dots+(-1)^{m-2}n_{2}+(-1)^{m-1}n_{1}.$

Furthermore, let $\ell_{s}=n_{s}-n_{s-1}$. Since the unknot is the only
L-space knot with $n(K)=0$ (whose dual knot is the unknot in $S^{3}$), we will
only consider knots with $n(K)\geqslant 1$ for our computational result.

L-space knots have a knot Floer complex where differentials form a “staircase”
. The knot Floer complex consists of $2m+1$ generators
$x_{0},x^{1}_{1},x^{2}_{1},...,x^{1}_{m},x^{2}_{m}$. The $(i,j)$-filtration of
$x_{0},x^{1}_{m}$ are $(0,0)$, $(-n(K),g(K)-n(K))$, respectively. The
filtration of $x^{t}_{m-2s}$ and $x^{t}_{m-(2s+1)}$ differ only in
$i$-filtration by $\ell_{m-2s}$ and the filtration of $x^{t}_{m-(2s+1)}$ and
$x^{t}_{m-(2s+2)}$ differ only in $j$-filtration by $\ell_{m-(2s+1)}$. If $m$
is odd, the differentials are

$\displaystyle\partial(x_{0})$ $\displaystyle=x_{1}^{1}+x_{1}^{2}$
$\displaystyle\partial(x_{s}^{t})$
$\displaystyle=x_{s-1}^{t}+x_{s+1}^{t}\text{ for }s>0\text{ even},\
t\in\\{1,2\\}$

whereas if $m$ is even, the differentials are

$\displaystyle\partial(x_{1}^{t})$ $\displaystyle=x_{0}+x_{2}^{t}$
$\displaystyle\partial(x_{s}^{t})$
$\displaystyle=x_{s-1}^{t}+x_{s+1}^{t}\text{ for }s>1\text{ odd},\
t\in\\{1,2\\}.$

Observe that $n(K)$ is the total length of the horizontal arrows in the top
half of the complex.

Let $K$ be an L-space knot in $S^{3}$. Consider
$\operatorname{CFK}^{\infty}(S^{3}_{1}(K),\tilde{K})$, where $\tilde{K}$ is
the core circle of the surgery solid torus in $1$-surgery on $K$. We are now
ready to state the following computational result.

######  Theorem 3.1.

Assume $K$ is an L-space knot whose knot complex has $2m+1$ generators and
$n(K)=n\geqslant 1$. If $m$ is odd, then
$\operatorname{CFK}^{\infty}(S^{3}_{1}(K),\tilde{K})$ is filtered chain
homotopy equivalent to $C_{n}\oplus A$ as a filtered chain complex, where $A$
is an acyclic complex, and the complex $C_{n}$ is generated by three
generators whose $(i,j)$-filtrations are $(n-1,n),(0,0)$ and $(n,n-1)$
respectively, both generators in filtration $(n-1,n)$ and $(n,n-1)$ having an
arrow going to filtration $(0,0).$

If $m$ is even, $\operatorname{CFK}^{\infty}(S^{3}_{1}(K),\tilde{K})$ is
homotopy equivalent to $C_{0}\oplus A$ as a filtered chain complex, where $A$
is an acyclic complex, and the complex $C_{0}$ consists of one generator with
$(i,j)$-filtration $(0,0)$.

######  Remark 3.2.

In fact, the complex $C_{n}$ we have described in Theorem 3.1 and Figure 1 is
the connected knot complex of
$\operatorname{CFK}^{\infty}(S^{3}_{1}(K),\tilde{K})$, for L-space knot $K$
whose knot complex has $2m+1$ generators where $m$ is odd and $n(K)=n$. Since
we do not need the connected knot complex to explain the computational result,
we will only formally define it in a later section. For the precise definition
and more details see Section $6$.

$n$$n-1$$n-1$$n$$(0)$ Figure 1. The filtered chain complex $C_{n}$, where the
number in parenthesis marks the grading. For an L-space knot $K$ whose knot
complex has $2m+1$ generators where $m$ is odd and $n(K)=n$,
$\operatorname{CFK}^{\infty}(S^{3}_{1}(K),\tilde{K})$ is filtered chain
homotopy equivalent to $C_{n}\oplus A$, where $A$ is an acyclic complex.

Before we move on to the method used in the proof of Theorem 3.1, we first
explain why Theorem 3.1 implies Proposition 1.4 (1), (2). For $n>1$, let
$M_{n}$ denote $+1$-surgery on $T_{2,4n-1}$, the $(2,4n-1)$-torus knot, let
$\tilde{T}_{2,4n-1}$ denote the core of the surgery in $M_{n}$, let
$Y_{n}=M_{n}\mathbin{\\#}-M_{n}$, and let $K_{n}\subset Y_{n}$ denote the
connected sum of $\tilde{T}_{2,4n-1}$ in $M_{n}$ with the unknot in $-M_{n}$.

###### Proof of Proposition 1.4 (1), (2).

The knot $T_{2,4n-1}$ has a symmetrized Alexander polynomial

$-1+\sum_{i=1}^{2n-1}(-1)^{i-1}(t^{i}+t^{-i}).$

In particular, $n(T_{2,4n-1})=n$ and there are $4n-1$ generators in the knot
Floer complex. Applying Theorem 3.1, we see
$\operatorname{CFK}^{\infty}(M_{n},\tilde{T}_{2,4n-1})$ is generated by 3
generators whose $(i,j)$-filtrations are $(-1,0),(0,-1)$ and $(-n,-n)$
respectively. Both generators in filtration $(-1,0)$ and $(0,-1)$ have an
arrow going to filtration $(-n,-n).$ Consider the generator at filtration
$(0,-1)$ and denote it by $\xi$. The homology class of $\xi$ in
$\operatorname{HF}^{+}(M_{n})$ is nontrivial in both
$\operatorname{im}(\rho_{*}\circ\iota_{-1*})$ and
$\operatorname{im}(\rho_{*}\circ v_{-1*})$ that is in the image of
$U^{N}\operatorname{HF}^{+}(M_{n})$ for all $N\gg 0$, and moreover
$v^{\prime}_{-1*}([\xi])$ is nontrivial. (See Definition 2.1.)

From this observation, it is straightforward to verify that
$\tau(M_{n},\tilde{T}_{2,4n-1})=\nu(M_{n},\tilde{T}_{2,4n-1})=\nu^{\prime}(M_{n},\tilde{T}_{2,4n-1})=-1$,
and so $\varepsilon(M_{n},\tilde{T}_{2,4n-1})=0$. Connecting sum with the
unknot in $-M_{n}$ doesn’t change the value of $\tau$ or $\varepsilon$, so it
follows $\tau(Y_{n},K_{n})=-1$ and $\varepsilon(Y_{n},K_{n})=0$. ∎

######  Remark 3.3.

By Theorem 3.1, if $K$ is an L-space knot in $S^{3}$, the only parameters that
determine the connected knot complex of
$\operatorname{CFK}^{\infty}(S^{3}_{1}(K),\tilde{K})$ (See Definition 6.7) are
the value of $n(K)$ and the number of generators in
$\operatorname{CFK}^{\infty}(S^{3},K)$. Hence our choice of knots is not
particularly special. In fact, suppose we have a family of knots in $S^{3}$
such that for each knot, after a filtered change of basis, the knot Floer
complex consists of acyclic summands and a positive staircase with $2m+1$
generators where $m$ is odd. Then as long as $n(K)$ of the staircase (the
total length of the horizontal arrows in the top half of the staircase) is
unbounded for this family of knots, they could be used to construct the
$\mathbb{Z}^{\infty}$ subgroup.

Next, we explain how to refine the construction such that the infinite
generating pairs are inside $\operatorname{ker}\psi$, where $\psi$ is the
natural map from $\widehat{\mathcal{C}}_{\mathbb{Z}}$ to
$\widehat{\mathcal{C}}_{\mathbb{Z},\mathrm{top}}$. The following theorem is
suggested to the author by JungHwan Park. The construction of the contractible
$4$ manifold where the core of surgery bounds a slice disk in the proof is
described to the author by Chuck Livingston.

First, we need a result from Hedden, Kim and Livingston’s work [HKL16]. Let
$D$ denote the positively-clasped untwisted Whitehead double of the right-
handed trefoil, $T_{2,3},$ and let $D_{k}$ denote $kD$.

######  Proposition 3.4 (Proposition 6.1 in [HKL16]).

The chain complex $\operatorname{CFK}^{\infty}(S^{3},D_{k})$ is filtered chain
homotopy equivalent to the chain complex
$\operatorname{CFK}^{\infty}(S^{3},T_{2,2k+1})\oplus A,$ where $A$ is an
acyclic complex.

Since $D$ has Alexander polynomial one, by Freedman and Quinn’s work [FQ90],
$D$ is topologically slice and so is $D_{k}$.

###### Proof of Theorem 1.7..

For $n>1$, set $M^{\prime}_{n}=S^{3}_{+1}(D_{2n-1})$, set
$Y^{\prime}_{n}=M^{\prime}_{n}\mathbin{\\#}-M^{\prime}_{n}$, let
$\tilde{D}_{2n-1}$ denote the core of the surgery in $M^{\prime}_{n}$ , and
let $J_{n}\subset Y^{\prime}_{n}$ be the connected sum of $\tilde{D}_{2n-1}$
with the unknot in $-M^{\prime}_{n}$. Since
$\operatorname{CFK}^{\infty}(S^{3},D_{2n-1})$ is filtered chain homotopy
equivalent to the chain complex
$\operatorname{CFK}^{\infty}(S^{3},T_{2,4n-1})\oplus A,$ where $A$ is an
acyclic complex, chosing $(Y^{\prime}_{n},J_{n})$ provides us the same family
of knot complexes $C_{n}$ as $(Y_{n},K_{n})$ according to Theorem 3.1.

It remains to prove $(Y^{\prime}_{n},J_{n})$ is topologically homology
concordant to $(S^{3},O)$, where $O$ is the unknot in $S^{3}$. In fact,
$M^{\prime}_{n}=S^{3}_{+1}(D_{2n-1})$ bounds a contractible $4$ manifold where
$\tilde{D}_{2n-1}$ bounds a locally flat disk. The construction is as follows.
Attach a $2$-handle to $B^{4}$ along $D_{2n-1}$ with framing $1$ to form
$W_{n}$. There is a $2$-sphere inside $W_{n}$ built from the core of the $2$
handle and the slice disk of $D_{2n-1}$. Since the framing of the attaching
$2$-handle is $1$, the normal bundle of this $2$-sphere has Euler number $1$
and its boundary is $S^{3}$. Cut out the neighbourhood of this $2$-sphere and
replace it with $B^{4}$, yielding a new $4$ manifold $Z_{n}$. The boundary of
$Z_{n}$ is $S^{3}_{+1}(D_{2n-1})$ and $Z_{n}$ is contractible (To see this,
first note that $Z_{n}$ is simply connected, and by Mayer-Vietoris sequence
one can show it has the same homology as a point. Therefore $Z_{n}$ is
contractible according to Whitehead Theorem). For $+1$ surgery, the core of
surgery is isotopic to the zero framed longtitude in the complement of the
knot, which bounds a locally flat disk in $B^{4}$ disjoint from the slicing
disk for $D_{2n-1}$. Thus, $\tilde{D}_{2n-1}$ bounds a locally flat disk in
$Z_{n}$. So $J_{n}$ bounds a topologically slice disk inside
$Z_{n}\mathbin{\\#}-Z_{n}$ and $(Y^{\prime}_{n},J_{n})$ is topologically
homology concordant to $(S^{3},O)$.

∎

The next two sections are dedicated to the proof of Theorem 3.1.

##  4\. The Filtered Mapping cone formula

In this section, we briefly describe the filtered mapping cone formula and
include an example of its application. We assume the reader is familiar with
the surgery mapping cone formula by Ozsváth and Szabó [OS08].

The filtered mapping cone formula from [HL19], is a refinement of the surgery
mapping cone formula by Ozsváth and Szabó. Let
$C=\operatorname{CFK}^{\infty}(Y,K)$ be the reduced filtered knot Floer
homology for a knot $K$ in integer homology sphere $Y$. The filtered mapping
cone formula puts extra filtrations on the mapping cone formula for
$\operatorname{HF}^{+}(Y_{1}(K))$, which in turns gives the knot Floer complex
$\operatorname{CFK}^{\infty}(Y_{1}(K),\tilde{K})$, where $\tilde{K}$ is the
core of the surgery solid torus of the $+1$-surgery on $K$.

Let $C=\bigoplus_{i,j\in\mathbb{Z}}C(i,j)$. For each integer $s$, let
$A_{s}^{\infty}$ and $B_{s}^{\infty}$ each denote a copy of the chain complex
$C$, and write
$A_{s}^{\infty}=\bigoplus_{i,j\in\mathbb{Z}}A_{s}^{\infty}(i,j)$ and
$B_{s}^{\infty}=\bigoplus_{i,j\in\mathbb{Z}}B_{s}^{\infty}(i,j)$. We define a
pair of $\mathbb{Z}$-filtrations $\mathcal{I}$ and $\mathcal{J}$ on each of
these complexes as follows:

$\displaystyle\mathcal{I}(A_{s}^{\infty}(i,j))$ $\displaystyle=\max(i,j-s)$
$\displaystyle\mathcal{J}(A_{s}^{\infty}(i,j))$ $\displaystyle=\max(i+s-1,j)$
$\displaystyle\mathcal{I}(B_{s}^{\infty}(i,j))$ $\displaystyle=i$
$\displaystyle\mathcal{J}(B_{s}^{\infty}(i,j))$ $\displaystyle=i+s-1.$

Let $A_{s}^{-}$ (resp. $B_{s}^{-}$) denote the subcomplex of $A_{s}^{\infty}$
(resp. $B_{s}^{\infty}$) with $\mathcal{I}<0$, let $A_{s}^{+}$ (resp.
$B_{s}^{+}$) denote the quotient, and let $\hat{A}_{s}$ (resp. $\hat{B}_{s}$)
be the subcomplex of $A_{s}^{+}$ (resp. $B_{s}^{+}$) with $\mathcal{I}=0$.
This definition of $A_{s}^{\circ}$ and $B_{s}^{\circ}$ coincides with the
definition by Ozsváth and Szabó.

Next we define the maps $v_{s}^{\circ}\colon A_{s}^{\circ}\to B_{s}^{\circ}$
and $h_{s}^{\circ}\colon A_{s}^{\circ}\to B_{s+1}^{\circ}$. Let
$v_{s}^{\infty}$ be the identity map of $C$, and let $h_{s}^{\infty}$ be
$\phi$ composed with multiplication by $U^{s}$, where $\phi$ is a
$U$-equivariant, grading-preserving chain homotopy equivalence $\phi\colon
C\to C$, which restricts to a homotopy equivalence between the subcomplexes
$C\\{j\leq s\\}$ and $C\\{i\leq s\\}$. Each of these maps is filtered with
respect to both $\mathcal{I}$ and $\mathcal{J}$. In particular,
$v_{s}^{\infty}$ (resp. $h_{s}^{\infty}$) takes the subcomplex $A_{s}^{-}$
into $B_{s}^{-}$ (resp. $B_{s+1}^{-}$), and induces a map $v_{s}^{+}\colon
A_{s}^{+}\to B_{s}^{+}$ (resp. $h_{s}^{+}\colon A_{s}^{+}\to B_{s+1}^{+}$),
which agrees with the definition in [OS08]. Moreover, each of $v_{s}^{\infty}$
and $h_{s}^{\infty}$ is homogeneous of degree $-1$ with respect to the Maslov
grading. Let

$\Psi^{\infty}_{1-g,g}\colon\bigoplus_{s=1-g}^{g}A_{s}^{\infty}\to\bigoplus_{s=2-g}^{g}B_{s}^{\infty}$

be the map given by the sum of the maps $v_{s}^{\infty}\colon
A_{s}^{\infty}\to B_{s}^{\infty}$ ($s=2-g,\dots,g$) and $h_{s}^{\infty}\colon
A_{s}^{\infty}\to B_{s+1}^{\infty}$ ($s=1-g,\dots,g-1$), and let
$\mathbb{X}^{\infty}$ denote the mapping cone of $\Psi^{\infty}_{1-g,g}$.
Filtration functions $\mathcal{I}$ and $\mathcal{J}$ give
$\mathbb{X}^{\infty}$ the structure of a doubly filtered chain complex with an
action of $\mathbb{F}[U,U^{-1}]$.

The following is implied by the main result from [HL19]:

######  Theorem 4.1 (Theorem 1.1 in [HL19]).

The chain complex $\mathbb{X}^{\infty}$ is filtered quasi-isomorphic to
$\operatorname{CFK}^{\infty}(Y_{1}(K),\tilde{K})$, where the filtrations
$\mathcal{I}$ and $\mathcal{J}$ on $\mathbb{X}^{\infty}$ correspond to $i$ and
$j$ on $\operatorname{CFK}^{\infty}(Y_{1}(K),\tilde{K})$.

For a filtered basis of $\mathbb{X}^{\infty}$, we have
$\partial^{\infty}=\partial+\partial^{\prime}$, where $\partial$ consists of
the terms that preserve both $\mathcal{I}$ and $\mathcal{J}$ filtrations and
$\partial^{\prime}$ consists of the terms that lower at least one of the
filtrations.

In order to compute invariants such as $\tau$ and $\varepsilon$ more
effectively, one wants to obtain a _reduced_ complex for
$\operatorname{CFK}^{\infty}(Y_{1}(K),\tilde{K})$, namely a complex where each
term in the differential strictly lowers at least one of the filtrations.
There is a “cancellation” mathod (See Proposition 11.57 in [LOT18]) that
allows us to perform a filtered change of basis to obtain a reduced complex of
$\mathbb{X}^{\infty}$. We will describe its procedure as follows. In each
summand $\mathbb{X}(0,s)$, one can pick a basis $\\{y_{i}\\}$ for
$\operatorname{im}(\partial)$, and then select elements
$x_{i}\in\mathbb{X}(0,s)$ such that $\partial(x_{i})=y_{i}$. Then
$\partial^{\infty}(x_{i})=y_{i}+{}$ terms in lower filtration levels. The
subcomplex of $\mathbb{X}^{\infty}$ spanned (over $\mathbb{F}[U,U^{-1}]$) by
all the $\\{x_{i},\partial^{\infty}(x_{i})\\}$ is acyclic, and quotienting
$\mathbb{X}^{\infty}$ by this subcomplex yields a reduced complex for
$\operatorname{CFK}^{\infty}(Y_{1}(K),\tilde{K})$. The generators for this
quotient complex (over $\mathbb{F}[U,U^{-1}]$) are in one-to-one
correspondence with the generators (over $\mathbb{F}$) of
$\widehat{\operatorname{HFK}}(Y_{1}(K),\tilde{K})$, and the differential is
induced from the terms in $\partial^{\prime}$ which strictly lower one of the
filtrations. From the above analysis, we see that it is practical to start
from each individual complex $\mathbb{X}(0,s)$ to compute
$\widehat{\operatorname{HFK}}(Y_{1}(K),\tilde{K})$. We identify the generators
in a reduced complex for $\operatorname{CFK}^{\infty}(Y_{1}(K),\tilde{K})$
with the generators of $\widehat{\operatorname{HFK}}(Y_{1}(K),\tilde{K})$ and
compute the induced differentials. Next, we compute an example utilizing this
method.

### An example:
$\operatorname{CFK}^{\infty}{(S^{3}_{+1}(T_{3,5}),\tilde{T}_{3,5})}$

As an example, we compute
$\operatorname{CFK}^{\infty}{(S^{3}_{+1}(T_{3,5}),\tilde{T}_{3,5})}$ using the
filtered mapping cone formula, where $T_{3,5}$ is the $(3,5)$-torus knot, and
$\tilde{T}_{3,5}$ is the core circle of the surgery solid torus in $1$-surgery
on $T_{3,5}$. See [HLL18] Section $6$ for a more complicated example.

$x^{1}_{3}(0)$$x^{1}_{2}(1)$$x^{1}_{1}(0)$$x_{0}(1)$$x^{2}_{1}(0)$$x^{2}_{2}(1)$$x^{2}_{3}(0)$
Figure 2. The complex $\operatorname{CFK}^{\infty}(T_{3,5})$. Numbers in
parenthesis indicate Maslov gradings.

Starting from $\operatorname{CFK}^{\infty}{(T_{3,5})}$, here we use the same
notations for generators as in [HM17]. Since $g(T_{3,5})=4$, the complex
$\mathbb{X}^{\infty}$ is a mapping cone

$\bigoplus_{s=-3}^{4}A_{s}^{\infty}\to\bigoplus_{s=-2}^{4}B_{s}^{\infty},$

We will work out
$\widehat{\operatorname{CFK}}{(S^{3}_{+1}(T_{3,5}),\tilde{T}_{3,5},1)}$
explicitly; the remaining computations are similar and left to the reader.
Reader can find the information of all generators in Table 1. To start, recall
$\widehat{\operatorname{CFK}}{(S^{3}_{+1}(T_{3,5}),\tilde{T}_{3,5},1)}$ is
given by the mapping cone

(1) $A_{1}\\{i\leqslant 0,j=1\\}\oplus A_{2}\\{i=0,j\leqslant
1\\}\xrightarrow{(h_{1},\ v_{2})}B_{2}\\{i=0\\}.$

We depict $A_{1}$, $A_{2}$, and $B_{2}$ in Figures 8, 3, and 3 respectively.
The generators of different $A_{i}$ and $B_{i}$ are distinguished by the
indice outside the parentheses. The generators of $A_{i}$ are labeled with an
$i$-subscript outside the parentheses and the generators of $B_{i}$ are
labeled with a prime and an $i$-subscript outside the parentheses.

$U^{3}(x^{1}_{3})_{1}$$U^{3}(x^{1}_{2})_{1}$$U(x^{1}_{1})_{1}$

$U(x^{1}_{1})_{2}$$U^{2}(x_{0})_{2}$$U^{2}(x^{2}_{1})_{2}$$U^{4}(x^{2}_{2})_{2}$$U^{4}(x^{2}_{3})_{2}$$U(x^{1}_{1})_{2}$$U^{2}(x_{0})_{2}$$U^{2}(x^{2}_{1})_{2}$$U^{4}(x^{2}_{2})_{2}$$U^{4}(x^{2}_{3})_{2}$$U(x^{1}_{1})_{2}$$U^{2}(x_{0})_{2}$$U^{2}(x^{2}_{1})_{2}$$U^{4}(x^{2}_{2})_{2}$$U^{4}(x^{2}_{3})_{2}$$U(x^{1}_{1})_{2}$$U^{2}(x_{0})_{2}$$U^{2}(x^{2}_{1})_{2}$$U^{4}(x^{2}_{2})_{2}$$U^{4}(x^{2}_{3})_{2}$$U(x^{1}_{1})_{2}$$U^{2}(x_{0})_{2}$$U^{2}(x^{2}_{1})_{2}$$U^{4}(x^{2}_{2})_{2}$$U^{4}(x^{2}_{3})_{2}$

$(x^{1}_{3})^{\prime}_{2}$$U(x^{1}_{2})^{\prime}_{2}$$U(x^{1}_{1})^{\prime}_{2}$$U^{2}(x_{0})^{\prime}_{2}$$U^{2}(x^{2}_{1})^{\prime}_{2}$$U^{4}(x^{2}_{2})^{\prime}_{2}$$U^{4}(x^{2}_{3})^{\prime}_{2}$$(x^{1}_{3})^{\prime}_{2}$$U(x^{1}_{2})^{\prime}_{2}$$U(x^{1}_{1})^{\prime}_{2}$$U^{2}(x_{0})^{\prime}_{2}$$U^{2}(x^{2}_{1})^{\prime}_{2}$$U^{4}(x^{2}_{2})^{\prime}_{2}$$U^{4}(x^{2}_{3})^{\prime}_{2}$$(x^{1}_{3})^{\prime}_{2}$$U(x^{1}_{2})^{\prime}_{2}$$U(x^{1}_{1})^{\prime}_{2}$$U^{2}(x_{0})^{\prime}_{2}$$U^{2}(x^{2}_{1})^{\prime}_{2}$$U^{4}(x^{2}_{2})^{\prime}_{2}$$U^{4}(x^{2}_{3})^{\prime}_{2}$$(x^{1}_{3})^{\prime}_{2}$$U(x^{1}_{2})^{\prime}_{2}$$U(x^{1}_{1})^{\prime}_{2}$$U^{2}(x_{0})^{\prime}_{2}$$U^{2}(x^{2}_{1})^{\prime}_{2}$$U^{4}(x^{2}_{2})^{\prime}_{2}$$U^{4}(x^{2}_{3})^{\prime}_{2}$$(x^{1}_{3})^{\prime}_{2}$$U(x^{1}_{2})^{\prime}_{2}$$U(x^{1}_{1})^{\prime}_{2}$$U^{2}(x_{0})^{\prime}_{2}$$U^{2}(x^{2}_{1})^{\prime}_{2}$$U^{4}(x^{2}_{2})^{\prime}_{2}$$U^{4}(x^{2}_{3})^{\prime}_{2}$

Figure 3. Top left, $A_{1}$. Top right, $A_{2}$. Bottom, $B_{2}$. The dark
grey regions comprise
$\widehat{\operatorname{CFK}}{(S^{3}_{+1}(T_{3,5}),\tilde{T}_{3,5},1)}$.

From Figure 3 one can read off that
$\widehat{\operatorname{CFK}}{(S^{3}_{+1}(T_{3,5}),\tilde{T}_{3,5},1)}$
consists $4$ direct summands under the basis we choose. We summarize them in
Figure 4.

$U^{3}(x^{1}_{2})_{1}$$U^{3}(x^{1}_{3})_{1}$$U^{4}(x^{2}_{2})^{\prime}_{2}$$U^{4}(x^{2}_{3})^{\prime}_{2}$$U^{4}(x^{2}_{2})_{2}$$U^{4}(x^{2}_{3})_{2}$

$U(x^{1}_{1})_{1}$$U^{2}(x_{0})^{\prime}_{2}$$U^{2}(x^{2}_{1})^{\prime}_{2}$$U^{2}(x_{0})_{2}$$U^{2}(x^{2}_{1})_{2}$

$U(x^{1}_{2})^{\prime}_{2}$$U(x^{1}_{1})^{\prime}_{2}$$U(x^{1}_{1})_{2}$

$(x^{1}_{3})^{\prime}_{2}$

Figure 4. The four direct summands in
$\widehat{\operatorname{CFK}}{(S^{3}_{+1}(T_{3,5}),\tilde{T}_{3,5},1)}$

The first summand is acyclic whereas the other three summands each generates
one dimensional homology.

It follows that
$\widehat{\operatorname{HFK}}{(S^{3}_{+1}(T_{3,5}),\tilde{T}_{3,5},1)}$ is
generated by

$\\{U(x^{1}_{1})_{1}+U^{2}(x_{0})^{\prime}_{2},\quad
U(x^{1}_{1})_{2}+U(x^{1}_{2})^{\prime}_{2},\quad(x^{1}_{3})^{\prime}_{2}\\}$

The next step is to pick a reduced basis for
$\operatorname{CFK}^{\infty}{(S^{3}_{+1}(T_{3,5}),\tilde{T}_{3,5})}$. We
summarize all generators and their $\operatorname{CFK}^{\infty}$ differentials
in Table 1. The notations we choose here, even though cumbersome, will be
helpful in the more general case. Note that the isolated generator
$g_{j-1}:=(x^{1}_{3})^{\prime}_{j}$ in $B_{j}$ always survives as a generator
of Alexander grading $j-1$ for $j\in\\{-2,...,4\\}$; we choose pairs of image
and preimage under map $\partial$ for $j\in\\{-2,...,4\\}$:

$\partial U(x^{1}_{2})^{\prime}_{j}=U(x^{1}_{1})^{\prime}_{j},\quad\partial
U^{2}(x_{0})^{\prime}_{j}=U^{2}(x^{2}_{1})^{\prime}_{j},\quad\partial
U^{4}(x^{2}_{2})^{\prime}_{j}=U^{4}(x^{2}_{3})^{\prime}_{j}$

and compute

$\displaystyle\partial^{\infty}(x^{1}_{2})^{\prime}_{j}$
$\displaystyle=(x^{1}_{3})^{\prime}_{j}+(x^{1}_{1})^{\prime}_{j}$
$\displaystyle\partial^{\infty}(x_{0})^{\prime}_{j}$
$\displaystyle=(x^{1}_{1})^{\prime}_{j}+(x^{2}_{1})^{\prime}_{j}$
$\displaystyle\partial^{\infty}(x^{2}_{2})^{\prime}_{j}$
$\displaystyle=(x^{2}_{1})^{\prime}_{j}+(x^{2}_{3})^{\prime}_{j}$

After quotienting by the $\mathbb{F}[U,U^{-1}]$-submodule $S$ generated by

$\\{(x^{1}_{2})^{\prime}_{j},\quad(x_{0})^{\prime}_{j},\quad(x^{2}_{2})^{\prime}_{j},\quad,(x^{1}_{3})^{\prime}_{j}+(x^{1}_{1})^{\prime}_{j},\quad(x^{1}_{1})^{\prime}_{j}+(x^{2}_{1})^{\prime}_{j},\quad(x^{2}_{1})^{\prime}_{j}+(x^{2}_{3})^{\prime}_{j}\\}$

it readily follows that
$(x^{1}_{3})^{\prime}_{j}=(x^{1}_{1})^{\prime}_{j}=(x^{2}_{1})^{\prime}_{j}=(x^{2}_{3})^{\prime}_{j}=g_{j-1}$.

Alexander gr. |  | Generator | Maslov gr. | $\partial^{\infty}$ |
---|---|---|---|---|---
$4$ | $\beta_{4}$ | $(x^{1}_{3})_{4}$ | $12$ | $(x^{1}_{3})^{\prime}_{4}$ | $=g_{3}$
$3$ | $g_{3}$ | $(x^{1}_{3})^{\prime}_{4}$ | $11$ | $0$ |
$2$ | $g_{2}$ | $(x^{1}_{3})^{\prime}_{3}$ | $5$ | $0$ |
$2$ | $\alpha_{2}$ | $U(x^{1}_{1})_{3}+U(x^{1}_{2})^{\prime}_{3}$ | $4$ | $U(x^{1}_{3})^{\prime}_{3}+U^{4}(x^{2}_{1})^{\prime}_{4}$ | $=Ug_{2}+U^{4}g_{3}$
$1$ | $g_{1}$ | $(x^{1}_{3})^{\prime}_{2}$ | $1$ | $0$ |
$1$ | $\beta_{1}$ | $U(x^{1}_{1})_{1}+U^{2}(x_{0})^{\prime}_{2}$ | $-2$ | $U(x^{1}_{1})^{\prime}_{1}+U^{2}(x^{1}_{1})^{\prime}_{2}$ | $=Ug_{0}+U^{2}g_{1}$
$1$ | $\alpha_{1}$ | $U(x^{1}_{1})_{2}+U(x^{1}_{2})^{\prime}_{2}$ | $0$ | $U(x^{1}_{3})^{\prime}_{2}+U^{3}(x^{2}_{1})^{\prime}_{3}$ | $=Ug_{1}+U^{3}g_{2}$
$0$ | $g_{0}$ | $(x^{1}_{3})^{\prime}_{1}$ | $-1$ | $0$ |
$-1$ | $g_{-1}$ | $(x^{1}_{3})^{\prime}_{0}$ | $-1$ | $0$ |
$-1$ | $\beta_{-1}$ | $U^{2}(x^{2}_{1})_{-1}+U(x^{1}_{2})^{\prime}_{0}$ | $-2$ | $U(x^{1}_{3})^{\prime}_{0}+U^{2}(x^{2}_{1})^{\prime}_{-1}$ | $=Ug_{-1}+U^{2}g_{-2}$
$-1$ | $\alpha_{-1}$ | $U^{2}(x^{2}_{1})_{0}+U^{2}(x_{0})^{\prime}_{0}$ | $-4$ | $U^{2}(x^{1}_{1})^{\prime}_{0}+U^{2}(x^{1}_{1})^{\prime}_{1}$ | $=U^{2}g_{-1}+U^{3}g_{-3}$
$-2$ | $g_{-2}$ | $(x^{1}_{3})^{\prime}_{-1}$ | $1$ | $0$ |
$-2$ | $\beta_{-2}$ | $U^{3}(x^{2}_{1})_{-2}+U(x^{1}_{2})^{\prime}_{-1}$ | $0$ | $U(x^{1}_{3})^{\prime}_{-1}+U^{3}(x^{2}_{1})^{\prime}_{-2}$ | $=Ug_{-2}+U^{3}g_{-3}$
$-3$ | $g_{-3}$ | $(x^{1}_{3})^{\prime}_{-2}$ | $5$ | $0$ |
$-4$ | $\alpha_{-4}$ | $U^{4}(x^{2}_{3})_{-3}$ | $4$ | $U(x^{1}_{3})^{\prime}_{-2}$ | $=Ug_{-3}$
Table 1. Summary of the generators of $\mathbb{X}^{\infty}$ which survive in
the reduced complex for
$\operatorname{CFK}^{\infty}{(S^{3}_{+1}(T_{3,5}),\tilde{T}_{3,5})}$.

$U^{6}\beta_{4}$$U^{6}g_{3}$$U^{2}\alpha_{2}$$U^{3}g_{2}$$\alpha_{1}$$Ug_{1}$$U^{-1}\beta_{1}$$g_{0}$$U^{-2}\alpha_{-1}$$g_{-1}$$U^{-1}\beta_{-1}$$Ug_{-2}$$\beta_{-2}$$U^{3}g_{-3}$$U^{2}\alpha_{-4}$

$U^{6}\beta_{4}$$U^{6}g_{3}$$U^{2}\alpha_{2}+U^{6}\beta_{4}$$U^{3}g_{2}$$\alpha_{1}+U^{2}\alpha_{2}+U^{6}\beta_{4}$$Ug_{1}$$U^{-1}\beta_{1}+\alpha_{1}+U^{2}\alpha_{2}+U^{6}\beta_{4}$$g_{0}$$U^{-2}\alpha_{-1}+U^{-1}\beta_{-1}+\beta_{-2}$$+U^{2}\alpha_{-4}$$g_{-1}$$U^{-1}\beta_{-1}+\beta_{-2}+U^{2}\alpha_{-4}$$Ug_{-2}$$\beta_{-2}+U^{2}\alpha_{-4}$$U^{3}g_{-3}$$U^{2}\alpha_{-4}$

Figure 5. The reduced complex for
$\operatorname{CFK}^{\infty}{(S^{3}_{+1}(T_{3,5}),\tilde{T}_{3,5})}$, drawn in
the $(i,j)$-plane. The right figure is after the change of basis.

We perform a change of basis that yields Figure 5. On the top half of the
graph, we replace $U^{2}\alpha_{2}$ by $U^{2}\alpha_{2}+U^{6}\beta_{4}$,
replace $\alpha_{1}$ by $\alpha_{1}+U^{2}\alpha_{2}+U^{6}\beta_{4}$ and
$U^{-1}\beta_{-1}$ by
$U^{-1}\beta_{-1}+\alpha_{1}+U^{2}\alpha_{2}+U^{6}\beta_{4}$. We change the
basis for the bottom half of the graph in a similar way, and keep the rest
generators unchanged. Quotienting all the acyclic summands from Figure 5
leaves us with a complex with $3$ generators from which one can readily read
out that $\tau(S^{3}_{+1}(T_{3,5}),\tilde{T}_{3,5})=-1$ and
$\varepsilon(S^{3}_{+1}(T_{3,5}),\tilde{T}_{3,5})=0$.

##  5\. Computation for general L-Space knots

In this section we prove Theorem 3.1. First, recall from Section 3 that if $K$
is an L-space knot, then the Alexander polynomial of $K$ is of the form

$\Delta_{K}(t)=(-1)^{m}+\sum_{i=1}^{m}(-1)^{m-i}(t^{n_{i}}+t^{-n_{i}})$

for a sequence of positive integers $0<n_{1}<n_{2}<\dots<n_{m}.$ Here,
$n_{m}=g(K)$ is the genus of $K$. Let $n(K)\geqslant 0$ be the quantity

$n(K):=n_{m}-n_{m-1}+\dots+(-1)^{m-2}n_{2}+(-1)^{m-1}n_{1}.$

Furthermore, let $\ell_{s}=n_{s}-n_{s-1}$.

We compute a reduced complex for
$\operatorname{CFK}^{\infty}(S^{3}_{1}(K),\tilde{K})$ according to the method
we described in Section 4 (See the paragraph after Theorem 4.1). Recall that
we identify the generators in the reduced complex of
$\operatorname{CFK}^{\infty}(S^{3}_{1}(K),\tilde{K})$ with the generators in
$\widehat{\operatorname{HFK}}(S^{3}_{1}(K),\tilde{K})$ and compute the induced
differential where each term lowers at least one of the $\mathcal{I}$ and
$\mathcal{J}$ filtrations. Throughout this section, when we say a generator in
the reduced complex of $\operatorname{CFK}^{\infty}(S^{3}_{1}(K),\tilde{K})$
has Alexander filtration level $j$, without specification it means that in
$\widehat{\operatorname{HFK}}(S^{3}_{1}(K),\tilde{K})$ the generator has
Alexander grading $\mathcal{J}=j$.

Each generator in $\operatorname{CFK}^{\infty}(S^{3}_{1}(K),\tilde{K})$ has a
coordinate in $\mathbb{Z}\oplus\mathbb{Z}$ given by
$(\mathcal{I},\mathcal{J})$. Define $\prec$ to be the natural partial order in
$\mathbb{Z}\oplus\mathbb{Z}$, where $(a,b)\prec(c,d)$ if and only if
$a\leqslant c$ and $b\leqslant d$.

Theorem 3.1 follows immediately from the following observations. Indices are
assigned according to the same rule as in Section 4: the generators of $A_{j}$
are labeled with an $j$-subscript outside the parentheses and the generators
of $B_{j}$ are labeled with a prime and an $j$-subscript outside the
parentheses.

######  Proposition 5.1.

Given an L-space knot $K$ with Alexander polynomial

$\Delta_{K}(t)=(-1)^{m}+\sum_{i=1}^{m}(-1)^{m-i}(t^{n_{i}}+t^{-n_{i}}),$

we can choose a basis such that in the reduced complex
$\operatorname{CFK}^{\infty}(S^{3}_{1}(K),\tilde{K})$ the following are
satisfied.

  1. (i)

$g_{j}:=(x^{1}_{m})^{\prime}_{j+1}$ is a generator in Alexander filtration
level $j$ for $-n_{m}+1\leqslant j\leqslant n_{m}-1$. We call all $g_{j}$
lower corner generators, and the rest generators upper corner generators. We
abuse notation and also call their images under $\mathbb{F}[U,U^{-1}]$-actions
lower or upper corner generators, respectively.

  2. (ii)

There are at most two upper corner generators in filtration level $j$, which,
if they exist, we denote by $\alpha_{j}$ and $\beta_{j}$, where

$\displaystyle\partial^{\infty}\alpha_{j}$
$\displaystyle=U^{a_{1}}g_{j}+U^{a_{2}}g_{j+1},\qquad-n_{m}+1\leqslant
j\leqslant n_{m}-1$ $\displaystyle\partial^{\infty}\beta_{j}$
$\displaystyle=U^{b_{1}}g_{j}+U^{b_{2}}g_{j-1},\qquad-n_{m}+1\leqslant
j\leqslant n_{m}-1$ $\displaystyle\partial^{\infty}\beta_{n_{m}}\ \
=g_{n_{m}-1},$ $\displaystyle\partial^{\infty}\alpha_{-n_{m}}=Ug_{-n_{m}+1}.$

where $a_{i},b_{i},i\in\\{1,2\\}$ are some constants (also depend on $j$).
Each $g_{j}$ is in the image of exactly two upper corner generators.

  3. (iii)

Given two upper corner generators $u_{j},u_{j^{\prime}}$ in $j,j^{\prime}$
filtration level respectively with $j>j^{\prime}\geqslant 0$, suppose for some
constants $t$ and $s$, $U^{t}u_{j}$ and $U^{s}u_{j^{\prime}}$ have the same
Maslov grading and their coordinates are $(i_{1},j_{1}),(i_{2},j_{2})$
respectively, then $(i_{1},j_{1})\prec(i_{2},j_{2})$.

  4. (iv)

Suppose $\alpha_{j},\beta_{j}$ both exist in filtration level $j\geqslant 0$.
If for some constants $t$ and $s$, $U^{t}\alpha_{j}$ and $U^{s}\beta_{j}$ have
the same Maslov grading and their coordinates are
$(i_{1},j_{1}),(i_{2},j_{2})$ respectively, then
$(i_{1},j_{1})\prec(i_{2},j_{2})$.

Before proving this result, we show how it implies Theorem 3.1.

###### Proof of Theorem 3.1.

Starting from the top half of the complex,
$\beta_{n_{m}}:=(x^{1}_{m})_{n_{m}}$ is the only generator in Alexander level
$n_{m}$. According to (ii), $\partial^{\infty}\beta_{n_{m}}=g_{n_{m}-1}$ and
since $g_{n_{m}-1}$ is in the image of exactly two upper corner generators, we
have $\partial^{\infty}(\text{another upper corner
generator})=g_{n_{m}-1}+U^{c}g_{n_{m}-2},$ where $c$ is some constant.
Similarly $g_{n_{m}-2}$ is in the image of exactly two upper corner
generators, so we can find the next upper corner generator that has
$g_{n_{m}-2}$ in the differential. Notice that the next upper corner generator
we find in this way will have non increasing $j$ filtration level due to the
relation in (ii). Repeating the process until we reach filtration level $0$,
this determines a sequence of all the upper corner generators with filtration
level $j\geqslant 0$ where adjacent elements in the sequence share a same
lower corner generator in the differential.

We then perform a change of basis similar to the one in Figure 5: for each
$\beta_{j}$ where $j\geqslant 0$, replace it with the sum of itself, all the
$\alpha_{j}^{\prime}$ with $j^{\prime}\geqslant j$ and all the
$\beta_{j}^{\prime}$ with $j^{\prime}>j$, multiplying each term by $U$ or
$U^{-1}$ if necessary such that they have the same Maslov grading; for each
$\alpha_{j}$ where $j\geqslant 0$, replace it with the sum of itself and all
the $\alpha_{j}^{\prime}$ and $\beta_{j}^{\prime}$ with $j^{\prime}>j$,
multiplying each term by $U$ or $U^{-1}$ if necessary such that they have the
same Maslov grading. By (iii) and (iv) this is a filtered change of basis. The
symmetry of knot complex allows us to change the basis for the bottom half of
the complex in a similar way. Quotienting out all acyclic summands reduces
$\operatorname{CFK}^{\infty}(S^{3}_{1}(K),\tilde{K})$ to a complex with $3$
generators. A direct computation shows that for $m$ odd case, the quotient
complex consists of $\beta_{1},g_{0},\alpha_{-1}$ and is determined by the
relations (after the quotient) :

$\partial^{\infty}\alpha_{-1}=U^{n(K)}g_{0},\quad\partial^{\infty}\beta_{1}=U^{n(K)-1}g_{0}$

For the $m$ even case, the quotient complex consists of
$\beta_{0},g_{0},\alpha_{0}$ with relations

$\partial^{\infty}\beta_{0}=U^{n(k)}g_{0},\quad\partial^{\infty}\alpha_{0}=U^{n(k)}g_{0}$

and a further change of basis yields a complex with one generator. ∎

The rest of the section is dedicated to prove the Proposition 5.1. In order to
prove (i), simply notice $(x^{1}_{m})^{\prime}_{j}$ is always in the kernel of
$\partial$, and never in the image of $v_{j},h_{j-1}$.

Before proceeding to the proof of (ii), we choose a reduced basis that will
simplify the computation. We demonstrate the case when $m$ is odd here, while
the even case is analogous. Recall that $\ell_{s}=n_{s}-n_{s-1}$, and let
$L_{s}:=\ell_{m}+\ell_{m-2}+...+\ell_{s+2}$ where $m-s$ is a positive even
integer. Choose pairs of image and preimage of map $\partial$ for
$j\in\\{-n_{m}+2,..,n_{m}\\}$:

$\displaystyle\partial U^{L_{2i-1}}(x^{1}_{2i})^{\prime}_{j}$
$\displaystyle=U^{L_{2i-1}}(x^{1}_{2i-1})^{\prime}_{j}\quad 1\leqslant
i\leqslant\frac{m-1}{2}$ $\displaystyle\partial U^{n(K)}(x_{0})^{\prime}_{j}$
$\displaystyle=U^{n(K)}(x^{2}_{1})^{\prime}_{j}$ $\displaystyle\partial
U^{n(K)+\sum^{i}_{1}{\ell_{2k}}}(x^{2}_{2i})^{\prime}_{j}$
$\displaystyle=U^{n(K)+\sum^{i}_{1}{\ell_{2k}}}(x^{2}_{2i+1})^{\prime}_{j}\quad
1\leqslant i\leqslant\frac{m-1}{2}$

and compute

$\displaystyle\partial^{\infty}(x^{1}_{2i})^{\prime}_{j}$
$\displaystyle=(x^{1}_{2i+1})^{\prime}_{j}+(x^{1}_{2i-1})^{\prime}_{j}\quad
1\leqslant i\leqslant\frac{m-1}{2}$
$\displaystyle\partial^{\infty}(x_{0})^{\prime}_{j}$
$\displaystyle=(x^{1}_{1})^{\prime}_{j}+(x^{2}_{1})^{\prime}_{j}$
$\displaystyle\partial^{\infty}(x^{2}_{2i})^{\prime}_{j}$
$\displaystyle=(x^{2}_{2i+1})^{\prime}_{j}+(x^{2}_{2i-1})^{\prime}_{j}\quad
1\leqslant i\leqslant\frac{m-1}{2}$

After quotienting by the $\mathbb{F}[U,U^{-1}]$-submodule $S$ generated by

$\\{(x^{1}_{2i})^{\prime}_{j},\quad\partial^{\infty}(x^{1}_{2i})^{\prime}_{j},\quad(x_{0})^{\prime}_{j},\quad,\partial^{\infty}(x_{0})^{\prime}_{j},\quad(x^{2}_{2i})^{\prime}_{j},\quad\partial^{\infty}(x^{2}_{2i})^{\prime}_{j}\\}\quad
1\leqslant i\leqslant\frac{m-1}{2}$

it readily follows that
$(x^{1}_{2i+1})^{\prime}_{j}=(x^{2}_{2i+1})^{\prime}_{j}=g_{j-1}$ for
$0\leqslant i\leqslant\frac{m-1}{2}$. Similarly, in the case where $m$ is even
we get
$(x^{1}_{2i})^{\prime}_{j}=(x_{0})^{\prime}_{j}=(x^{2}_{2i})^{\prime}_{j}=g_{j-1}$
for $1\leqslant i\leqslant\frac{m}{2},\quad j\in\\{-n_{m}+2,...,n_{m}\\}$.

To show (ii), we observe there are $4$ different cases depending on the choice
of Alexander filtration $j$ of the generators in
$\widehat{\operatorname{CFK}}(S^{3}_{1}(K),\tilde{K})$, see Figure 6 . Recall
that $\widehat{\operatorname{CFK}}(S^{3}_{1}(K),\tilde{K},n_{m})$ is given by
$A_{n_{m}}\\{i^{\prime}\leqslant 0,j^{\prime}=n_{m}\\}$, and for $j<n_{m},$
$\widehat{\operatorname{CFK}}(S^{3}_{1}(K),\tilde{K},j)$ is given by the
mapping cone (Here $i^{\prime}$ and $j^{\prime}$ are used to denote the $i$
and $j$ filtration in $\operatorname{CFK}^{\infty}(S^{3},K)$, respectively. )

$A_{j}\\{i^{\prime}\leqslant 0,j^{\prime}=j\\}\oplus
A_{j+1}\\{i^{\prime}=0,j^{\prime}\leqslant j\\}\xrightarrow{(h_{j},\
v_{j+1})}B_{j+1}\\{i^{\prime}=0\\}.$

For the following computation, we only care about the case when $j\geqslant 0$
and adopt the convention $x^{1}_{0}:=x_{0}=:x^{2}_{0}$.

  1. (a)

Figure 6 represents the case when $n_{s-1}<j<n_{s}$ where $m-s$ is a positive
even integer, note that the mapping cone consists of two summands that
generate homology: the isolated generator $g_{j}:=(x^{1}_{m})^{\prime}_{j+1}$
in $B_{j+1}$, and a summand with $5$ generators consists of the right most
generator in $A_{j}\\{i^{\prime}\leqslant
0,j^{\prime}=j\\},U^{L_{s}+(n_{s}-j)}(x^{1}_{s})_{j}$, $2$ generators in
$B_{j+1},U^{L_{s}+n_{s}}(x^{2}_{s-1})^{\prime}_{j+1}$ and
$U^{L_{s}+n_{s}}(x^{2}_{s})^{\prime}_{j+1}$, and $2$ generators in
$A_{j+1},U^{L_{s}+n_{s}}(x^{2}_{s-1})_{j+1}$ and
$U^{L_{s}+n_{s}}(x^{2}_{s})_{j+1}$. In this case
$U^{L_{s}+(n_{s}-j)}(x^{1}_{s})_{j}+U^{L_{s}+n_{s}}(x^{2}_{s-1})^{\prime}_{j+1}$
is the only upper corner generator , which consists of elements from $A_{j}$
and $B_{j+1}$. We call this kind of generator a $\beta$ type generator, and
label them $\beta_{j}$. Direct computation yields
$\partial^{\infty}\beta_{j}=U^{L_{s}+(n_{s}-j)}g_{j-1}+U^{L_{s}+n_{s}}g_{j}$.

  2. (b)

Figure 6 represents the case when $j=n_{s-1}$ where $m-s=2i,i\in\mathbb{N}$.
The isolated generator $g_{n_{s-1}}$ is the only generator in this case.

  3. (c)

Figure 6 represents the case when $n_{s}<j<n_{s+1}$ where $m-s=2i,i>0$. Aside
from $g_{j}$, the only generator is
$U^{L_{s}}(x^{1}_{s})_{j+1}+U^{L_{s}}(x^{1}_{s+1})^{\prime}_{j+1}$, which
consists of elements from $A_{j+1}$ and $B_{j+1}$. We call this kind of
generator an $\alpha$ type generator, and label them $\alpha_{j}$.
$\partial^{\infty}\alpha_{j}=U^{L_{s}+j+1}g_{j+1}+U^{L_{s}}g_{j}$.

  4. (d)

Figure 6 represents the case when $j=n_{s}$ where $m-s=2i,i>0$. We have in
this case three generators
$\beta_{n_{s}}=U^{L_{s}}(x^{1}_{s})_{n_{s}}+U^{L_{s}+n_{s}}(x^{2}_{s-1})^{\prime}_{n_{s}+1}$,
$\alpha_{n_{s}}:=U^{L_{s}}(x^{1}_{s})_{n_{s}+1}+U^{L_{s}}(x^{1}_{s+1})^{\prime}_{n_{s}+1}$
and $g_{n_{s}}$. We compute
$\partial^{\infty}\beta_{n_{s}}=U^{L_{s}}g_{n_{s}-1}+U^{L_{s}+n_{s}}g_{n_{s}}$,$\partial^{\infty}\alpha_{n_{s}}=U^{L_{s}+n_{s}+1}g_{n_{s}+1}+U^{L_{s}}g_{n_{s}}$.

The above computation along with $\partial^{\infty}\beta_{n_{m}}=g_{n_{m}-1}$
gives explicit relations for all generators on the top half. After multiplying
the upper corner generators by $U$ and $U^{-1}$ if necessary to make them have
the same Maslov grading, we analyse the four cases for $j$ as follows

  * –

case (a), $g_{j}$ is in $\partial^{\infty}U^{a_{1}}\beta_{j+1}$ and in
$\partial^{\infty}U^{a_{2}}\beta_{j}$, for some $a_{1},a_{2}\in\mathbb{Z}$;

  * –

case (b), $g_{j}$ is in $\partial^{\infty}U^{b_{1}}\beta_{j+1}$ and in
$\partial^{\infty}U^{b_{2}}\alpha_{j-1}$, for some $b_{1},b_{2}\in\mathbb{Z}$;

  * –

case (c), $g_{j}$ is in $\partial^{\infty}U^{c_{1}}\alpha_{j}$ and in
$\partial^{\infty}U^{c_{2}}\alpha_{j-1}$, for some $c_{1},c_{2}\in\mathbb{Z}$;

  * –

case (d), $g_{j}$ is in $\partial^{\infty}U^{d_{1}}\alpha_{j}$ and in
$\partial^{\infty}U^{d_{2}}\beta_{j}$, for some $d_{1},d_{2}\in\mathbb{Z}$;

This completes the proof of (ii).

$j$$U^{L_{s}}x^{1}_{s}$$U^{L_{s}}x^{1}_{s-1}$$U^{L_{s}}x^{1}_{s-2}$$U^{L_{s-2}}x^{1}_{s-1}$$n_{s-1}$$n_{s}$

$j$$U^{L_{s}}x^{1}_{s}$$U^{L_{s}}x^{1}_{s-1}$$U^{L_{s}}x^{1}_{s-2}$$U^{L_{s-2}}x^{1}_{s-1}$$n_{s-1}$$n_{s}$

$U^{L_{s}x^{1}_{s+1}}$$U^{L_{s}x^{1}_{s}}$$j$$n_{s+1}$$n_{s}$

$U^{L_{s}x^{1}_{s+1}}$$U^{L_{s}x^{1}_{s}}$$j$$n_{s+1}$$n_{s}$

Figure 6. Four different situations depending on $j$ filtration of
$\widehat{\operatorname{CFK}}(S^{3}_{1}(K),\tilde{K},j)$. The dotted line
shows $A_{j}\\{i^{\prime}\leqslant 0,j^{\prime}=j\\}$.

We prove (iii) and (iv) together. From the final part of the discussion in the
proof of (ii) it is clear there are $4$ cases where adjacent upper corner
generators share a same lower corner generator in the differential, see Figure
7.

The proof is to verify in each of the $4$ cases the conclusion is correct. We
will carry out the computation in case (a); see Figure 6 and Figure 7. The
other three cases are similar and left for the reader.

In this case $j$ satisfies $n_{s-1}<j<n_{s}$ where $m-s=2i,i\in\mathbb{N}$,

$\displaystyle\beta_{j}$
$\displaystyle=U^{L_{s}+(n_{s}-j)}(x^{1}_{s})_{j}+U^{L_{s}+n_{s}}(x^{2}_{s-1})^{\prime}_{j+1},$
$\displaystyle\partial^{\infty}\beta_{j}$
$\displaystyle=U^{L_{s}+(n_{s}-j)}g_{j-1}+U^{L_{s}+n_{s}}g_{j},$
$\displaystyle\beta_{j+1}$
$\displaystyle=U^{L_{s}+(n_{s}-j-1)}(x^{1}_{s})_{j+1}+U^{L_{s}+n_{s}}(x^{2}_{s-1})^{\prime}_{j+2},$
$\displaystyle\partial^{\infty}\beta_{j+1}$
$\displaystyle=U^{L_{s}+(n_{s}-j-1)}g_{j}+U^{L_{s}+n_{s}}g_{j+1}.$

Thus $U^{-(L_{s}+n_{s}-j-1)}\beta_{j+1}$ and $U^{-(L_{s}+n_{s})}\beta_{j}$
have the same Maslov grading. Clearly we have
$(L_{s}+n_{s}-j-1,L_{s}+n_{s})\prec(L_{s}+n_{s},L_{s}+n_{s}+j)$.

$U^{a_{1}}\beta_{j+1}$$U^{a_{2}}\beta_{j}$$g_{j}$

$U^{b_{1}}\beta_{j+1}$$U^{b_{2}}\alpha_{j-1}$$g_{j}$

$U^{c_{1}}\alpha_{j}$$U^{c_{2}}\alpha_{j-1}$$g_{j}$

$U^{d_{2}}\beta_{j}$$U^{d_{1}}\alpha_{j}$$g_{j}$

Figure 7. The four possible situations where adjacent upper corner generators
share a same lower corner generator in the differential. Each dotted line
consists of points with coordinates $(i,j)$ where $i-j=$ constant.

##  6\. Connected knot floer complex

In this section, we review the local equivalence studied in [Zem19a] and
define the connected knot Floer complex, using the same argument in the
construction of the connected Heegaard Floer homology in [HHL18].

###  6.1. Knot Floer homology

In the previous parts of the paper, we look at the knot complex in the ring
$\mathbb{F}[U]$, but now it is necessary to move to a refined version as we
want the extra grading information it encodes. Let
$\mathbb{F}[\mathcal{U},\mathcal{V}]$ be the polynomial ring with $2$
variables over finite field $\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$. Here we
denote the new variables $\mathcal{U},\mathcal{V}$ to distinguish from the
variable of $\mathbb{F}[U]$. We start from reviewing some background of knot
Floer homology over the ring $\mathbb{F}[\mathcal{U},\mathcal{V}]$.

For a knot $K$ in a integer homology sphere $Y$, let
$\mathcal{H}=(\Sigma,\boldsymbol{\boldsymbol{\alpha}},\boldsymbol{\boldsymbol{\beta}},w,z)$
be a doubly-pointed Heegaard diagram compatible with $(Y,K)$. Define
$\operatorname{CFK}_{\mathcal{R}}(K)$ to be the chain complex freely generated
over the ring $\mathcal{R}=\mathbb{F}[\mathcal{U},\mathcal{V}]$ by
$\mathbf{x}\in\mathbb{T}_{\boldsymbol{\boldsymbol{\alpha}}}\cap\mathbb{T}_{\boldsymbol{\boldsymbol{\beta}}}$
with differential

$\partial\mathbf{x}=\sum_{\mathbf{y}\in\mathbb{T}_{\boldsymbol{\boldsymbol{\alpha}}}\cap\mathbb{T}_{\boldsymbol{\boldsymbol{\beta}}}}\sum_{\begin{subarray}{c}\phi\in\pi_{2}(\mathbf{x},\mathbf{y})\\\
\mu(\phi)=1\end{subarray}}\mathcal{U}^{n_{w}(\phi)}\mathcal{V}^{n_{z}(\phi)}\mathbf{y},$

where, as usual, $\pi_{2}(\mathbf{x},\mathbf{y})$ denotes Whitney disks
connecting $\mathbf{x}$ to $\mathbf{y}$, and $\mu(\phi)$ denotes the Maslov
index of $\phi$.

Define the relative bigradings $\operatorname{gr}_{U}$ and
$\operatorname{gr}_{V}$ as follow:

$\displaystyle\operatorname{gr}_{U}(\mathbf{x},\mathbf{y})$
$\displaystyle=\mu(\phi)-2n_{w}(\phi)$
$\displaystyle\operatorname{gr}_{V}(\mathbf{x},\mathbf{y})$
$\displaystyle=\mu(\phi)-2n_{z}(\phi).$

where $\operatorname{gr}_{U}$ is the previous Maslov grading and
$\operatorname{gr}_{V}$ is the grading after exchanging the role of two base
points in the Heegaard Floer data. We often refer to $\operatorname{gr}_{U}$
as $U$-grading and $\operatorname{gr}_{V}$ $V$-grading. Define the relative
Alexander grading by

$A(\mathbf{x},\mathbf{y})=\frac{1}{2}(\operatorname{gr}_{U}(\mathbf{x},\mathbf{y})-\operatorname{gr}_{V}(\mathbf{x},\mathbf{y}))=n_{z}(\phi)-n_{w}(\phi).$

$x$$\mathcal{U}y$$\mathcal{V}z$$\mathcal{U}\mathcal{V}x$$\mathcal{U}^{2}\mathcal{V}y$$\mathcal{U}\mathcal{V}^{2}z$$\mathcal{U}^{2}\mathcal{V}^{2}x$$\mathcal{U}^{3}\mathcal{V}^{2}y$$\mathcal{U}^{2}\mathcal{V}^{3}z$

$\mathcal{V}x$$y$$\mathcal{V}^{2}z$$\mathcal{U}\mathcal{V}^{2}x$$\mathcal{U}\mathcal{V}y$$\mathcal{U}\mathcal{V}^{3}z$$\mathcal{U}^{2}\mathcal{V}^{3}x$$\mathcal{U}^{2}\mathcal{V}^{2}y$$\mathcal{U}^{2}\mathcal{V}^{4}z$$\mathcal{U}^{3}\mathcal{V}^{3}y$

Figure 8.  Knot complex $\operatorname{CFK}_{\mathcal{R}}(T_{2,3})$, where
$T_{2,3}$ is the right handed trefoil. Figure 8 shows the Alexander grading
$0$ complex $\operatorname{CFK}_{\mathcal{R}}(K,0)$, while Figure 8 shows the
Alexander graing $1$ complex $\operatorname{CFK}_{\mathcal{R}}(K,1)$.

Note that $\mathcal{U}$ lowers $\operatorname{gr}_{U}$ by $2$, preserves
$\operatorname{gr}_{V}$, and lowers $A$ by $1$, while $\mathcal{V}$ preserves
$\operatorname{gr}_{U}$, lowers $\operatorname{gr}_{V}$ by $2$, and increases
$A$ by $1$. Importantly, the differential preserves Alexander grading. Hence
the complex $\operatorname{CFK}_{\mathcal{R}}(K)$ splits as an
$\mathbb{F}[\mathcal{U},\mathcal{V}]$-module over the Alexander grading:

$\operatorname{CFK}_{\mathcal{R}}(K)=\bigoplus_{s\in\mathbb{Z}}\operatorname{CFK}_{\mathcal{R}}(K,s),$

The chain complex $\operatorname{CFK}_{\mathcal{R}}(K,s)$ is isomorphic to the
complex $A^{-}_{s}$ (See the beginning of Section 4 for the definition). One
can think of $\operatorname{CFK}_{\mathcal{R}}(K)$ as a direct sum of
$A^{-}_{s}$. We have

$\displaystyle\mathcal{U}$
$\displaystyle\colon\operatorname{CFK}_{\mathcal{R}}(K,s)\to\operatorname{CFK}_{\mathcal{R}}(K,s-1)$
$\displaystyle\mathcal{V}$
$\displaystyle\colon\operatorname{CFK}_{\mathcal{R}}(K,s)\to\operatorname{CFK}_{\mathcal{R}}(K,s+1).$

We identify $\operatorname{CFK}^{\infty}(K)$ with
$\operatorname{CFK}_{\mathcal{R}}(K,0)\otimes\mathbb{F}[\mathcal{U},\mathcal{V},(\mathcal{U}\mathcal{V})^{-1}]$,
where $\mathbb{F}[\mathcal{U},\mathcal{V},(\mathcal{U}\mathcal{V})^{-1}]$ is
the localization of $\mathbb{F}[\mathcal{U},\mathcal{V}]$ at the ideal
$(\mathcal{U}\mathcal{V})$. While previously multiplication by $U$ is the
chain isomorphism that shift the complex down the diagonal by $1$ unit, now
this action is refined to be multiplication by $\mathcal{U}\mathcal{V}$. The
complex $\operatorname{CFK}_{\mathcal{R}}(K)$ is generated by elements without
$\mathcal{U},\mathcal{V}$ decoration.

###  6.2. Self-local equivalence

Local equivalence encodes the information of homology concordance. This fact
allows us to extend various invariants originally defined only for knots in
$S^{3}$ to homology concordance invariants for knots in arbitrary integer
homology spheres. To say it more precisely, we have the following:

######  Definition 6.1 (Definition 2.4 in [Zem19a]).

For knots $K_{1},K_{2}$ in integer homology spheres $Y_{1},Y_{2}$
respectively, if there exist $\mathbb{F}[\mathcal{U},\mathcal{V}]$-equivariant
chain maps

$f\colon\operatorname{CFK}_{\mathcal{R}}(Y_{1},K_{1})\rightarrow\operatorname{CFK}_{\mathcal{R}}(Y_{2},K_{2})\qquad\textup{
and }\qquad
g\colon\operatorname{CFK}_{\mathcal{R}}(Y_{2},K_{2})\rightarrow\operatorname{CFK}_{\mathcal{R}}(Y_{1},K_{1})$

such that $f$ and $g$ preserve absolute $\mathcal{U},\mathcal{V}$ gradings and
induce isomorphisms on $(\mathcal{U}\mathcal{V})^{-1}H_{*}$, then
$\operatorname{CFK}_{\mathcal{R}}(Y_{1},K_{1})$ and
$\operatorname{CFK}_{\mathcal{R}}(Y_{2},K_{2})$ are locally equivalent.

Here for a complex $C$, $(\mathcal{U}\mathcal{V})^{-1}H_{*}(C)$ is defined to
be
$H_{*}(C\otimes_{\mathbb{F}[\mathcal{U},\mathcal{V}]}\mathbb{F}[\mathcal{U},\mathcal{V},(\mathcal{U}\mathcal{V})^{-1}])$,
where $\mathbb{F}[\mathcal{U},\mathcal{V},(\mathcal{U}\mathcal{V})^{-1}]$ is
the localization of $\mathbb{F}[\mathcal{U},\mathcal{V}]$ at the ideal
$(\mathcal{U}\mathcal{V})$. For any knot $K$ in an integer homology sphere
$Y$, there is a relatively bi-graded isomorphism between
$(\mathcal{U}\mathcal{V})^{-1}H_{*}(\operatorname{CFK}_{\mathcal{R}}(Y,K))$
and $\mathbb{F}[\mathcal{U},\mathcal{V},(\mathcal{U}\mathcal{V})^{-1}].$ We
will denote local equivalence by $\simeq$. It is easy to verify that $\simeq$
is an equivalence relation.

######  Proposition 6.2 (Theorem 1.5 in [Zem19a]).

If $(Y_{1},K_{1})$ and $(Y_{2},K_{2})$ are homology concordant, then

$\operatorname{CFK}_{\mathcal{R}}(Y_{1},K_{1})\simeq\operatorname{CFK}_{\mathcal{R}}(Y_{2},K_{2}).$

######  Definition 6.3.

Let $\mathfrak{J}$ be the set of local equivalence class of chain complexes
over the ring $\mathbb{F}[\mathcal{U},\mathcal{V}]$. The operation over
$\mathfrak{J}$ is induced by tensor product of chain complexes. For complex
$C\in\mathfrak{J}$, define the inverse of $C$ to be the dual complex $C^{*}$
and the identity element is $\mathbb{F}[\mathcal{U},\mathcal{V}]$.

######  Proposition 6.4 (Proposition 2.6 in [Zem19a]).

$\mathfrak{J}$ is a well-defined abelian group.

Taking into the consideration of the above proposition, we will use additive
notion to indicate the tensor product of the chain complexes.

Using the relation of local equivalence from a complex to itself, we can
define the connected knot complex, which will be our key ingredient in the
proof. We start by recalling the definition of self-local equivalence.

######  Definition 6.5 (Definition 3.1 in [HHL18]).

Let $C=\operatorname{CFK}_{\mathcal{R}}(Y,K)$, where $K$ is a knot in an
integer homology sphere $Y$. An absolute bigrading preserving chain map

$f\colon C\longrightarrow C$

is a self-local equivalence if $f$ induces an isomorphism on
$(\mathcal{U}\mathcal{V})^{-1}H_{*}$.

The set of self-local equivalences come with a preorder defined by comparing
the kernel. A self-local equivalence $f$ is called maximal if for any other
self-local equivalence $g$, ker $f$ $\subset$ ker $g$ implies ker $f$ $=$ ker
$g$. Maximal self-local equivalences always exist since $C$ is finitely
generated.

######  Lemma 6.6 (Lemma 3.4 in [HHL18]).

If $f,g\colon C\rightarrow C$ are maximal self-local equivalences of $C$, then
$f|_{\mathrm{im}g}\colon\mathrm{im}g\rightarrow\mathrm{im}f$ is an isomorphism
of chain complexes.

###### Proof.

Consider $f\circ g\colon C\rightarrow C.$ It is clearly a self-local
equivalence, so ker $f\circ g\subset$ ker $g$. But on the other hand ker
$g\subset$ ker $f\circ g$, so ker $f\circ g=$ ker $g$. Hence $f|_{\text{im}g}$
is injective. Parallelly $g|_{\text{im}f}$ is injective. Since both im $g$ and
im $f$ are finitely generated complex, it follows that $f|_{\text{im}g}$ is
indeed an isomorphism of chain complexes. ∎

As a result, the following is well-defined.

######  Definition 6.7 (Definition 3.9 in [HHL18]).

Let $C=\operatorname{CFK}_{\mathcal{R}}(Y,K)$, where $K$ is a knot in an
integer homology sphere $Y$ and $f$ a maximal self-local equivalence. The
connected complex is defined to be im $f$, denoted $C_{\mathrm{conn}}$ .

######  Proposition 6.8 (Lemma 3.6 in [HHL18]).

Suppose $C_{\mathrm{conn}}$ is the connected complex of
$C=\operatorname{CFK}_{\mathcal{R}}(Y,K)$, then $C$ is isomorphic to
$C_{\mathrm{conn}}\oplus A$ as chain complex, where $A$ is a subcomplex such
that
$A\otimes_{\mathbb{F}[\mathcal{U},\mathcal{V}]}\mathbb{F}[\mathcal{U},\mathcal{V},(\mathcal{U}\mathcal{V})^{-1}]$
has trivial homology.

###### Proof.

Suppose $f$ is a maximal self-local equivalence, then by Lemma 6.6,
$f|_{\mathrm{im}f}$ is injective. Let $A=\mathrm{ker}f$ and we aim to show $C$
is isomorphic to $\mathrm{im}f\oplus\mathrm{ker}f$ as chain complex. There is
a standard algebra argument to prove this. For $m\in C$,
$(n,p)\in\mathrm{im}f\oplus\mathrm{ker}f$, define maps in both directions:

$\displaystyle(n,p)$ $\displaystyle\longrightarrow
p+\big{(}f|_{\mathrm{im}f}\big{)}^{-1}(n)$ $\displaystyle m$
$\displaystyle\longrightarrow(f(m),\big{(}f|_{\mathrm{im}f}\big{)}^{-1}\circ
f(m)+m).$

This gives desired isomorphism. The fact that
$A\otimes_{\mathbb{F}[\mathcal{U},\mathcal{V}]}\mathbb{F}[\mathcal{U},\mathcal{V},(\mathcal{U}\mathcal{V})^{-1}]$
has trivial homology follows from the definition of self-local equivalence. ∎

######  Proposition 6.9 (Proposition 3.10 in [HHL18]).

Let $C=\operatorname{CFK}_{\mathcal{R}}(Y_{1},K_{1})$ and
$C^{\prime}=\operatorname{CFK}_{\mathcal{R}}(Y_{2},K_{2})$ be locally
equivalent knot complexes, then $C_{\mathrm{conn}}$ and
$C^{\prime}_{\mathrm{conn}}$ are isomorphic as chain complexes.

###### Proof.

Since $C$ is local equivalent to $C^{\prime}$, there exists bigrading
preserving, $\mathbb{F}[\mathcal{U},\mathcal{V}]$-equivariant chain maps

$\displaystyle F\colon C\longrightarrow C^{\prime}$ $\displaystyle G\colon
C^{\prime}\longrightarrow C$

such that $F$ and $G$ induce isomorphisms on
$(\mathcal{U}\mathcal{V})^{-1}H_{*}$.

Suppose $f$ and $g$ are maximal self-local equivalence of $C$ and $C^{\prime}$
respectively. Then $f\circ G\circ g\circ F$ is a self-local equivalence of
$C$. So ker $G\circ g\circ F\circ f\subset$ ker $f$. But on the other hand,
ker $f\subset$ ker $G\circ g\circ F\circ f$. We have ker $G\circ g\circ F\circ
f=$ ker $f$. In particular, $g\circ
F|_{\mathrm{im}f}\colon\mathrm{im}f\rightarrow\mathrm{im}g$ is injective.
Parallelly, $f\circ
G|_{\mathrm{im}g}\colon\mathrm{im}g\rightarrow\mathrm{im}f$ is injective.
Since both im $g$ and im $f$ are finitely generated, $g\circ
F|_{\mathrm{im}f}$ and $f\circ G|_{\mathrm{im}g}$ are isomorhisms of chain
complexes. ∎

##  7\. Linear Independence of $(Y_{n},K_{n})$

In this section, we prove the linear independence of the previously
constructed pairs $(Y_{n},K_{n})$ in the group
$\widehat{\mathcal{C}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$ using
connected knot complex. Recall the definition of pairs $(Y_{n},K_{n})$: For
$n>1$, let $M_{n}$ denote $+1$-surgery on $T_{2,4n-1}$, the $(2,4n-1)$-torus
knot, let $Y_{n}=M_{n}\mathbin{\\#}-M_{n}$, and let $K_{n}\subset Y_{n}$
denote the connected sum of the core of the surgery in $M_{n}$ with the unknot
in $-M_{n}$. We remind the reader we use $\simeq$ to denote local equivalence,
and since throughout the section the computation is carried out in the group
$\mathfrak{J}$, we will use additive notation for tensor product of chain
complexes over the ring $\mathbb{F}[\mathcal{U},\mathcal{V}]$.

Let $C_{n}$ denote the connected knot complex of
$\operatorname{CFK}_{\mathcal{R}}(-Y_{n},-K_{n})$, and $C_{n}^{*}$ the
connected knot complex of $\operatorname{CFK}_{\mathcal{R}}(Y_{n},K_{n})$.
Recall $C_{n}^{*}$ is isomorphic to $(C_{n})^{*}$ as a chain complex.

We recapitulate the properties regarding pairs $(Y_{n},K_{n})$ that we have
studied over the preivous sections (See Theorem 3.1), using the language of
the chosen ring $\mathbb{F}[\mathcal{U},\mathcal{V}]$ for the knot Floer
complex.

######  Proposition 7.1.

The connected knot complex $C_{n}$, where $n\geqslant 1$, is generated by
$x_{0},x_{1}$ and $y_{1}$ with bigradings

$\displaystyle\operatorname{gr}(x_{0})=(-2,0)$
$\displaystyle\operatorname{gr}(x_{1})=(0,-2)$
$\displaystyle\operatorname{gr}(y_{1})=(-2n+1,-2n+1).$

and the differential

$\partial
y_{1}=\mathcal{U}^{n-1}\mathcal{V}^{n}x_{0}+\mathcal{U}^{n}\mathcal{V}^{n-1}x_{1}.$

The connected knot complex $C_{n}^{*}$, where $n\geqslant 1$, is generated by
$x_{0}^{*},x_{1}^{*}$ and $y_{1}^{*}$ with bigradings

$\displaystyle\operatorname{gr}(x_{0}^{*})=(2,0)$
$\displaystyle\operatorname{gr}(x_{1}^{*})=(0,2)$
$\displaystyle\operatorname{gr}(y_{1}^{*})=(2n-1,2n-1).$

and the differentials

$\partial\mathcal{U}x_{0}^{*}=\partial\mathcal{V}x_{1}^{*}=\mathcal{U}^{n}\mathcal{V}^{n}y_{1}^{*}.$

Morever, for $n\geqslant 2$, we also have

  1. (1)

$\varepsilon(C_{n})=\varepsilon(C_{n}^{*})=0;$

  2. (2)

$\tau(C_{n})=1,\quad\tau(C_{n}^{*})=-1.$

$\mathcal{U}x_{0}^{*}$$\mathcal{V}x_{1}^{*}$$\mathcal{U}^{n}\mathcal{V}^{n}y_{1}^{*}$

$\mathcal{U}^{-n}\mathcal{V}^{-n}y_{1}$$\mathcal{U}^{-1}x_{0}$$\mathcal{V}^{-1}x_{1}$

Figure 9. Figure 9 shows $C_{n}^{*}$. Figure 9 shows $C_{n}$ in the preimage
of $\mathcal{U}^{n}\mathcal{V}^{n}$.

Since the complex $C_{1}$ and $C_{1}^{*}$ are isomorphic to the knot complex
of right and left handed trefoil respectively, for the rest of the section we
will only consider $C_{n}$ and $C_{n}^{*}$ with $n\geqslant 2$. Our goal is to
prove the following statement, which will in turn finish the proof of
Proposition 1.4.

######  Proposition 7.2.

If there exist nonnegative integers $p_{n},q_{n},N$ , such that

(2) $\sum_{n=2}^{N}(p_{n}C_{n}+q_{n}C_{n}^{*})\simeq C$

for some knot complex $C$ of a knot in $S^{3}$, then $p_{n}=q_{n}$ for all
$n\geqslant 2$.

Note that $C_{n}+C_{n}^{*}\simeq\mathbb{F}[\mathcal{U},\mathcal{V}]$. When not
all $p_{n}=q_{n}$ we can rewrite Equation (2) as

$\sum_{i=1}^{k}C_{n_{i}}+\sum_{i=1}^{l}C_{m_{i}}^{*}\simeq C$

where $2\leqslant n=n_{1}\leqslant..\leqslant n_{k}$ , $2\leqslant
m=m_{1}\leqslant..\leqslant m_{l}$. Without the loss of generality we are
assuming $n>m$. A straightforward computation shows that

$\displaystyle\varepsilon(C)=$
$\displaystyle\varepsilon\big{(}\sum_{i=1}^{k}C_{n_{i}}+\sum_{i=1}^{l}C_{m_{i}}^{*}\big{)}=0$
$\displaystyle\tau(C)=$
$\displaystyle\tau\big{(}\sum_{i=1}^{k}C_{n_{i}}+\sum_{i=1}^{l}C_{m_{i}}^{*}\big{)}=k-l$

If $l\neq k$ this contradicts the fact that $C$ is the knot complex of a knot
in $S^{3}$. Hence we care only about the case

(3) $\sum_{i=1}^{k}C_{n_{i}}+\sum_{i=1}^{k}C_{m_{i}}^{*}\simeq C$

where $2\leqslant n=n_{1}\leqslant..\leqslant n_{k}$ , $2\leqslant
m=m_{1}\leqslant..\leqslant m_{k}$ and $n>m$.

For a chain complex $C$ over $\mathbb{F}[\mathcal{U},\mathcal{V}]$, let
$C_{\mathrm{vert}}:=C/(\mathcal{U}=0)\otimes\mathbb{F}[\mathcal{V},\mathcal{V}^{-1}]$.
Define the vertical homology $H_{\mathrm{vert}}$ to be
$H_{*}(C_{\mathrm{vert}})$.

######  Proposition 7.3.

If $C$ is the knot complex of a knot in $S^{3}$, then $H_{\mathrm{vert}}(C)$
is one-dimensional. If $C$ is locally equivalent to a knot complex of a knot
in $S^{3}$, then $H_{\mathrm{vert}}(C_{\mathrm{conn}})$ is one-dimensional.

###### Proof.

Given a knot complex $C$ for a knot in $S^{3}$,
$C/(\mathcal{U}=0)\otimes\mathbb{F}[\mathcal{V},\mathcal{V}^{-1}]$ recovers
$\widehat{\operatorname{CF}}(S^{3})$, whose homology is one-dimensional. If
$C$ is locally equivalent to a knot complex of a knot in $S^{3}$, then
$C_{\mathrm{conn}}$ is isomorphic to a connected knot complex of a knot in
$S^{3}$. So $H_{\mathrm{vert}}(C_{\mathrm{conn}})$ is one-dimensional. ∎

The strategy of our proof for Proposition 7.2 is to show Equation (3) never
holds true by comparing the connected complex on both sides. Let
$C^{\prime}=\sum_{i=1}^{k}C_{n_{i}}+\sum_{i=1}^{k}C_{m_{i}}^{*}$. By the way
of contradiction, if Equation (3) holds true, it would imply that
$C^{\prime}_{\mathrm{conn}}$ is isomorphic to $C_{\mathrm{conn}}$ as a chain
complex. In particular,
$H_{\mathrm{vert}}(C^{\prime}_{\mathrm{conn}})=H_{\mathrm{vert}}(C_{\mathrm{conn}})$.
Hence it suffices to prove dim
$(H_{\mathrm{vert}}(C^{\prime}_{\mathrm{conn}}))>1$.

Figure 10. Complex $3C^{*}_{2}+C^{*}_{3}$, which contains the inverse saw-edge
$\mathfrak{C}^{*}(4,2)$ as a subcomplex (generators of $\mathfrak{C}^{*}(4,2)$
are depicted in darker color).

As an effort to discuss the objects at hand precisely, we introduce the
following definitions.

######  Definition 7.4.

Define a saw-edge $\mathfrak{C}(k,n)$ of length $k$ with tooth size $n$, where
$k\geqslant 1$ and $n\geqslant 2$, to be a chain complex over
$\mathbb{F}[\mathcal{U},\mathcal{V}]$, generated by $x_{i}^{(k)},i=0,...,k$
and $y_{i}^{(k)},i=1,...,k.$

Bigradings are given by

$\displaystyle\operatorname{gr}(x^{(k)}_{i})=(-2(k-i),-2i)$
$\displaystyle\operatorname{gr}(y^{(k)}_{i})=(-2(n+k-1)+1,-2(n-1+i)+1).$

while the differentials are

$\partial
y^{(k)}_{i}=\mathcal{U}^{n}\mathcal{V}^{n-1}x^{(k)}_{i}+\mathcal{U}^{n-1}\mathcal{V}^{n}x^{(k)}_{i-1}.\qquad\qquad\qquad$

Define an inverse saw-edge $\mathfrak{C}^{*}(k,n)$ of length $k$ with tooth
size $n$ , where $k\geqslant 1$ and $n\geqslant 2$, to be simply the dual
complex of the saw-edge $\mathfrak{C}(k,n)$. Dualizing the generators of
$\mathfrak{C}(k,n)$, the inverse saw-edge $\mathfrak{C}^{*}(k,n)$ is generated
by $x_{i}^{(k),*},i=0,...,k$ and $y_{i}^{(k),*},i=1,...,k.$

Bigradings are given by

$\displaystyle\operatorname{gr}(x^{(k),*}_{i})=(2(k-i),2i)$
$\displaystyle\operatorname{gr}(y^{(k),*}_{i})=(2(n+k-1)-1,2(n-1+i)-1).$

while the differentials are

$\partial
x^{(k),*}_{i}=\begin{cases}\mathcal{U}^{n-1}\mathcal{V}^{n}y^{(k),*}_{1},&\quad
i=0,\\\
\mathcal{U}^{n}\mathcal{V}^{n-1}y^{(k),*}_{i}+\mathcal{U}^{n-1}\mathcal{V}^{n}y^{(k),*}_{i+1},&\quad
i=1,...k-1,\\\ \mathcal{U}^{n}\mathcal{V}^{n-1}y^{(k),*}_{k}&\quad
i=k.\end{cases}$

$x^{(2)}_{2}$$x^{(2)}_{1}$$x^{(2)}_{0}$$y^{(2)}_{2}$$y^{(2)}_{1}$$d_{1}$$b_{1}$$c_{1}$$a_{1}$
Figure 11.  Complex $C_{2}+C_{3}$, where generators are labelled abstractly,
missing their actual $\mathcal{U},\mathcal{V}$ decorations. Using the
notations from Proposition 7.5, $a_{1},b_{1},c_{1},d_{1}$ generate subcomplex
$\mathfrak{D}$, where $b_{1}$ plays the role of $w_{2}$.

Next, we want to study some key properties of the complex
$\sum_{i=1}^{k}C_{n_{i}}$. Roughly speaking, $\sum_{i=1}^{k}C_{n_{i}}$
consists of two parts, the saw-edge $\mathfrak{C}(k,n)$ and a subcomplex we
call $\mathfrak{D}$. We can pick a basis for $\sum_{i=1}^{k}C_{n_{i}}$ over
$\mathbb{F}[\mathcal{U},\mathcal{V}]$, such that on the module level,
$\sum_{i=1}^{k}C_{n_{i}}$ decomposites as a direct sum
$\mathfrak{C}(k,n)\oplus\mathfrak{D}$, where $\mathfrak{C}(k,n)$ is generated
by $x^{(k)}_{i},i=0,...,k$ and $y^{(k)}_{i},i=1,...,k$, whereas $\mathfrak{D}$
is generated by $w_{i},i=1,...,k$ and other generators. While as a chain
complex, the saw-edge $\mathfrak{C}(k,n)$ is no longer a direct summand, there
are generators we denote by $w_{i},i=1,...,k$ in $\mathfrak{D}$, such that the
image of
$\partial\big{(}(\mathcal{U}\mathcal{V})^{h_{i}}y^{(k)}_{i}+w_{i}\big{)}$ is
inside $\mathfrak{C}(k,n)$. More precisely, we have
$\partial\big{(}(\mathcal{U}\mathcal{V})^{h_{i}}y^{(k)}_{i}+w_{i}\big{)}=(\mathcal{U}\mathcal{V})^{h_{i}}\big{(}\mathcal{U}^{n}\mathcal{V}^{n-1}x^{(k)}_{i}+\mathcal{U}^{n-1}\mathcal{V}^{n}x^{(k)}_{i-1}\big{)},i=1,...,k$
where $h_{i}$ is a nonnegative integer. We will slightly abuse the notation
and write
$\partial\big{(}y^{(k)}_{i}+(\mathcal{U}\mathcal{V})^{-h_{i}}w_{i}\big{)}=\mathcal{U}^{n}\mathcal{V}^{n-1}x^{(k)}_{i}+\mathcal{U}^{n-1}\mathcal{V}^{n}x^{(k)}_{i-1},i=1,...,k$
instead.

######  Proposition 7.5.

There exists a basis for the complex $\sum_{i=1}^{k}C_{n_{i}}$, where
$2\leqslant n=n_{1}\leqslant..\leqslant n_{k}$, and a subcomplex
$\mathfrak{D}$, such that the following are satisfied:

  1. (1)

on the module level, $\sum_{i=1}^{k}C_{n_{i}}$ is isomorphic to
$\mathfrak{C}(k,n)\oplus\mathfrak{D}$ as a module;

  2. (2)

$\sum_{i=1}^{k}C_{n_{i}}/\mathfrak{D}$ is isomorphic to $\mathfrak{C}(k,n)$ as
a chain complex;

  3. (3)

each $x^{(k)}_{i}\in\mathfrak{C}(k,n),i=0,...,k$ generates
$(\mathcal{U}\mathcal{V})^{-1}H_{*}\big{(}\sum_{i=1}^{k}C_{n_{i}}\big{)};$

  4. (4)

for each $y^{(k)}_{i}\in\mathfrak{C}(k,n),i=1,...,k$, there exists some
$w_{i}\in\mathfrak{D}$, such that

$\partial\big{(}y^{(k)}_{i}+(\mathcal{U}\mathcal{V})^{-h_{i}}w_{i}\big{)}=\mathcal{U}^{n}\mathcal{V}^{n-1}x^{(k)}_{i}+\mathcal{U}^{n-1}\mathcal{V}^{n}x^{(k)}_{i-1},$

where $h_{i}$ is a nonnegative integer, and $\partial$ is the differential of
the chain complex $\sum_{i=1}^{k}C_{n_{i}}$.

Note that in (3),(4) above when we say
$x^{(k)}_{i},y^{(k)}_{i}\in\mathfrak{C}(k,n)$ it specifically means that we
see $\mathfrak{C}(k,n)$ as a submodule of $\sum_{i=1}^{k}C_{n_{i}}$, take
elements $x^{(k)}_{i},y^{(k)}_{i}$ from $\mathfrak{C}(k,n)$ and equip them
with the differential of the whole complex $\sum_{i=1}^{k}C_{n_{i}}$. In order
to prove Proposition 7.5 we require a lemma. In the following lemma, we will
construct a basis $x^{(k+1)}_{i},i=0,...,k+1,y^{(k+1)}_{i},i=1,...,k+1$ and
$\\{a_{i},b_{i},c_{i},d_{i}\\}_{i=1,...,k}$ that generates
$C^{\prime\prime}:=\mathfrak{C}(k,n)+C_{\ell}$ over
$\mathbb{F}[\mathcal{U},\mathcal{V}],$ where $\ell\geqslant n$. On the module
level, $C^{\prime\prime}$ decomposites as a direct sum
$\mathfrak{C}(k+1,n)\oplus\mathfrak{D}_{0}$, where $\mathfrak{C}(k+1,n)$ is
generated by $x^{(k+1)}_{i},i=0,...,k+1$ and $y^{(k+1)}_{i},i=1,...,k+1$,
while $\mathfrak{D}_{0}$ is generated by
$\\{a_{i},b_{i},c_{i},d_{i}\\}_{i=1,...,k}$.

######  Lemma 7.6.

There exists a basis for the complex
$C^{\prime\prime}:=\mathfrak{C}(k,n)+C_{\ell},$ where $\ell\geqslant n$, and a
subcomplex $\mathfrak{D}_{0}$, such that

  1. (1)

on the module level,
$C^{\prime\prime}=\mathfrak{C}(k+1,n)\oplus\mathfrak{D}_{0}$ as a module;

  2. (2)

$C^{\prime\prime}/\mathfrak{D}_{0}$ is isomorphic to $\mathfrak{C}(k+1,n)$ as
a chain complex;

  3. (3)

each $x^{(k+1)}_{i}\in\mathfrak{C}(k+1,n),i=0,...,k+1$ generates
$(\mathcal{U}\mathcal{V})^{-1}H_{*}\big{(}C^{\prime\prime}\big{)};$

  4. (4)

there exists $w_{k+1}\in\mathfrak{D}_{0}$ such that

$\partial\big{(}y^{(k+1)}_{k+1}+(\mathcal{U}\mathcal{V})^{-\ell+n}w_{k+1}\big{)}=\mathcal{U}^{n}\mathcal{V}^{n-1}x^{(k+1)}_{k+1}+\mathcal{U}^{n-1}\mathcal{V}^{n}x^{(k+1)}_{k},$

and we have

$\partial
y^{(k+1)}_{i}=\mathcal{U}^{n}\mathcal{V}^{n-1}x^{(k)}_{i}+\mathcal{U}^{n-1}\mathcal{V}^{n}x^{(k)}_{i-1},i=1,...,k$

where $\partial$ is the differential of the chain complex $C^{\prime\prime}$.

Here $x^{(k+1)}_{i},y^{(k+1)}_{i}$ are all elements of $\mathfrak{C}(k+1,n)$,
seen as a submodule of $C^{\prime\prime}$, equipped with the differential of
the whole complex $C^{\prime\prime}$.

###### Proof of Lemma 7.6.

Suppose complex $\mathfrak{C}(k,n)$ is generated by $x_{i}^{(k)},i=0,...,k$
and $y_{i}^{(k)},i=1,...,k.$ with differentials $\partial
y^{(k)}_{i}=\mathcal{U}^{n}\mathcal{V}^{n-1}x^{(k)}_{i}+\mathcal{U}^{n-1}\mathcal{V}^{n}x^{(k)}_{i-1},$
and complex $C_{\ell}$ is generated by $x^{(1)}_{0},x^{(1)}_{1}$ and
$y^{(1)}_{1}$ with differential $\partial
y_{1}^{(1)}=\mathcal{U}^{\ell-1}\mathcal{V}^{\ell}x_{0}^{(1)}+\mathcal{U}^{\ell}\mathcal{V}^{\ell-1}x_{1}^{(1)}.$
We will construct a basis for $C^{\prime\prime}=\mathfrak{C}(k,n)+C_{\ell}.$
Set

$\displaystyle x^{(k+1)}_{i}=\begin{cases}x^{(k)}_{k}\otimes
x^{(1)}_{1},\quad&i=k+1,\\\ x^{(k)}_{i}\otimes
x^{(1)}_{0},\quad&i=0,1,...,k;\end{cases}$ $\displaystyle
y^{(k+1)}_{i}=\begin{cases}y^{(k)}_{k}\otimes x^{(1)}_{1},\quad&i=k+1,\\\
y^{(k)}_{i}\otimes x^{(1)}_{0},\quad&i=1,...,k;\end{cases}$ $\displaystyle
a_{i}=y^{(k)}_{i}\otimes y^{(1)}_{1},\quad i=1,...,k;$ $\displaystyle
b_{i}=\begin{cases}x^{(k)}_{k}\otimes
y^{(1)}_{1}+(\mathcal{U}\mathcal{V})^{\ell-n}y^{(k)}_{k}\otimes
x^{(1)}_{1},\quad&i=k,\\\ y^{(k)}_{i+1}\otimes x^{(1)}_{0}+y^{(k)}_{i}\otimes
x^{(1)}_{1},\quad&i=1,...,k-1;\end{cases}$ $\displaystyle
c_{i}=x^{(k)}_{i-1}\otimes
y^{(1)}_{1}+(\mathcal{U}\mathcal{V})^{\ell-n}y^{(k)}_{i}\otimes
x^{(1)}_{0},i=1,...,k;$ $\displaystyle d_{i}=x^{(k)}_{i}\otimes
x^{(1)}_{0}+x^{(k)}_{i-1}\otimes x^{(1)}_{1},i=1,...,k.$

Differentials are

$\displaystyle\partial
y^{(k+1)}_{i}=\begin{cases}\mathcal{U}^{n}\mathcal{V}^{n-1}x^{(k+1)}_{k+1}+\mathcal{U}^{n-1}\mathcal{V}^{n}x^{(k+1)}_{k}+\mathcal{U}^{n-1}\mathcal{V}^{n}d_{k},\quad&i=k+1,\\\
\mathcal{U}^{n}\mathcal{V}^{n-1}x^{(k+1)}_{i}+\mathcal{U}^{n-1}\mathcal{V}^{n}x^{(k+1)}_{i-1},\quad&i=1,...,k;\end{cases}$
$\displaystyle\partial
a_{i}=\begin{cases}\mathcal{U}^{n}\mathcal{V}^{n-1}b_{k}+\mathcal{U}^{n-1}\mathcal{V}^{n}c_{k},\quad&i=k,\\\
\mathcal{U}^{\ell}\mathcal{V}^{\ell-1}b_{i}+\mathcal{U}^{n}\mathcal{V}^{n-1}c_{i+1}+\mathcal{U}^{n-1}\mathcal{V}^{n}c_{i},\quad&i=1,...,k-1;\end{cases}$
$\displaystyle\partial
b_{i}=\begin{cases}\mathcal{U}^{\ell-1}\mathcal{V}^{\ell}d_{k},\quad&i=k,\\\
\mathcal{U}^{n}\mathcal{V}^{n-1}d_{i+1}+\mathcal{U}^{n-1}\mathcal{V}^{n}d_{i},\quad&i=1,...,k-1;\end{cases}$
$\displaystyle\partial c_{i}=\mathcal{U}^{\ell}\mathcal{V}^{\ell-1}d_{i},\quad
i=1,...,k.$

Let $\mathfrak{D}_{0}$ be the subcomplex generated by
$\\{a_{i},b_{i},c_{i},d_{i}\\}_{i=1,...,k}$. Conditions (1),(2) and (3) are
all straightforward to verify. To show (4), let $w_{k+1}=b_{k}$, it then
follows that
$\partial\big{(}y^{(k+1)}_{k+1}+(\mathcal{U}\mathcal{V})^{-\ell+n}w_{k+1}\big{)}=\mathcal{U}^{n}\mathcal{V}^{n-1}x^{(k+1)}_{k+1}+\mathcal{U}^{n-1}\mathcal{V}^{n}x^{(k+1)}_{k}.$
This completes the proof of Lemma 7.6. ∎

######  Remark 7.7.

In Lemma 7.6 if $\ell=n$, $\mathfrak{D}_{0}$ is actually a direct summand and
$\mathfrak{C}(k,n)+C_{n}\simeq\mathfrak{C}(k+1,n).$

###### Proof of Proposition 7.5.

We will prove the statement by induction over $k$.
$C_{n}\simeq\mathfrak{C}(1,n)$ follows directly from Proposition 7.1.

Suppose the proposition holds for $\sum_{i=1}^{k}C_{n_{i}}$, where
$\sum_{i=1}^{k}C_{n_{i}}=\mathfrak{C}(k,n)\oplus\mathfrak{D}$ as a module.
Consider $\widetilde{C}:=\sum_{i=1}^{k}C_{n_{i}}+C_{\ell},$ where $2\leqslant
n=n_{1}\leqslant..\leqslant n_{k}\leqslant\ell.$ From Lemma 7.6 we know that
on the module level $\big{(}\mathfrak{C}(k,n)\oplus\mathfrak{D}\big{)}\otimes
C_{\ell}=\big{(}\mathfrak{C}(k,n)\otimes
C_{\ell}\big{)}\oplus\big{(}\mathfrak{D}\otimes
C_{\ell}\big{)}=\mathfrak{C}(k+1,n)\oplus\mathfrak{D}_{0}\oplus\big{(}\mathfrak{D}\otimes
C_{\ell}\big{)}$. Note that this gives a basis of $\widetilde{C}$ that extends
the basis of $C^{\prime\prime}$, which has been chosen in the proof of Lemma
7.6.

Define $\mathfrak{D}_{1}:=\mathfrak{D}+C_{\ell}$ as chain complexes. Note that
$\mathfrak{D}_{1}$ has underlying module $\mathfrak{D}\otimes C_{\ell}$, which
appeared in the above expression, now further equipped with the subcomplex
differential.

Define $\widetilde{\mathfrak{D}}:=\mathfrak{D}_{0}\oplus\mathfrak{D}_{1}$
first as a module, then equip it with the differential of the chain complex
$\widetilde{C}$. Observe that $\widetilde{\mathfrak{D}}$ is a subcomplex of
$\widetilde{C}$ because both $\mathfrak{D}_{0}$ and $\mathfrak{D}_{1}$ are
subcomplexes. We will prove that the complex $\widetilde{C}$ together with the
subcomplex $\widetilde{\mathfrak{D}}$ under the basis chosen above satisfy all
$4$ requirements in Proposition 7.5.

Obviously, $\widetilde{C}=\mathfrak{C}(k+1,n)\oplus\widetilde{\mathfrak{D}}$
as a module. This completes (1).

To prove (2), observe that $\widetilde{C}/\widetilde{\mathfrak{D}}$ is
isomorphic to $\big{(}\widetilde{C}/\mathfrak{D}_{1}\big{)}/\mathfrak{D}_{0}$
as a chain complex. By definition,
$\widetilde{C}/\mathfrak{D}_{1}=\big{(}\sum_{i=1}^{k}C_{n_{i}}+C_{\ell}\big{)}/(\mathfrak{D}+C_{\ell})$.
The natural map between
$\big{(}\sum_{i=1}^{k}C_{n_{i}}+C_{\ell}\big{)}/(\mathfrak{D}+C_{\ell})$ and
$\sum_{i=1}^{k}C_{n_{i}}/\mathfrak{D}+C_{\ell}$ is a chain complex
isomorphism. By induction hypothesis $\sum_{i=1}^{k}C_{n_{i}}/\mathfrak{D}$ is
isomorphic to $\mathfrak{C}(k,n)$ and finally according to Lemma 7.6,
$\big{(}\mathfrak{C}(k,n)+C_{\ell}\big{)}/\mathfrak{D}_{0}$ is isomorphic to
$\mathfrak{C}(k+1,n)$ as a chain complex, finishing the proof of (2).

From our construction of $\mathfrak{C}(k+1,n)$ it is not hard to see

$\displaystyle x^{(k+1)}_{i}=\begin{cases}x^{(k)}_{k}\otimes
x^{(1)}_{1},\quad&i=k+1,\\\ x^{(k)}_{i}\otimes
x^{(1)}_{0},\quad&i=0,1,...,k;\end{cases}$

generates $(\mathcal{U}\mathcal{V})^{-1}H_{*}\big{(}\widetilde{C}\big{)}$,
proving (3).

In order to show (4), suppose there exists some $w_{i}\in\mathfrak{D}$ and
nonnegative $s_{i}$ for each $i$ such that

$\partial\big{(}y^{(k)}_{i}+(\mathcal{U}\mathcal{V})^{-h_{i}}w_{i}\big{)}=\mathcal{U}^{n}\mathcal{V}^{n-1}x^{(k)}_{i}+\mathcal{U}^{n-1}\mathcal{V}^{n}x^{(k)}_{i-1}.$

For $i=1,...,k,$

$\displaystyle\partial\big{(}y^{(k+1)}_{i}+(\mathcal{U}\mathcal{V})^{-h_{i}}w_{i}\otimes
x^{(1)}_{0}\big{)}$
$\displaystyle=\mathcal{U}^{n}\mathcal{V}^{n-1}x^{(k)}_{i}\otimes
x^{(1)}_{0}+\mathcal{U}^{n-1}\mathcal{V}^{n}x^{(k)}_{i-1}\otimes x^{(1)}_{0}$
$\displaystyle=\mathcal{U}^{n}\mathcal{V}^{n-1}x^{(k+1)}_{i}+\mathcal{U}^{n-1}\mathcal{V}^{n}x^{(k+1)}_{i-1}$

If $i=k+1$, then

$\displaystyle\partial\big{(}y^{(k+1)}_{k+1}+(\mathcal{U}\mathcal{V})^{-h_{i}}w_{i}\otimes
x^{(1)}_{1}+(\mathcal{U}\mathcal{V})^{-(l-n)}b_{k}\big{)}$
$\displaystyle=\mathcal{U}^{n}\mathcal{V}^{n-1}x^{(k)}_{i}\otimes
x^{(1)}_{1}+\mathcal{U}^{n-1}\mathcal{V}^{n}x^{(k)}_{k}\otimes x^{(1)}_{0}$
$\displaystyle=\mathcal{U}^{n}\mathcal{V}^{n-1}x^{(k+1)}_{k+1}+\mathcal{U}^{n-1}\mathcal{V}^{n}x^{(k+1)}_{k}$

satisfying the requirement of (4). ∎

######  Proposition 7.8.

$\mathfrak{C}^{*}(k,m)$ is a subcomplex of $\sum_{i=1}^{k}C_{m_{i}}^{*},$
where $2\leqslant m=m_{1}\leqslant..\leqslant m_{k}.$ In particular,
$\sum^{k}_{i=0}\mathcal{U}^{k-i}\mathcal{V}^{i}x^{(k),*}_{i}$ is a generator
of $(\mathcal{U}\mathcal{V})^{-1}H_{*}$.

###### Proof.

Simply take the dual complex of $\sum_{i=1}^{k}C_{m_{i}}.$ Since there is no
incoming arrow to $\mathfrak{C}(k,m)$, $\mathfrak{C}^{*}(k,m)$ is a subcomplex
of $\sum_{i=1}^{k}C_{m_{i}}^{*}.$ It is also easy to verify that
$\sum^{k}_{i=0}\mathcal{U}^{k-i}\mathcal{V}^{i}x^{(k),*}_{i}$ generates
$(\mathcal{U}\mathcal{V})^{-1}H_{*}$. ∎

###### Proof of Proposition 7.2.

Consider the complex
$C^{\prime}=\sum_{i=1}^{k}C_{n_{i}}+\sum_{i=1}^{k}C_{m_{i}}^{*}$. We fix a
basis for $\sum_{i=1}^{k}C_{n_{i}}$ according to Proposition 7.5. Similarly
fix a basis for $\sum_{i=1}^{k}C_{m_{i}}$ and take the dual basis. This gives
a basis for $C^{\prime}$. We will suppress the $(k)$ from upper indices since
both saw-edge complexes have length $k$.

Suppose $f$ is a maximal self-local equivalence of $C^{\prime}$. In particular
$f$ is $\mathcal{U},\mathcal{V}$-equivariant, preserves bigradings and maps
one generator of $(\mathcal{U}\mathcal{V})^{-1}H_{*}$ to another. Observe that
there is no vertical arrow in $C^{\prime}$, therefore all generators in
$C^{\prime}_{\mathrm{conn}}$ survive into
$H_{\mathrm{vert}}(C^{\prime}_{\mathrm{conn}})$. We only need to show
$C^{\prime}_{\mathrm{conn}}=$ im$f$ contains more than one generators of
homology.

Recall that for each $y_{i}\in\sum_{i=1}^{k}C_{n_{i}},i=1,...,k$, there exists
some $w_{i}\in\sum_{i=1}^{k}C_{n_{i}}$, such that

$\partial\big{(}y_{i}+(\mathcal{U}\mathcal{V})^{-h_{i}}w_{i}\big{)}=\mathcal{U}^{n}\mathcal{V}^{n-1}x_{i}+\mathcal{U}^{n-1}\mathcal{V}^{n}x_{i-1},$

where $h_{i}$ is a nonnegative integer. Define

$\widetilde{y}_{i}:=w_{i}+(\mathcal{U}\mathcal{V})^{h_{i}}y_{i},$

and let $h:=$ max $\\{h_{i}\colon i=1,...,k\\}.$

Set

$\displaystyle\alpha=\sum^{k}_{i=1}(\mathcal{U}\mathcal{V})^{h-h_{i}}\widetilde{y_{i}}\otimes
y_{i}^{*}+(\mathcal{U}\mathcal{V})^{h+n-m}\sum^{k}_{i=0}x_{i}\otimes
x^{*}_{i},$
$\displaystyle\beta=\sum^{k}_{i=0}\mathcal{U}^{k-i}\mathcal{V}^{i}x_{0}\otimes
x^{*}_{i}.$

We want to point out that $\alpha$ is an actual generator of the chain
complex, meaning $\alpha$ contains a non-trivial component with no
$\mathcal{U},\mathcal{V}$ decoration. This fact turns out to be necessary in
order for the proof to work. On the other hand, $\beta$ is not an actual
generator of the chain complex.

We compute the bigradings of $\alpha$ and $\beta$:

$\displaystyle\operatorname{gr}(\alpha)=(-2(m+h-n),-2(m+h-n)),$
$\displaystyle\operatorname{gr}(\beta)=\operatorname{gr}(x_{0})=(-2k,0).$

Since tensoring the complex $C^{\prime}$ by $C_{m}+C_{m}^{*}$ doesn’t change
its local equivalence type, we can assume that $k$ is large enough such that
$-2k<-2(m+h-n)$.

In fact, $\alpha$ and $\beta$ are both generators of
$(\mathcal{U}\mathcal{V})^{-1}H_{*}$. Here we slightly abuse the notation:
when we refer to an element in $C^{\prime}$ we actually mean its counterpart
in
$C^{\prime}\otimes_{\mathbb{F}[\mathcal{U},\mathcal{V}]}\mathbb{F}[\mathcal{U},\mathcal{V},(\mathcal{U}\mathcal{V})^{-1}].$
It is a simple computation that both $\alpha$ and $\beta$ are in ker
$\partial$. To show they are not in the image, first assume
$\partial\big{(}\sum z\otimes r\big{)}=\beta$, where each
$z\in\sum_{i=1}^{k}C_{n_{i}}$ and each $r\in\sum_{i=1}^{k}C_{m_{i}}^{*}$. Non-
trivial components of $\partial\big{(}\sum z\otimes r\big{)}$ are either of
the form $(\partial z)\otimes r$ or $z\otimes(\partial r)$. Consider the term
$\mathcal{U}^{k-i}\mathcal{V}^{i}x_{0}\otimes x^{*}_{i}$ in $\beta$. Since
each $x_{i}$ is in ker $\partial$, $x_{i}^{*}$ never occurs in the image of
$\partial$ as a non-trivial component. Hence
$\mathcal{U}^{k-i}\mathcal{V}^{i}x_{0}\otimes x^{*}_{i}$ is of the form
$\sum(\partial z)\otimes r$. It follows that $x_{0}$ has to be in the image, a
contradiction. Supposing $\partial\big{(}\sum z\otimes r\big{)}=\alpha$,
similarly, each term
$(\mathcal{U}\mathcal{V})^{h+n-m}\sum^{k}_{i=0}x_{i}\otimes x^{*}_{i}$ in
$\alpha$ is of the form $\sum(\partial z)\otimes r$. Since $y_{i}$ never
occurs in the image as a non-trivial component, each term
$(\mathcal{U}\mathcal{V})^{h-h_{i}}\widetilde{y_{i}}\otimes y_{i}^{*}$ is of
the form $\sum z\otimes(\partial r).$ A brief examination shows that the only
suitable elements for $z\otimes r$ are $\widetilde{y_{i}}\otimes x_{j}^{*}$.
However, there does not exist any combination of
$\partial\big{(}\widetilde{y_{i}}\otimes x_{j}^{*}\big{)}$ equal to $\alpha$.
This proves that both $\alpha$ and $\beta$ generate
$(\mathcal{U}\mathcal{V})^{-1}H_{*}$.

According to the property that self-local equivalence $f$ maps one generator
of $(\mathcal{U}\mathcal{V})^{-1}H_{*}$ to another, we know $f(\alpha)$ and
$f(\beta)$ are both non-trivial in $C^{\prime}_{\mathrm{conn}}$. However,
since $f$ preserves bigradings, $f(\alpha)\neq f(\beta)$. Moreover, there does
not exist any positive integers $s,t$ such that
$f(\alpha)=\mathcal{U}^{s}\mathcal{V}^{t}f(\beta)$ or
$f(\beta)=\mathcal{U}^{s}\mathcal{V}^{t}f(\alpha)$. We conclude that
$C^{\prime}_{\mathrm{conn}}$ contains at least $2$ generators of homology.

∎

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