# Is the geography of Heegaard Floer homology restricted or the L-space
conjecture false?

Antonio Alfieri and Fraser Binns  Centre de recherche Mathématique. Montreal,
Canada<EMAIL_ADDRESS>Princeton University. Princeton, New
Jersey, USA<EMAIL_ADDRESS>
###### Abstract.

In a recent note F. Lin showed that if a rational homology sphere $Y$ admits a
taut foliation then the Heegaard Floer module $\operatorname{HF}^{-}(Y)$
contains a copy of $\mathbb{F}[U]/U$ as a summand [19]. This implies that
either the $L$-space conjecture is false or that Heegaard Floer homology
satisfies a geography restriction. We verify that Lin’s geography restriction
holds for a wide class of rational homology spheres. Indeed, we show that the
Heegaard Floer module $\operatorname{HF}^{-}(Y)$ may satisfy a stronger
geography restriction.

FB was supported by the Simons Grant New structures in low-dimensional
topology.

##  1\. Introduction

Heegaard Floer homology is a package of invariants introduced by Ozsváth and
Szabó in 2001 [30]. It assigns to each three-manifold $Y$ an
$\mathbb{F}[U]$-module, $\operatorname{HF}^{-}(Y)$, that can be used to study
– for example – surgery problems, contact structures on three-manifolds, and
the symplectic topology of four-manifolds; see [20, 22, 21, 31, 24].

While a quarter of a century has passed since it was defined, there are still
many mysterious open questions regarding Heegaard Floer homology. One of these
asks if the structure of the module $\operatorname{HF}^{-}(Y)$ is related to
the existence of taut foliations on $Y$.

Recall that a foliation of a three-manifold $Y$ is a decomposition of $Y$ into
a disjoint union of $2$-dimensional sub-manifolds (leaves) that locally looks
like the decomposition of $\mathbb{R}^{3}$ into horizontal planes. Every
three-manifold $Y$ admits a foliation. However, it is not necessarily the case
that every manifold admits a foliation if additional constrains are required –
for example if the foliation is asked to be “Reebless” or “taut”. In the other
direction, if a manifold admits such a foliation, this tells us something
about the topology of $Y$ [26, 38].

A foliation is _taut_ if there exists a simple closed curve that intersects
every leaf. In the 1980’s Gabai [8, 9, 10] showed that every three-manifold
with $b_{1}>0$ admits a taut foliation. At the time of writing it is unknown
which _rational homology spheres_ (connected, oriented, three manifolds with
$b_{1}=0$) admit taut foliations. Understanding this problem is one of the
greatest open challenges of three-manifold topology.

There is a conjectural relationship between rational homology spheres that
admit taut foliations and those whose Heegaard Floer homology is free as
follows:

######  Conjecture 1.1 (Boyer, Gordon and Watson [6], Juhász [17]: the
$L$-space conjecture).

For an irreducible rational homology sphere $Y$ the following statements are
equivalent:

  1. (1)

$Y$ is an $L$-space – i.e. $\operatorname{HF}^{-}(Y)$ is free.

  2. (2)

$Y$ does not support a taut foliation,

  3. (3)

$\pi_{1}(Y)$ has no left-ordering; that is an ordering $<$ such that: $a<b$
iff $ga<gb$.

Note that the equivalence of (1) and (3) would pin down a property of the
fundamental group characterizing $L$-spaces, thus answering one of the
fundamental questions in Ozsváth and Szabó’s problem list [28, Question 11].

######  Question 1.2.

Is there a topological classification of $L$-spaces?

In this paper we advance the hypothesis that the $L$-space conjecture is
linked to another fundamental problem in Heegaard Floer theory:

######  Question 1.3 (The Geography Question).

Which $\mathbb{F}[U]$-modules are realized as the Heegaard Floer group
$\operatorname{HF}^{-}(Y)$ of some three-manifold $Y$?

It is immediate from the definition that $\operatorname{HF}^{-}(Y)$ is
finitely generated for any three-manifold manifold $Y$. Indeed, Ozsváth and
Szabó showed that for a rational homology sphere $Y$,
$\operatorname{HF}^{-}(Y)$ is of rank $|H_{1}(Y;\mathbb{Z})|$ as a $U$-module
[30, Theorem 10.3].

###  1.1. Lin’s theorem about taut foliations

The $L$-space conjecture holds for graph manifolds, some families of knot
surgeries, and some branched coverings – see [5, 12, 4, 18], for example.
Nevertheless there is no consensus on whether the $L$-space holds true in full
generality. The only part of the conjecture which is known to be true in
general is the following: if a rational homology sphere $Y$ admits a taut
foliation then the Heegaard Floer group $\operatorname{HF}^{-}(Y)$ has non-
trivial torsion [29, Theorem 1.4]. Recently Lin proved the following
refinement of this result.

######  Theorem 1.4 (Lin [19]).

Suppose $Y$ admits a taut foliation. Then the Heegaard Floer module
$\operatorname{HF}^{-}(Y)$ contains a copy of $\mathbb{F}[U]/U$ as an
$\mathbb{F}[U]$-summand.

For a rational homology sphere $Y$, we say that $\operatorname{HF}^{-}(Y)$
satisfies the _Lin geography restriction_ if $\operatorname{HF}^{-}(Y)$ has
either no $U$-torsion or an $\mathbb{F}[U]/U\simeq\mathbb{F}$ summand. In what
follows we use Heegaard Floer surgery formulae [23, 37, 39] to prove that a
broad class of rational homology spheres have Heegaard Floer homology
satisfying the Lin geography restriction. Thus we present evidence for the
following conjecture.

######  Conjecture 1.5.

The Heegaard Floer module $\operatorname{HF}^{-}(Y)$ satisfies the Lin
geography restriction for all rational homology spheres $Y$.

It follows from Lin’s theorem, Theorem 1.4, that either Conjecture 1.5 is
true, or the $L$-space conjecture is false. Note that for a large class of
graph manifolds the Lin geography restriction was first verified by Bodnar,
and Plamenevskaya [2].

###  1.2. Statement of the main results

We shall make use of the following definition.

######  Definition 1.6.

An $\mathbb{F}[U]$-module $M$ satisfies the strong geography restriction if it
contains a direct summand

$\mathbb{F}[U]/U^{\ell}\oplus\mathbb{F}[U]/U^{\ell-1}\oplus\dots\oplus\mathbb{F}[U]/U^{2}\oplus\mathbb{F}[U]/U$

where $\ell=\min\\{\ell\geq 0:U^{\ell}\cdot M_{\text{red}}=0\\}$.

Our first theorem concerns surgeries on knots.

######  Theorem 1.7.

If $Y$ is surgery on a knot in $S^{3}$ then the Heegaard Floer module
$\operatorname{HF}^{-}(Y)$ satisfies the strong geography restriction.

In particular if $Y$ is surgery on a knot then either $Y$ is an $L$-space, or
$\operatorname{HF}^{-}(Y)$ has an $\mathbb{F}[U]/U$-summand, i.e. it satisfies
the Lin geography restriction. This answers a question that was recently asked
by Baldwin on his research blog [16]. Our second theorem covers large
surgeries on links.

######  Theorem 1.8.

If $Y$ is large surgery on a link in $S^{3}$ then the Heegaard Floer module
$\operatorname{HF}^{-}(Y)$ satisfies the strong geography condition.

Here “large surgery on $L$” means that the framing matrix of $L$ is “larger”
than the smallest hyperbox containing the link Floer homology polytope of $L$
– see Definition 3.6 for further details. Recall that the link Floer homology
polytope of $L$ contains equivalent information to the Thurston norm of $L$,
under mild hypothesese [36, Theorem 1.1]. In the case of knots “large” simply
means at least $2g(K)-1$.

######  Remark 1.9.

Theorem 1.7 and Theorem 1.8 hold true in the more general case of null-
homologous knots and links in any $L$-space.

Given the above results it is natural to ask the following question.

######  Question 1.10.

Does $\operatorname{HF}^{-}(Y)$ satisfy the strong geography restriction for
every rational homology sphere $Y$?

Of course, given that every rational homology sphere is integral surgery on
_some_ link, removing the hypothesis that $Y$ is _large_ surgery on a link in
the statement of Theorem 1.8 would provide an affirmative answer to Question
1.10. Indeed, we know many examples of links – for instance torus links, and
connected sums of Hopf links – whose small surgeries satisfy at least the Lin
geography restriction.

###  1.3. The realization problem

Theorem 1.8 leads naturally to the following question.

######  Question 1.11.

Which rational homology spheres arise as large surgery along links in $S^{3}$?

There are relatively few results in this direction. Gordon-Luecke showed that
a reducible integral homology sphere can never be surgery on a knot [11].
Examples of irreducible rational homology spheres which are not surgery on a
knot in $S^{3}$ have been given by Hom-Lidman-Karakurt [14, Theorem 1.1] ,
Hom-lidman [15, Theorem 1], and Auckley [1]. Other examples have been given by
Doig [7, Corollary 5], Hoffman-Walsh [13, Theorem 4.4] and Oszváth-Szabó [27,
Section 10.2].

######  Proposition 1.12.

Suppose an integral homology sphere $Y$ is large surgery on a link. Then
$\operatorname{HF}^{-}(Y)\cong\mathbb{F}[U]\oplus\mathbb{F}^{\oplus n}$ for
some $n$.

For the Brieskorn homology spheres $\Sigma(2,2n+1,4n+3)$ Rustamov [40] showed
that $\operatorname{HF}^{-}(\Sigma(2,2n+1,4n+3))$ has an $\mathbb{F}[U]/U^{2}$
summand for $n\geq 3$.

######  Corollary 1.13.

For $n\geq 3$, $\Sigma(2,2n+1,4n+3)$ is not large surgery on a link in
$S^{3}$.

### Acknowledgments

The authors would like express their gratitude to Subhankar Dey, Franceso Lin,
Duncan McCoy, Peter Ozsváth, Jacob Rasmussen, Charles Stine, András Stipsicz,
and Zoltan Szabó for a number of informative conversations.

##  2\. Three-manifolds arising as surgery on knots

###  2.1. Heegaard Floer groups

Recall that Heegaard Floer homology [30] associates to a pointed Heegaard
diagram
$\mathcal{H}=(\Sigma,\alpha_{1}\dots\alpha_{g},\beta_{1},\dots,\beta_{g},z)$
of a three-manifold $Y$ a $\mathbb{Q}$-graded, finitely generated, free chain
complex $\text{CF}^{-}(\mathcal{H})$ over $\mathbb{F}[U]$. The differential
counts $J$-holomorphic disks in the symmetric product
$\text{Sym}^{g}(\Sigma)$.

Since the chain homotopy type of $\text{CF}^{-}(\mathcal{H})$ does not depend
on $\mathcal{H}$ but only on the three-manifold $Y$ it is denoted by
$\text{CF}^{-}(Y)$. We shall also consider the localization
$\text{CF}^{\infty}(Y)=\text{CF}^{-}(Y)\otimes\mathbb{F}[U,U^{-1}]$ and the
quotient $\text{CF}^{+}(Y)=\text{CF}^{\infty}(Y)/\text{CF}^{-}(Y)$. Note that
the homologies of these chain complexes are related by an exact sequence

$\dots\to\operatorname{HF}^{-}(Y)\to\operatorname{HF}^{\infty}(Y)\to\operatorname{HF}^{+}(Y)\to\dots\
,$

and that they decompose as direct sums along $\text{spin}^{c}$ structures:

$\text{CF}^{\circ}(Y)=\bigoplus_{\mathfrak{s}\in\text{Spin}^{c}(Y)}\text{CF}^{\circ}(Y,\mathfrak{s})\
.$

This gives rise to a splitting
$\operatorname{HF}^{\circ}(Y)=\bigoplus_{\mathfrak{s}\in\text{Spin}^{c}(Y)}\operatorname{HF}^{\circ}(Y,\mathfrak{s})$
in homology, where $\circ$ denotes any of the symbols $-,\infty,+$.

######  Remark 2.1.

The group $\operatorname{HF}^{-}(Y)$ is a finitely generated, graded
$\mathbb{F}[U]$-module. Since $\mathbb{F}[U]$ is a principle ideal domain,
$\operatorname{HF}^{-}(Y)$ can be decomposed as a direct sum of elementary
modules of the form:

  1. (1)

$T^{-}=\mathbb{F}[U]$ so called towers, and

  2. (2)

$T^{-}_{k}:=F[U]/U^{k}$, $k\in\mathbb{Z}^{\geq 0}$.

Note that for a rational homology sphere there is exactly one tower for each
$\text{spin}^{c}$ structure, that is $\text{rk}_{\mathbb{F}[U]}\
\operatorname{HF}^{-}(Y)=|\text{Spin}^{c}(Y)|=|H_{1}(Y;\mathbb{Z})|$ [30,
Theorem 10.3].

######  Remark 2.2.

The group $\operatorname{HF}^{+}(Y)$ is complete torsion when considered as a
module over $\mathbb{F}[U]$. That is: $U^{i}\cdot\alpha=0$ for all
$\alpha\in\operatorname{HF}^{+}(Y)$, and $i>>0$. Nevertheless, there is a
decomposition
$\operatorname{HF}^{+}(Y)=F\oplus\operatorname{HF}^{+}_{\text{red}}(Y)$ where
$F$ is a direct sum of a few copies of
$T^{+}=\mathbb{F}[U,U^{-1}]/U\cdot\mathbb{F}[U]$ , and
$\operatorname{HF}_{\text{red}}^{+}(Y)\simeq\text{Tor}(\operatorname{HF}^{-}(Y))$.
When working with the $\operatorname{HF}^{+}$ version we prefer the notation
$T^{+}_{k}$.

###  2.2. Knot Floer groups: one complex to rule them all

Heegaard Floer homology also associates to each knot $K\subset S^{3}$ a
filtered chain complex that can be used to compute the Heegaard Floer homology
groups of all surgeries along $K$.

######  Definition 2.3.

A knot type chain complex is a finitely-generated, graded,
$\mathbb{Z}\oplus\mathbb{Z}$-filtered chain complex

$C_{*}=\left(\bigoplus_{\mathbf{x}\in B}\mathbb{F}[U,U^{-1}],\ \partial\
\right)$

over the ring $\mathbb{F}[U,U^{-1}]$ satisfying the following properties

  * •

$\partial$ is $\mathbb{F}[U,U^{-1}]$-linear, and given a basis element
$\mathbf{x}\in B$,
$\partial\mathbf{x}=\sum_{\mathbf{y}}n_{\mathbf{x},\mathbf{y}}U^{m_{\mathbf{x},\mathbf{y}}}\cdot\mathbf{y}$
for suitable coefficients $n_{\mathbf{x},\mathbf{y}}\in\mathbb{F}$, and non-
negative exponents $m_{\mathbf{x},\mathbf{y}}\geq 0$,

  * •

multiplication by $U$ drops the absolute $\mathbb{Q}$-grading (Maslov grading)
$M$ by two, and the two $\mathbb{Z}$-filtrations (denoted by $\mathcal{F}$ and
$j$, and called the Alexander and algebraic filtrations) by one.

  * •

there is a grading-preserving isomorphism
$H_{*}(C_{*},\partial)\simeq\mathbb{F}[U,U^{-1}]$.

A knot $K\subset S^{3}$ can be encoded by the choice of an additional
basepoint $w$ in a suitable Heegaard diagram $\mathcal{H}$ of $S^{3}$. As
explained in [33] and [39], this gives rise to a filtration $\mathcal{F}_{K}$
of $\text{CF}^{\infty}(\mathcal{H})$ and hence to a knot type complex

$\operatorname{CFK}^{\infty}(K)=(\text{CF}^{\infty}(S^{3}),\mathcal{F}_{K},j)\
,$

where $j$ denotes the filtration with $j(\mathbf{x})=0$ for every intersection
point $\mathbf{x}$ of the Heegaard diagram. A knot type chain complex is
usually presented by a planar drawing. This is done according to the following
procedure:

  * •

represent each $\mathbb{F}$-generator $U^{m}\cdot\mathbf{x}$ of $C_{*}$,
$\mathbf{x}\in B$, as a point on the planar lattice
$\mathbb{Z}\times\mathbb{Z}\subset\mathbb{R}^{2}$ in position
$(i,j)=\left(A(\mathbf{x})-m,j(\mathbf{x})-m\right)\in\mathbb{Z}\times\mathbb{Z}$,

  * •

label each $\mathbb{Z}_{2}$-generator $U^{m}\cdot\mathbf{x}$ of $C_{*}$ with
its Maslov grading $M(\mathbf{x})-2m\in\mathbb{Z}$,

  * •

connect two $\mathbb{F}$-generators $U^{n}\cdot\mathbf{x}$ and
$U^{m}\cdot\mathbf{y}$ with an arrow if in the differential of
$U^{n}\cdot\mathbf{x}$ the coefficient of $U^{m}\cdot\mathbf{y}$ is non-zero.

With these conventions in place any region of the plane
$S\subset\mathbb{R}^{2}$ identifies a sub-vector space
$C_{*}S=\text{Span}_{\mathbb{F}}\\{U^{m}\cdot\mathbf{x}\in
C_{*}:\left(A(\mathbf{x})-m,j(\mathbf{x})-m\right)\in S\\}$ of $C_{*}$. Note
that for $C_{*}S$ to be a sub-complex of $C_{*}$ the regions $S$ needs to
satisfy a certain convexity condition:

######  Definition 2.4.

A region of the Euclidean plane $S\subset\mathbb{R}^{2}$ is south-west if it
is closed, non-empty, and such that: $(\overline{x},\overline{y})\in
S\Rightarrow\\{(x,y)\ |\ x\leq\overline{x},y\leq\overline{y}\\}\subseteq S.$

The other case in which $C_{*}S$ is a chain complex is when $S=S_{1}\setminus
S_{2}$ can be expressed as the difference of two south-west regions
$S_{2}\subset S_{1}$, and $C_{*}S$ is identified with quotient chain complex
$C_{*}S_{1}/C_{*}S_{2}$. Note that the latter is not free: if $x\in S_{1}$ is
a generator and $U^{i}\cdot x\in S_{2}$ then $U^{i}\cdot x=0$ in $C_{*}S$.

Thus we get chain complexes $C_{*}^{-}=C_{*}\\{j\leq 0\\}$, and
$C_{*}^{+}=\\{j\geq 0\\}=C_{*}/C_{*}^{-}$. In the case of knot Floer homology
these chain complexes compute $\operatorname{HF}^{-}(Y)=\mathbb{F}[U]$ and
$\operatorname{HF}^{+}(Y)=\mathbb{F}[U,U^{-1}]/U\mathbb{F}[U]$ respectively.

###  2.3. Review of the knot surgery formula

We now review the homological algebra governing the knot surgery formula.

Let $C_{*}$ be a knot type chain complex. For $s\in\mathbb{Z}$ let
$A_{s}^{+}=C_{*}\\{\max(i,j-s)\geq 0\\}$ and $B^{+}_{s}=C_{*}\\{j\geq
0\\}=B^{+}$. We define maps $v_{s}^{+}:A_{s}^{+}\to B^{+}$ and
$h_{s}^{+}:A_{s}^{+}\to B^{+}$. The first is the quotient

$v_{s}^{+}:A_{s}^{+}=C_{*}\\{\max(i,j-s)\geq 0\\}\to C_{*}\\{i\geq 0\\}=B^{+}$

with kernel $C_{*}\\{j\geq s\text{ and }i<0\\}$. The definition of $h_{s}^{+}$
is slightly more complicated and requires the choice of some additional data
as in the following definition.

######  Definition 2.5 (Flip map).

A flip map is a chain homotopy equivalence $n:C_{*}\\{j\geq 0\\}\to
C_{*}\\{i\geq 0\\}$. A master type complex is a pair $(C_{*},n)$ where $C_{*}$
is a knot type complex, and $n$ is a flip map of $C_{*}$.

Note that if $C_{*}=\operatorname{CFK}^{\infty}(K)$ is the knot Floer complex
of some knot in $S^{3}$, $C_{*}\\{i\geq 0\\}$ and $C_{*}\\{j\geq 0\\}$
represent two chain complexes for $\text{CF}^{+}(S^{3})$ with respect to the
choice of two different basepoints on the same Heegaard diagram. The flip map
$n:C_{*}\\{j\geq 0\\}\to C_{*}\\{i\geq 0\\}$ is induced by a sequence of
Heegaard moves taking one basepoint into the other. We call
$(\operatorname{CFK}^{\infty}(K),n)$ the _knot Floer master complex_.

For a master type complex $(C_{*},n)$ we define $h_{s}^{+}:A_{s}^{+}\to B^{+}$
as the composition

$h^{+}_{s}:A_{s}^{+}=C_{*}\\{\max(i,j-s)\geq 0\\}\to C_{*}\\{j\geq s\\}\to
C_{*}\\{j\geq 0\\}\to C_{*}\\{i\geq 0\\}=B^{+}$

where the first map is the projection $\pi:C_{*}\\{\max(i,j-s)\geq 0\\}\to
C_{*}\\{j\geq s\\}$, the second map $C_{*}\\{j\geq s\\}\to C_{*}\\{j\geq 0\\}$
is multiplication by $U^{s}$, and the last one is the flip map
$n:C_{*}\\{j\geq s\\}\to C_{*}\\{i\geq 0\\}$.

Finally we set

$\mathbb{A}^{+}=\bigoplus_{s\in\mathbb{Z}}A^{+}_{s}\ \ \ \text{ and }\ \ \
\mathbb{B}^{+}=\bigoplus_{s\in\mathbb{Z}}B^{+}_{s}\ .$

We have maps $v^{+}=\sum_{s}v_{s}^{+}:\mathbb{A}^{+}\to\mathbb{B}^{+}$, and
$h^{+}=\sum_{s}h_{s}^{+}:\mathbb{A}^{+}\to\mathbb{B}^{+}$. We define the map
$D_{p}^{+}:\mathbb{A}^{+}\to\mathbb{B}^{+}$ as the sum
$D_{p}^{+}:v^{+}+\sigma_{p}\circ h^{+}$, where
$\sigma_{p}:\bigoplus_{s\in\mathbb{Z}}B^{+}\to\bigoplus_{s\in\mathbb{Z}}B^{+}$
denotes the shift map
$\sigma_{p}:(\dots,a_{s},\dots)\mapsto(\dots,b_{s}=a_{s-p},\dots)$. Thus:

$D^{+}_{p}:(\dots,a_{s},\dots)\mapsto(\dots,b_{s}=v_{s}^{+}(a_{s})+h_{s-p}^{+}(a_{s-p}),\dots)\
.$

Taking the mapping cone of the map $D_{p}^{+}:\mathbb{A}^{+}\to\mathbb{B}^{+}$
we get a chain complex
$\mathbb{X}^{+}(p)=\left(\mathbb{A}^{+}\oplus\mathbb{B}^{+}\
,\partial_{\text{cone}}=\partial+D_{p}^{+}\right)$ that in [32] Ozsváth and
Szabó show to be quasi-isomorphic to $\text{CF}^{-}(S^{3}_{p}(K))$.

There is also a rational surgery formula. Set
$\mathbb{A}_{i}^{+}=\underset{s\in\mathbb{Z}}{\bigoplus}(s,A^{+}_{\lfloor\frac{i+ps}{q}\rfloor}(K))$
and ${\mathbb{B}^{+}=\underset{s\in\mathbb{Z}}{\bigoplus}(s,B^{+})}$. Let
$\mathbb{X}^{+}_{i,p/q}$ be the chain complex defined as the mapping cone of
the map $D^{+}_{i,p/q,j}:\mathbb{A}_{i}^{+}\to\mathbb{B}^{+}$, which we define
by $(s,A_{\lfloor\frac{i+ps}{q}\rfloor})\to(s,B^{+})+(s+1,B^{+})$ by
$\psi_{m}:(s,x)\mapsto((s,v_{m}^{+}(x)),(s+1,h_{m}^{+}(x)),)$ and extending
linearly. In [37] Ozsváth and Szabó show that
$\operatorname{HF}^{+}(S^{3}_{p/q}(K))$ is isomorphic to
$H_{*}(\mathbb{X}^{+}_{i,p/q})$.

###  2.4. Modules with similar skylines

The following definition and lemma will be used repeatedly in the remainder of
the paper.

######  Definition 2.6.

Let $A$ be a finitely generated $\mathbb{F}[U]$ module with

$A\simeq\underset{1\leq i\leq n}{\bigoplus}T^{-}\oplus\underset{1\leq j\leq
m}{\bigoplus}T_{h(j)}^{-}\ .$

where $h(j)>0$ for all $j$. If $B$ is a finitely generated $\mathbb{F}[U]$
module we say that $A$ and $B$ have similar skylines if

$B\simeq\underset{1\leq i\leq n}{\bigoplus}T^{-}\oplus\underset{1\leq j\leq
m}{\bigoplus}T_{h^{\prime}(j)}^{-}\oplus\mathbb{F}^{k}$

for some function $h^{\prime}$ with $|h^{\prime}(j)-h(j)|\leq 1$ for all $j$
and $k\geq 0$.

######  Remark 2.7.

Note that “having similar skylines” is a reflexive relation: if $B$ has a
similar skyline to $A$ then $A$ has a similar skyline to $B$.

We say that an $\mathbb{F}[U]$-module $M$ is _of $\operatorname{HF}^{+}$-type_
if

$M\simeq\underset{1\leq i\leq n}{\bigoplus}T^{+}\oplus\underset{1\leq j\leq
m}{\bigoplus}T_{h(j)}^{+},$

for some $m,n,h$. In this case we write

$M_{\text{red}}=\underset{1\leq j\leq m}{\bigoplus}T_{h(j)}^{+}$

for the so called _reduced_ part. Note that for any three-manifold
$\operatorname{HF}^{+}(Y)$ is of $\operatorname{HF}^{+}$-type. There is a
natural notion of similar skylines for modules of
$\operatorname{HF}^{+}$-type, defined exactly as in the case of finitely
generated $\mathbb{F}[U]$-modules.

######  Lemma 2.8.

Let $n\in\mathbb{Z}_{\geq 0}$. Suppose that we have an exact triangle

${\mathbb{F}^{n}}$${B}$${A}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{h}$
(1)

of either finitely generated $\mathbb{F}[U]$-modules, or modules of
$\operatorname{HF}^{+}$-type. Then $A$ and $B$ have similar skylines.

###### Proof.

We treat the case of finitely generated $\mathbb{F}[U]$-modules. The case of
modules of $\operatorname{HF}^{+}$ type can be proved analogously.

Let $x$ be an element of $\mathbb{F}^{n}$. Observe that $Uf(x)=0$. Write
$\mathbb{F}^{n}\simeq\mathbb{F}^{n_{a}}\oplus\mathbb{F}^{n_{b}}$ and
$A=\underset{1\leq j\leq m_{a}}{\bigoplus}T_{h_{a}(j)}^{-}$, where we allow
$h_{a}(j)=\infty$, with $f$ the map that sends the first $n_{a}$ components of
$\mathbb{F}^{n}$ into the bottom generators of the first $n_{a}$ towers of
$\underset{1\leq j\leq n_{a}}{\bigoplus}T_{h_{a}(j)}^{-}$.

Consider now the maps $h$ restricted to each tower $T^{-}_{h_{a}(j)}$. Since
$h$ is $U$ equivariant we have that $U^{h(a_{j})-1}h(T_{h(a_{j})}^{-})=0$ for
$1\leq j\leq n_{a}$ and that $U^{h(a_{j})}h(T_{h(a_{j})}^{-})=0$ for
$n_{a}<j\leq m_{a}$. Since $h$ is injective when restricted to elements that
are not at the bottom of the first $m_{a}$ towers we can write
$B=\underset{1\leq j\leq
n_{a}}{\bigoplus}T_{h_{b}(j)}^{-}\oplus\underset{n_{a}<j\leq
m_{a}}{\bigoplus}T_{h_{b}(j)}^{-}\oplus\underset{m_{a}<j\leq
m_{b}}{\bigoplus}T_{h_{b}(j)}^{-}$, where $h_{b}(j)\geq h_{a}(j)-1$ for $1\leq
j\leq n_{a}$ and $h_{b}(j)\geq h_{a}(j)$ for $n_{a}<j\leq m_{a}$. Here we
again allow $h_{a}(j)=\infty$. Note that $g$ is $U$ equivariant and injective
on the complement of the image of $h$. It follows in turn that $h_{a}(j)\geq
h_{b}(j)\geq h_{a}(j)-1$ for $1\leq j\leq n_{a}$ and $h_{a}(j)+1\geq
h_{b}(j)\geq h_{a}(j)$ for $n_{a}<j\leq m_{a}$ and $h_{a}(j)=1$ for
$m_{a}<j\leq m_{b}$. The result follows. ∎

###  2.5. Large integer surgeries

In this section we prove that large integer surgeries on knots have Heegaard
Floer homology that satisfies the strong geography restriction.

As is customary we label $\text{spin}^{c}$ structures on the surgery three-
manifold $S^{3}_{n}(K)$ with integers $-|n|/2<s\leq|n|/2$ so that we can use
the Ozsváth-Szabó surgery formula to get an identification

$H_{*}(A^{+}_{s})\simeq\operatorname{HF}^{+}(S^{3}_{n}(K),s)\ .$

for any integer $n\geq 2g(K)-1$, where $g(K)$ denotes the Seifert genus of the
knot $K$.

######  Lemma 2.9.

Let $K\subset S^{3}$ be a knot, and $n\geq 2g(K)-1$. Then
$\operatorname{HF}^{+}(S^{3}_{n}(K),s)$ and
$\operatorname{HF}^{+}(S^{3}_{n}(K),s+1)$ have similar skylines.

###### Proof.

It follows from the surgery formula that for $n\geq 2g(K)-1$,
$\operatorname{HF}^{+}(S^{3}_{n}(K),m)$ is isomorphic to $H_{*}(A_{m})$ for
any $m\in\mathbb{Z}\cap(-|n|/2,|n|/2]$. Note that there is a short exact
sequence of chain complexes

${0}$${C(j=m,i<0)}$${A_{m}}$${A_{m+1}}$${0}$

which yields an exact triangle

${H_{*}(j=m,i<0)}$${H_{*}(A_{m+1})}$${H_{*}(A_{m})}$

That is, we have the following exact triangle

${H_{*}(j=m,i<0)}$${\operatorname{HF}^{+}(S^{3}_{n}(K),m+1)}$${\operatorname{HF}^{+}(S^{3}_{n}(K),m)}$
(2)

Observe that $H_{*}(j=m,i<0)\simeq\mathbb{F}^{k}$ for some
$k\in\mathbb{Z}_{\geq 0}$ so we can apply Lemma 2.8 and obtain the desired
result. ∎

######  Corollary 2.10.

Let $K\subset S^{3}$ be a knot. For any integer $n\geq 2g(K)-1$ the module

$\operatorname{HF}^{+}(S^{3}_{n}(K))\simeq\bigoplus_{-|n|/2<i\leq|n|/2}H_{*}(A^{+}_{s})$

satisfies the strong geography restriction.

###### Proof.

If $K$ is an $L$-space knot there is nothing to prove, otherwise there is a
$\text{spin}^{c}$ structure labelled by some integer $s$ (with
$-|n|/2<s\leq|n|/2$) containing a copy of $\mathbb{F}[U]/U^{N}$, where
$N=\min\\{k\geq 0\ |\ U^{k}\cdot\operatorname{HF}^{+}(S^{3}_{n}(K))=0\\}$, as
direct summand. Applying Lemma 2.9 repeatedly we conclude that

$\operatorname{HF}^{+}(S^{3}_{n}(K),s),\
\operatorname{HF}^{+}(S^{3}_{n}(K),s+1),\
\operatorname{HF}^{+}(S^{3}_{n}(K),s+2),\ \dots$

form a sequence where each term has a similar skyline to its successor.
Furthermore, since $A_{s}\simeq\text{CF}^{+}(S^{3})$ for $s\geq g(K)$, we have
that the modules in the sequence are eventually isomorphic to
$\operatorname{HF}^{+}(S^{3})\simeq T^{+}$. At this point we observe that the
intermediate value principle holds, that is: given a sequence of modules with
similar skylines that is eventually torsion-free, if the first module contains
a copy of $T^{+}_{N}=\mathbb{F}[U]/U^{N}$ then a copy of $T^{+}_{i}$ for every
$1\leq i<N$ should appear at some point along the sequence. It follows that

$T_{N}^{+}\oplus T_{N-1}^{+}\oplus\dots\oplus
T_{1}^{+}\subseteq\bigoplus_{s\leq
i\leq|n|/2}\operatorname{HF}^{+}(S^{3}_{n}(K),i)\ ,$

and we are done. ∎

###  2.6. Small integer surgeries

In this section we prove that the Lin geography restriction holds for integer
surgeries on non $L$-space knots. This can be viewed as a warm up for the next
section in which we show that the Heegaard Floer homology of any non-trivial
surgeries on any knot satisfies the strong geography restriction. Recall that
a knot $K\subset S^{3}$ is an $L$-space knot if $S^{3}_{n}(K)$ is an $L$-space
for some integer $n\geq 0$.

######  Lemma 2.11.

Let $K\subset S^{3}$ be a knot. If $L$ is not an $L$-space knot then the
module $\operatorname{HF}^{+}(S^{3}_{n}(K))$ satisfies the Lin geography
restriction.

###### Proof.

As explained in Section 2.3 the surgery formula expresses the Heegaard Floer
chain complex of the surgery $\text{CF}^{+}(S^{3}_{n},s)$ as the mapping cone
$\mathbb{X}^{+}(n)$ of a map $D_{n}:\mathbb{A}^{+}\to\mathbb{B}^{+}$. This has
an associated short exact sequence

${0}$${\mathbb{B}^{+}}$${\mathbb{X}^{+}(n)}$${\mathbb{A}^{+}}$${0}$

inducing an exact triangle

${\operatorname{HF}^{+}(S^{3}_{n}(K))}$${H_{*}(\mathbb{B}^{+})}$${H_{*}(\mathbb{A}^{+})}$$\scriptstyle{q_{*}}$$\scriptstyle{i_{*}}$$\scriptstyle{(D_{n})_{*}}$

whose connecting homomorphism is identified with the map
$(D_{n})_{*}:H_{*}(\mathbb{A}^{+})\to H_{*}(\mathbb{B}^{+})$ induced in
homology.

It is not hard to see that the map $(D_{n})_{*}$ is surjective. Indeed given
$b\in B^{+}_{j}\subset\mathbb{B}$ one can find a sequence
$(a_{s})_{s\in\mathbb{Z}}\in\mathbb{A}^{+}$ such that: $v_{j}(a_{j})=b$,
$v_{j+kn}(a_{j+kn})=h_{j+(k-1)n}(a_{j+(k-1)n})$, and $a_{s}=0$ otherwise.
Consequently, we have a short exact sequence:

${0}$${\operatorname{HF}^{+}(S^{3}_{n}(K))}$${H_{*}(\mathbb{A}^{+})}$${H_{*}(\mathbb{B}^{+})}$${0}$$\scriptstyle{(D_{n})_{*}}$

In particular $\operatorname{HF}^{+}(S^{3}_{n}(K))$ is identified with the
kernel of $(D_{n})_{*}$. Since $(D_{n})_{*}$ sends the reduced part of
$H_{*}(\mathbb{A}^{+})$ to zero, the result now follows from the proof of
Corollary 2.10 where it was shown that $H_{*}(\mathbb{A}^{+})$ contains an
$\mathbb{F}$-summand. ∎

###  2.7. Extension to the case of rational surgeries

We shall now extend the results from the previous sections, and prove Theorem
1.7.

Suppose that $K$ is a knot in $S^{3}$. Let $V_{i}$ and $H_{i}$ be the
invariants considered in [24], and
$M_{i}:=\min\\{V_{i},H_{i}\\}:\mathbb{Z}\to\mathbb{Z}_{\geq 0}$. Since they
will be used frequently in this section we recall some properties of the
invariants $M_{i}$.

######  Lemma 2.12.

For sufficiently large $|i|$, $M_{i}=0$.

###### Proof.

This follows from the fact that $V_{i}$ and $H_{-i}$ are zero for sufficiently
large $i$. ∎

######  Lemma 2.13.

$\lim_{i\to\infty}H_{i}=\lim_{i\to-\infty}V_{i}=\infty$.

###### Proof.

This follows from the definition of $V_{i}$ and $H_{i}$. ∎

######  Lemma 2.14.

$M_{-i}=M_{i}$.

###### Proof.

This follows from the definition of $M_{i}$ together with the fact that
$V_{i}=H_{-i}$. ∎

######  Lemma 2.15.

For $i\geq 0$ $M_{i}\geq M_{i+1}\geq M_{i}-1\geq 0$ and $M_{-i}\geq
M_{-i-1}\geq M_{-i}-1\geq 0$.

###### Proof.

Observe that after restricting to the infinite height towers in $H_{*}(A_{s})$
and $H_{*}(A_{s+1})$ we have that $(h_{s}^{+})_{*}=(h_{s+1}^{+})_{*}\circ j:$
where $j$ is either $0$ or multiplication by $U$. It follows that $H_{i}\leq
H_{i+1}\leq H_{i}+1$. Similarly $V_{i}\leq V_{i-1}\leq V_{i}+1$. Since
$M_{0}=V_{0}=H_{0}$ the result follows. ∎

Note that combining Lemma 2.15 and Lemma 2.12 we see that $M_{i}$ is bounded
from above. Indeed, Lemma 2.14 and the proof of Lemma 2.15 show that $M_{i}$
is bounded from above by $M_{0}$.

We now make some statements that already appeared in [25, Section 3.3]. The
rational surgery formula expresses the module
$\operatorname{HF}^{+}(S^{3}_{p/q},s)$ as the mapping cone of the map
${{D_{s}}:\mathbb{A}_{s}^{+}\to\mathbb{B}_{s}^{+}}$ we described at the end of
Section 2.3. It follows that there is an exact triangle:

${\operatorname{HF}^{+}(S^{3}_{p/q},s)}$${H_{*}(\mathbb{B}_{s}^{+})}$${H_{*}(\mathbb{A}_{s}^{+})}$$\scriptstyle{q_{*}}$$\scriptstyle{i_{*}}$$\scriptstyle{(D_{s})_{*}}$

where the connecting homomorphism can be identified with $(D_{s})_{*}$, the
map induced on homology by $D_{s}$, and $i_{*}$ and $q_{*}$ are the maps
induced by the relevant inclusion and quotient chain maps.

######  Lemma 2.16.

For a knot $K\subset S^{3}$ we have that
$\operatorname{HF}^{+}(S^{3}_{p/q}(K),s)\simeq\ker((D_{s})_{*})$.

###### Proof.

It follows from the construction of $D_{s}$ that $(D_{s})_{*}$ is surjective.
Thus $i_{*}$ is trivial and
$\operatorname{HF}^{+}(S^{3}_{p/q},s)\simeq\ker((D_{s})_{*})$. ∎

Set

$H_{\text{red},*}(\mathbb{A}_{s}^{+}):=\underset{i\in\mathbb{Z}}{\bigoplus}(H_{\text{red},*}(A^{+}_{\left\lfloor\frac{s+pi}{q}\right\rfloor}))$

and let $\ker((D_{s})_{*,T})$ denote the kernel of the restriction of
$(D_{s})_{*}$ to the infinite height towers of $H_{*}(\mathbb{A}^{+}_{s})$.

As in the case of the proof of Lemma 2.11, note that for any $s,p,i,q$, an
element $x\in H_{*}(A_{\lfloor\frac{s+ip}{q}\rfloor})$ can be extended
canonically to an element of $\tilde{x}\in\ker((D_{s})_{*})$, where the
components of $\tilde{x}$ are in the towers of
$H_{*}(A_{\lfloor\frac{s+jp}{q}\rfloor})$. This relies on the fact that for
any $n\in\mathbb{Z}^{\geq 0}$ there is a sufficiently large positive $s$
$h_{s}^{+}$ kills the bottom $n$ generators of $A_{s}^{+}$ and $v_{-s}^{+}$
kills the bottom $n$ generators of $A^{+}_{-s}$ – see Lemma 2.13, for
instance.

######  Lemma 2.17.

$\operatorname{HF}^{+}(S^{3}_{p/q},s)\simeq
H_{\text{red},*}(\mathbb{A}_{s}^{+})\oplus\ker((D_{s})_{*,T})$.

###### Proof.

The module $H_{*}(\mathbb{A}_{s}^{+})$ can be expressed as the direct sum of
$H_{\text{red},*}(\mathbb{A}_{s}^{+})$ and a “free” part consisting only of
$T^{+}$-summands, one for each
$A^{+}_{\left\lfloor\frac{s+pi}{q}\right\rfloor}$-summand. Furthermore, the
map $(D_{s})_{*}:H_{*}(\mathbb{A}_{s}^{+})\to H_{*}(\mathbb{B}_{s}^{+})$ maps
$H_{\text{red},*}(\mathbb{A}_{s}^{+})$ to zero. ∎

For each $s$, pick an $i_{s}\in\mathbb{Z}$ with
$M_{i_{s}}=\underset{i\in\mathbb{Z}}{\max}\\{M_{\left\lfloor\frac{s+ip}{q}\right\rfloor}\\}$.

######  Corollary 2.18.

$\operatorname{HF}^{+}_{\text{red}}(S^{3}_{p/q},s)\simeq
H_{\text{red},*}(\mathbb{A}_{s}^{+})\oplus\underset{i\neq
i_{s}\in\mathbb{Z}}{\bigoplus}T^{+}_{M_{\left\lfloor\frac{s+ip}{q}\right\rfloor}}$.

###### Proof.

Recall from Lemma 2.12 that $M_{i}=0$ for all but finitely many
$i\in\mathbb{Z}$. Given the previous lemma, it suffices to determine the
classes which are in the infinite height tower in $\ker((D_{s})_{*,T})$ .
Observe that in an arbitrary Maslov grading there are only finitely many
generators of $H_{*}(\mathbb{A}_{s}^{+})$. For any sufficiently large Maslov
grading $a$ in which $H_{*}(\mathbb{A}_{s}^{+})$, the sum of generators of
Maslov grading $a$ is in $\ker((D_{s})_{*})$. Let $x_{a}$ denote the sum of
generators of Maslov grading $a$. It follows that $x_{a}$ is in the tower of
$\ker((D_{s})_{*})$ for sufficiently large $a$. It follows that the bottom of
the tower coincides with one of the towers
$T_{M_{\left\lfloor\frac{s+i_{0}p}{q}\right\rfloor}}$ where $i_{0}$ is picked
so that
$M_{\left\lfloor\frac{s+i_{0}p}{q}\right\rfloor}=\underset{i\in\mathbb{Z}}{\max}\left\\{M_{\left\lfloor\frac{s+ip}{q}\right\rfloor}\right\\}$
or equivalently – given Lemma 2.15 – any $i_{0}$ which minimizes
$\left|\left\lfloor\frac{s+i_{0}p}{q}\right\rfloor\right|$. ∎

We now show that the two summands of

$\operatorname{HF}^{+}_{\text{red}}(S^{3}_{p/q})\simeq\left(\underset{0\leq
s<p}{\bigoplus}H_{\text{red},*}(\mathbb{A}_{s}^{+})\right)\oplus\left(\underset{0\leq
s<p}{\bigoplus}\left(\underset{i\neq
i_{s}\in\mathbb{Z}}{\bigoplus}T^{+}_{M_{\left\lfloor\frac{s+ip}{q}\right\rfloor}}\right)\right)$

in Corollary 2.18 satisfy the strong geography restriction. We begin with a
preparatory Lemma to treat the first summand:

######  Lemma 2.19.

Suppose that $H_{*}(A_{n})$ contains $m$ $T^{+}_{k}$ summands. Then
$H_{\text{red},*}(\mathbb{A}_{s}^{+})$ contains at least
$m\left|\left\\{i:\left\lfloor\dfrac{s+pi}{q}=n\right\rfloor\right\\}\right|$
many summands of the form $T^{+}_{k}$ .

###### Proof.

It follows from Lemma 2.17 that if $H_{\text{red},*}(\mathbb{A}^{+}_{s})$
contains $m$ $T^{+}_{k}$ summand then for each $i$ with
$\left\lfloor\dfrac{s+pi}{q}\right\rfloor=n$ we can find $m$ distinct
$T^{+}_{k}$ summands in $H_{*}(\mathbb{A}_{s}^{+})$. The result follows. ∎

Note that
$\left|\left\\{i:\left\lfloor\dfrac{s+pi}{q}=n\right\rfloor\right\\}\right|\neq
0$ for some $s$ and consequently that Lemma 2.19 always imply the existence of
at least one $T^{+}_{k}$ summand.

######  Corollary 2.20.

$\underset{0\leq s<p}{\bigoplus}H_{\text{red},*}(\mathbb{A}_{s}^{+})$
satisfies the strong geography restriction.

###### Proof.

Consider the largest finite height tower in
$H_{\text{red},*}(\mathbb{A}_{s}^{+})$, $T_{k}^{+}$. $T_{k}^{+}$ is supported
in $H_{\text{red},*}(A_{s}^{+})$ for some $s$. Observe that for sufficiently
large $s^{\prime}$, $H_{\text{red},*}(A_{s^{\prime}}^{+})$ has no finite
height towers. Observe too that $H_{\text{red},*}(A_{t}^{+})$ and
$H_{\text{red},*}(A_{t+1}^{+})$ have similar skylines for all $t$, as in the
proof of Lemma 2.9. It follows that for all $1\leq j\leq k$ there exists some
$t$ such that $H_{\text{red},*}(A_{t})$ contains a $T_{j}^{+}$ summand. The
result now follows by an application of Lemma 2.19, noting that for all $t$,
$H_{\text{red},*}(A_{t})$ occurs as a summand of $\underset{0\leq
s<p}{\bigoplus}H_{\text{red},*}(\mathbb{A}_{s}^{+})$. ∎

We now study $\underset{0\leq s<p}{\bigoplus}\left(\underset{i\neq
i_{s}\in\mathbb{Z}}{\bigoplus}T^{+}_{M_{\left\lfloor\frac{s+ip}{q}\right\rfloor}}\right)$.
The key property we use in the proof is that removing a subset of tallest
finite height towers from an $\mathbb{F}[U]$-module which satisfies the strong
geography restriction (if they exist) yields a new $\mathbb{F}[U]$-module
which still satisfies the strong geography restriction.

######  Lemma 2.21.

$\underset{0\leq s<p}{\bigoplus}\left(\underset{i\neq
i_{s}\in\mathbb{Z}}{\bigoplus}T^{+}_{M_{\left\lfloor\frac{s+ip}{q}\right\rfloor}}\right)$
satisfies the strong geography restriction.

###### Proof.

Observe that $s$ takes every value in the range $0\leq s\leq p-1$. It follows
that $\frac{s+ip}{q}$ takes the value of every fraction with denominator $q$
as $0\leq s\leq p-1$ and $i\in\mathbb{Z}$. Thus $\underset{0\leq
s<p}{\bigoplus}\left(\underset{i\in\mathbb{Z}}{\bigoplus}T^{+}_{M_{\left\lfloor\frac{s+ip}{q}\right\rfloor}}\right)$
satisfies the strong geography restriction, since for $k\geq 0$ $M_{K}\geq
M_{k+1}\geq M_{K}-1$ and $\lim_{|k|\to\infty}M_{k}=0$ – see Lemma 2.12 and
Lemma 2.15. However, we are considering the submodule where the $i$ values are
subject to the additional restraint that $i\neq i_{s}$, so we require some
additional analysis.

Fix $s\in\\{0,1,\dots p-1\\}$. Without loss of generality we may take
$\frac{p}{q}>0$ – since $S^{3}_{\frac{p}{q}}(K)\simeq
S^{3}_{-\frac{p}{q}}(-K)$. Observe that for each $s$ there is a unique $i$ for
which

$\displaystyle-\dfrac{p}{2q}\leq\dfrac{s+ip}{q}<\dfrac{p}{2q}$

It follows that there exists at least one $i$ for which

$\displaystyle-\left\lfloor\dfrac{p}{2q}\right\rfloor-1\leq\left\lfloor\dfrac{s+ip}{q}\right\rfloor\leq\left\lfloor\dfrac{p}{2q}\right\rfloor$
(3)

Observe then that $T_{M_{n}}^{+}$ appears in $\underset{0\leq
s<p}{\bigoplus}\left(\underset{i\neq
i_{s}\in\mathbb{Z}}{\bigoplus}T^{+}_{M_{\left\lfloor\frac{s+ip}{q}\right\rfloor}}\right)$
for all $n\geq\left\lfloor\dfrac{p}{2q}\right\rfloor$ and
$n\leq-\left\lfloor\dfrac{p}{2q}\right\rfloor-1$. Indeed, if for all $s$ there
is a unique $i$ which satisfies Equation 3, then that $i$ is $i_{s}$ and
$\underset{0\leq s<p}{\bigoplus}\left(\underset{i\neq
i_{s}\in\mathbb{Z}}{\bigoplus}T^{+}_{M_{\left\lfloor\frac{s+ip}{q}\right\rfloor}}\right)$
satisfies the strong geography restriction.

We now consider the case in which there exists more than one $i$ value
satisfying Equation 3. Without loss of generality we can take $i_{0}$ to be
the smallest solution to Equation 3. Observe that the set of solutions to
Equation 3 are then given by $\\{i_{0}+k:k\in\mathbb{Z},0\leq k\leq K\\}$ for
some fixed $K>0$. We seek to determine $i_{0}$. Observe that if $i_{0}$
satisfies Equation 3 and
$\left\lfloor\dfrac{s+i_{0}p}{q}\right\rfloor>-\left\lfloor\dfrac{p}{2q}\right\rfloor$
then for all $0<k\in\mathbb{Z}$ we have that

$\displaystyle\left\lfloor\dfrac{s+(i_{0}+k)p}{q}\right\rfloor$
$\displaystyle\geq\left\lfloor\dfrac{s+i_{0}p}{q}\right\rfloor+\left\lfloor\dfrac{kp}{q}\right\rfloor$
$\displaystyle>-\left\lfloor\dfrac{p}{2q}\right\rfloor+\left\lfloor\dfrac{kp}{q}\right\rfloor$
$\displaystyle\geq\left\lfloor\dfrac{p}{2q}\right\rfloor+\left\lfloor\dfrac{(k-1)p}{q}\right\rfloor$
$\displaystyle\geq\left\lfloor\dfrac{p}{2q}\right\rfloor$

Contradicting the assumption that $i_{0}$ is not a unique solution to Equation
3. It follows that
$\left\lfloor\dfrac{s+i_{0}p}{q}\right\rfloor\in\left\\{-\left\lfloor\dfrac{p}{2q}\right\rfloor,-1-\left\lfloor\dfrac{p}{2q}\right\rfloor\right\\}$.
Indeed, we see that if
$\left\lfloor\dfrac{s+i_{0}p}{q}\right\rfloor=-1-\left\lfloor\dfrac{p}{2q}\right\rfloor$
then
$\left\lfloor\dfrac{s+(i_{0}+k)p}{q}\right\rfloor=\left\lfloor\dfrac{p}{2q}\right\rfloor-1$
for all $0<k\leq K$, contradicting the definition of $i_{0}$. It follows that
$\left\lfloor\dfrac{s+i_{0}p}{q}\right\rfloor=-\left\lfloor\dfrac{p}{2q}\right\rfloor$
and
$\left\lfloor\dfrac{s+(i_{0}+k)p}{q}\right\rfloor=\left\lfloor\dfrac{p}{2q}\right\rfloor$
for all $0<k\leq K$.

Recalling then that $T_{M_{n}}^{+}$ appears in $\underset{0\leq
s<p}{\bigoplus}\left(\underset{i\neq
i_{s}\in\mathbb{Z}}{\bigoplus}T^{+}_{M_{\left\lfloor\frac{s+ip}{q}\right\rfloor}}\right)$
for all $n\geq\left\lfloor\dfrac{p}{2q}\right\rfloor$ we see that removing
$\underset{0\leq
s<p}{\bigoplus}T^{+}_{M_{\left\lfloor\frac{s+(i_{0}+k)p}{q}\right\rfloor}}$
from $\underset{0\leq
s<p}{\bigoplus}\left(\underset{i\in\mathbb{Z}}{\bigoplus}T^{+}_{M_{\left\lfloor\frac{s+ip}{q}\right\rfloor}}\right)$
removes some collection of tallest height towers, each of height $M_{\lfloor
p/q\rfloor}$. Thus even after removing these towers we obtain an
$\mathbb{F}[U]$-module which satisfies the strong geography restriction, as
desired. ∎

We can now prove Theorem 1.7.

###### Proof of Theorem 1.7.

Suppose $Y$ is $\frac{p}{q}\in\mathbb{Q}$ surgery on a knot $K$ in $S^{3}$. By
Lemma 2.17 $\operatorname{HF}^{+}(S^{3}_{p/q}(K))$ consists of two summands.
The two summands satisfy the strong geography restriction by Lemma 2.21 and
Corollary 2.20. The result follows. ∎

##  3\. Three-manifolds arising as surgery on links

A link $L=L_{1}\cup\dots\cup L_{\ell}\subset S^{3}$ with $\ell$ components can
be represented by a multi-pointed Heegaard diagram
$\mathcal{H}=(\Sigma,\alpha_{1}\dots\alpha_{k},\beta_{1},\dots,\beta_{k},z_{1},\dots
z_{\ell},w_{1},\dots w_{\ell})$. Similarly to the case of knots, this gives
rise to a multi-filtered chain complex
$(\operatorname{CFL}^{-}(\mathcal{H}),\mathcal{F}_{1},\dots.\mathcal{F}_{\ell})$
where: $\operatorname{CFL}^{-}(\mathcal{H})$ denotes the free
$\mathbb{F}[U_{1},\dots U_{\ell}]$-module generated by the intersection points
of the lagrangian tori $\mathbb{T}_{\alpha}$ and $\mathbb{T}_{\beta}$
associated to the $\alpha$\- and $\beta$-curves, equipped with differential

$\partial\mathbf{x}=\sum_{\mathbf{y}\in\mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta}}\
\ \sum_{\phi\in\pi_{2}(\mathbf{x},\mathbf{y}),\
\mu(\phi)=1}\\#\left(\frac{\mathcal{M}(\phi)}{\mathbb{R}}\right)U_{1}^{n_{z_{1}(\phi)}}\dots\
\ U_{\ell}^{n_{z_{\ell}(\phi)}}\cdot\mathbf{y}\ ,$

and $\mathcal{F}_{i}=\mathcal{F}_{L_{i}}$ denotes the filtration associated to
the $i^{\text{th}}$ component of the link $L$ extended so that multiplication
by the variable $U_{i}$ drops the filtration level $\mathcal{F}_{i}$ by one
[34].

###  3.1. Alexander filtration and $\text{spin}^{c}$ labelling

One can view that the multi-filtration
$\vec{\mathcal{F}}=(\mathcal{F}_{1}\dots,\mathcal{F}_{\ell})$ as taking value
in the lattice:

$\mathbb{H}(L)=\bigoplus_{i=1}^{\ell}\mathbb{H}(L)_{i}\ ,\ \ \ \ \ \
\mathbb{H}(L)_{i}=\mathbb{Z}+\frac{1}{2}\sum_{j\not=i}\text{lk}(L_{i},L_{j})\
.$

whose points parametrize relative $\text{spin}^{c}$ structures on
$S^{3}\setminus L$. Lattice points can also be interpreted as
$\text{spin}^{c}$ structures on any integer surgery
$S^{3}_{n_{1},\dots,n_{\ell}}(L)$, with the rule that two lattice points
$\mathbf{s}$ and $\mathbf{s}^{\prime}\in\mathbb{H}(L)$ define the same
$\text{spin}^{c}$ structure if their difference
$\mathbf{s}-\mathbf{s}^{\prime}$ lies in the span of the columns of the
linking matrix $\Lambda$:

$\Lambda_{ij}=\begin{cases}&lk(L_{i},L_{j})\ \ \ \text{ if }i\not=j\\\ &\ \ \
\ n_{i}\ \ \ \ \ \ \ \ \text{ otherwise }\end{cases}\ .$

To avoid ambiguity one can restrict to a fundamental domain
$P(\Lambda)\subset\mathbb{H}(L)$ of the action so that every point
$\mathbf{s}\in P(\Lambda)$ corresponds to a unique $\text{spin}^{c}$ structure
of $S^{3}_{n_{1},\dots,n_{\ell}}(L)$.

Sometimes instead of working with the fundamental domain $P(\Lambda)$ we shall
consider a small enlargement of it that we shall denote by
$\overline{P(\Lambda)}$. This consists of all the points in $P(\Lambda)$ union
all points $\mathbf{s}\in\mathbb{H}(L)$ whose coordinates differ from a point
in $P(\Lambda)$ by at most one. This can be interpreted as “the lattice
closure” of $P(\Lambda)$.

######  Example 3.1.

Consider $n$-surgery on a knot $K\subset S^{3}$. In the previous section we
chose $P(\Lambda)=(-n/2,n/2]\cap\mathbb{Z}$. In this case
$\overline{P(\Lambda)}=(-n/2-1,n/2+1]\cap\mathbb{Z}$.

######  Example 3.2.

For an algebraically split link $L=L_{1}\cup\dots L_{\ell}$ in $S^{3}$, that
is a link such that $\text{lk}(L_{i},L_{j})=0$ for all $i\not=j$, the linking
matrix is diagoinal

$\Lambda=\left(\begin{matrix}n_{1}&&\\\ &\ddots&\\\
&&n_{\ell}\end{matrix}\right)$

and one can pick $P(\Lambda)$ so that it has the shape of a hyperbox with
vertices at the points with coordinates $(\pm n_{1}/2,\dots,\pm n_{\ell}/2)$.

######  Example 3.3.

The linking matrix of a framed $2$-component link $L\subset S^{3}$ is a
$2\times 2$ matrix

$\Lambda=\left(\begin{matrix}n_{1}&\lambda\\\
\lambda&n_{2}\end{matrix}\right)$

where $\lambda=\text{lk}(L)$. In this case $P(\Lambda)$ has the shape of a
parallelogram with vertices at the points with coordinates

$\pm\frac{1}{2}\left(\begin{matrix}n_{1}\\\
\lambda\end{matrix}\right)\pm\frac{1}{2}\left(\begin{matrix}\lambda\\\
n_{2}\end{matrix}\right)\ .$

Its two sides have length $(n_{1}^{2}+\lambda^{2})^{1/2}$ and
$(n_{2}^{2}+\lambda^{2})^{1/2}$ respectively, and acute angle $\theta$ such
that:

$\cos\theta=\left(\frac{1}{n_{1}}+\frac{1}{n_{2}}\right)\cdot\frac{\lambda}{\left(\left(\frac{\lambda}{n_{1}}\right)^{2}+1\right)^{1/2}\cdot\
\ \left(\left(\frac{\lambda}{n_{2}}\right)^{2}+1\right)^{1/2}}\ .$

Note that as the framings $n_{1}$ and $n_{2}$ grow larger $P(\Lambda)$
approaches a rectangle ($\theta\to\pi/2$ as $n_{1},n_{1}\to\infty$) and its
sides grow in length.

For a general $\ell$-component link the domain $P(\Lambda)$ has the shape of a
parallelopiped with sides parallel to the column vectors of the linking matrix
$\Lambda$. Note that as the framings grow larger $P(\Lambda)$ approaches the
shape of an $\ell$-dimensional hyperbox. We shall take $P(\Lambda)$ to be the
unique domain with the points

$\pm\frac{1}{2}\sum_{i=1}^{\ell}\vec{\Lambda}_{i}$

as corners.

###  3.2. The Ozsváth-Manolescu surgery formula

Given $\mathbf{s}=(s_{1},\dots,s_{\ell})\in\mathbb{H}(L)$, as in the case of
knots, we define $A_{\mathbf{s}}^{-}=\mathfrak{A}^{-}(\mathcal{H},\mathbf{s})$
to be the subcomplex of $\operatorname{CFL}^{-}(\mathcal{H})$ spanned by all
generators $x=U_{1}^{m_{1}}\dots U_{\ell}^{m_{\ell}}\cdot\mathbf{x}$ with
$\mathbf{x}\in\mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta}$, and
$m_{1},\dots,m_{\ell}\geq 0$ such that
$\mathcal{F}_{i}(\mathbf{x})-m_{i}=\mathcal{F}_{i}(x)\leq s_{i}$ for
$i=1,\dots,\ell$.

######  Theorem 3.4 (Large surgery formula, Ozsváth & Manolescu [23]).

Let $\mathbf{n}=(n_{1},\dots,n_{\ell})\in\mathbb{Z}^{\ell}$, and
$\mathbf{s}\in P(\Lambda)$ a $\text{spin}^{c}$ structure. If $\mathbf{n}$ is
sufficiently large, that is $n_{i}>>0$ for $i\in\\{1,\dots,\ell\\}$, then
$\text{CF}^{-}(S^{3}_{\mathbf{r}}(L),\mathbf{s})$ and $A^{-}_{\mathbf{s}}$ are
quasi-isomorphic.

Note that multiplication by any of the variables $U_{1},\dots,U_{\ell}$ gives
rise to a chain map $A^{-}_{\mathbf{s}}\to A^{-}_{\mathbf{s}}$. It turns out
that these maps are chain homotopic to each other, and that in Theorem 3.4 the
map induced on homology can be identified with the $U$-action of the Heegaard
Floer group $\operatorname{HF}^{-}(S^{3}_{\mathbf{n}}(L),\mathbf{s})$.

###  3.3. A definition of large

Given a link $L$ we let $\mathcal{L}$ denote its link Floer polytope. This is
the convex hull of all those $\mathbf{s}\in\mathbb{H}(L)$ for which
$\widehat{\operatorname{HFL}}(L,\mathbf{s})\not=0$. In this context we will
denote by $Q=[-q_{1},q_{1}]\times[-q_{2},q_{2}]\dots[-q_{n},q_{\ell}]$ the
smallest hyperbox containing $\mathcal{L}$. Note that the link Floer polytope
can be expressed as the Minkowski sum of the dual of the Thurston polytope and
of a hyperbox of side length one, unless $L$ contains a split unlinked
component [36, Theorem 1.1]. In particular if $L$ does not have a split
unknotted component then $Q$ has non-vanishing volume.

######  Lemma 3.5.

If $\mathbf{n}>>0$ then $Q\subset P(\Lambda)$.

###### Proof.

For a given $\mathbf{n}$ the hyperbox $Q$ will be covered by a few different
translates of the fundamental domain $P(\Lambda)$. Let $\Lambda_{t}$ be the
matrix obtained from the linking matrix $\Lambda$ by increasing its diagonal
enteries by $t>0$. As $t\to+\infty$ the $\ell$-dimensional parallelogram
$P(\Lambda_{t})$ increase in size and eventually contains all the translates
of $P(\Lambda)$ covering the hyperbox $Q$. ∎

######  Definition 3.6.

$\mathbf{n}$-framed surgery on a link $L$ is large if
$Q\subset\overline{P(\Lambda)}$.

Note that if we take the framing $\mathbf{n}$ to be large in the sense of
Definition 3.6 then we can apply the large surgery formula stated in Theorem
3.4. This is explained implicitly by Manolesu-Ozsváth [23, P 112.], see also
the proof given by Borodzik-Gorsky for [3, Lemma 3.10]. To expand on this
point slightly, observe that Manolescu-Ozsváth’s general integer surgery
formula can be “horizontally truncated” – see [23, Section 10.1] – to the
large surgery formula in the case that $\Lambda$ is large in the sense of
Definition 3.6. The key observation that allows one to prove this is that
appropriate versions of the map from [23, Lemma 10.1] are isomorphisms for
$\mathbf{s}\not\in Q$.

###  3.4. Lemma 2.9 revisited

We now move towards proving Theorem 1.8. We start by introducing some
notation, definition, and lemmas.

######  Definition 3.7.

Let $\mathbf{s}=(s_{1},\dots,s_{\ell})$ and
$\mathbf{s}^{\prime}=(s_{1}^{\prime},\dots,s_{\ell}^{\prime})\in\mathbb{H}(L)$
be two lattice points. We say that $\mathbf{s}^{\prime}$ is adjacent to
$\mathbf{s}$ if $0\leq s_{i}^{\prime}-s_{i}\leq 1$ for $i=1,\dots,\ell$.

The key lemma is as follows.

######  Lemma 3.8.

Let $\mathbf{s}^{\prime}$ and $\mathbf{s}$ be two lattice points. If
$\mathbf{s}^{\prime}$ is adjacent to $\mathbf{s}$ then
$H_{*}(A^{-}_{\mathbf{s}^{\prime}})$ and $H_{*}(A^{-}_{\mathbf{s}})$ have
similar skylines.

###### Proof.

If $\mathbf{s}^{\prime}$ is adjacent to $\mathbf{s}$ then there is an
inclusion of sub-complexes $A^{-}_{\mathbf{s}}\subseteq
A^{-}_{\mathbf{s}^{\prime}}$. Furthermore
$H_{*}(A^{-}_{\mathbf{s}^{\prime}}/A^{-}_{\mathbf{s}})$ is annihilated by
multiplication by $U$. Thus we can look at the short exact sequence:

${0}$${A^{-}_{\mathbf{s}}}$${A^{-}_{\mathbf{s}^{\prime}}}$${A^{-}_{\mathbf{s}^{\prime}}/A^{-}_{\mathbf{s}}}$${0}$

and obtain an exact triangle of the form:

${\mathbb{F}^{k}\simeq
H_{*}(A^{-}_{\mathbf{s}^{\prime}}/A^{-}_{\mathbf{s}})}$${H_{*}(A^{-}_{\mathbf{s}})}$${H_{*}(A^{-}_{\mathbf{s}^{\prime}})}$
(4)

For some $k\geq 0$. It follows from Lemma 2.15 that
$H_{*}(A^{-}_{\mathbf{s}})$ and $H_{*}(A^{-}_{\mathbf{s}^{\prime}})$ have
similar skylines. ∎

We conclude this section with a discussion of integer homology spheres that
arise as large surgeries on links.

######  Lemma 3.9.

Suppose large surgery on an $\ell$-component link $L$ yields an integral
homology sphere. Then the link Floer polytope of $L$ is contained in
$[-1,1]^{\ell}$.

###### Proof.

Suppose $L$ and $\ell$ are as in the statement of the proposition and let
$S^{3}_{\mathbf{n}}(L)$ be a large integer homology sphere surgery on $L$.
Observe that the only lattice point that $P(\Lambda)$ contains is
$\mathbf{0}$. It follows that $\overline{P(\Lambda)}=[-1,1]^{\ell}$. By the
definition of large we have that $\mathcal{L}\subseteq
Q\subseteq[-1,1]^{\ell}$, as desired. ∎

###### Proof of Proposition 1.12.

Suppose $Y$ is as in the statement of the corollary. Write
$Y=S^{3}_{\mathbf{n}}(L)$ where $\mathbf{n}$ is large and $L$ has $\ell$
components. By Lemma 3.9 we have that $L$ has link Floer polytope supported in
$[-1,1]^{\ell}$. It follows that $H_{*}(A^{-}_{(1,1,\dots,1)})\simeq T^{-}$.
Since $Y$ is an integral homology sphere $P(\Lambda)$ contains only the origin
$\mathbf{0}\in\mathbb{H}(L)$. Observe that $\mathbf{0}$ is adjacent to
$(1,1,\dots,1)$ so the result follows from Lemma 3.8. ∎

###  3.5. Lattice paths

We shall consider paths of adjacent lattice points.

######  Definition 3.10.

A sequence of lattice points
$(\mathbf{s}_{0},\mathbf{s}_{1},\dots,\mathbf{s}_{k})$ is a lattice path if
each lattice point $\mathbf{s}_{j+1}$ in the sequence is adjacent to his
predecessor $\mathbf{s}_{j}$.

We now prove a preliminary version of Theorem 1.8.

######  Theorem 3.11.

Suppose that $L\subset S^{3}$ is a link with $\ell$ components. Then there
exists some $\mathbf{n}_{0}\in\mathbb{Z}^{\ell}_{\geq 0}$ such that for all
$\mathbf{n}\geq\mathbf{n}_{0}$ the surgery three-manifold
$S^{3}_{\mathbf{n}}(L)$ satisfies the Lin geography restriction.

###### Proof of Theorem 3.11.

Let $Q=[-q_{1},q_{1}]\times[-q_{2},q_{2}]\dots[-q_{n},q_{\ell}]$ be a hyperbox
containing $\mathcal{L}\subset\mathbb{H}(L)$, the link Floer polytope of $L$.
It is shown in [35] that $\operatorname{CFL}^{-}(L)$ is generated by module
elements $x\in\operatorname{CFL}^{-}(L)$ that have Alexander multi-grading
$\mathbf{s}\in\mathcal{L}$. Indeed,
$\vec{\mathcal{F}}(\mathbf{x})\in\mathcal{L}$ for all intersection points
$\mathbf{x}\in\mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta}$. Consequently
$H_{*}(A^{-}_{\mathbf{s}})\simeq\operatorname{HF}^{-}(S^{3})=\mathbb{F}[U]$
for all lattice points $\mathbf{s}=(s_{1},\dots,s_{\ell})$ such that

$\mathcal{L}\subset\\{(j_{1},\dots,j_{\ell})\in\mathbb{H}(L):s_{i}\leq
j_{i}\\}\ .$

Choose a lattice point $\mathbf{s}\in P(\Lambda)$ such that
$H_{*}(A_{\mathbf{s}}^{-})$ contains a direct summand $\mathbb{F}[U]/U^{n}$.
If there is no such $\mathbf{s}$ then $L$ is an $L$-space link and there is
nothing to prove. Starting from $\mathbf{s}$ form a lattice path
$(\mathbf{s}_{0}=\mathbf{s},\mathbf{s}_{1},\dots,\mathbf{s}_{k})$ where each
vector is obtained from its predecessor by increasing all components by one.
Taking $k\geq 1$ to be sufficiently large we can ensure that $\mathbf{s}_{k}$
lies in the semispace $\\{(j_{1},\dots,j_{\ell})\in\mathbb{H}(L):q_{i}\geq
j_{i}\\}$ so that $\mathcal{L}\subset A^{-}_{\mathbf{s}}$, and
$H_{*}(A^{-}_{\mathbf{s}})=\mathbb{F}[U]$.

Of course if $\mathbf{n}>>0$ then the lattice points in the path
$(\mathbf{s}_{0},\mathbf{s}_{1},\dots,\mathbf{s}_{k})$ all belong to
$P(\Lambda)$ and by Theorem 3.4 we have identifications

$T^{-}_{n}\subset\operatorname{HF}^{-}(S^{3}_{\mathbf{n}}(L),\mathbf{s}_{0})\simeq
H_{*}(A^{-}_{\mathbf{s}_{1}})\ ,\ \dots\ ,\
\operatorname{HF}^{-}(S^{3}_{\mathbf{n}}(L),\mathbf{s}_{k})\simeq
H_{*}(A^{-}_{\mathbf{s}_{k}})=\mathbb{F}[U]\ .$

At this point we can apply Lemma 3.8 to conclude that consecutive terms in
this sequence have similar skylines. Thus a term in the sequence has an
$\mathbb{F}$-summand and in turn we have that
$\operatorname{HF}^{-}(S^{3}_{\mathbf{n}}(L))$ satisfies the Lin geography
restriction. ∎

###  3.6. Proof of Theorem 1.8

We continue the discussion initiated in the previous section, maintaining the
same notation, with a simple Lemma.

######  Lemma 3.12.

Suppose that $\mathbf{s}\in\mathbb{H}(L)$ is some lattice point. Then there is
a lattice point $\mathbf{s}^{\prime}\in Q$ such that
$H_{*}(A^{-}_{\mathbf{s}})\simeq H_{*}(A^{-}_{\mathbf{s}^{\prime}})$.

###### Proof.

Suppose that $L$ is an $\ell$ component link. Let
$Q=[-q_{1},q_{1}]\times[-q_{2},q_{2}]\dots[-q_{\ell},q_{\ell}]$ and
$\mathbf{s}=(s_{1},s_{2},\dots s_{\ell})$. Suppose that $i$ is an index such
that $s_{i}\in\mathbb{R}\setminus[-q_{i},q_{i}]$. It suffices to show that
$H_{*}(A_{\mathbf{s}}^{-})\simeq H_{*}(A_{\mathbf{s}_{i}}^{-})$ where
$\mathbf{s}_{i}=(s_{1},\dots s_{i-1},q_{i}\dots s_{\ell})$ if $s_{i}$ is
positive or $\mathbf{s}_{i}=(s_{1},\dots s_{i-1},-q_{i}\dots s_{\ell})$ if
$s_{i}$ is negative.

_Case one: $s_{i}>0$._ We can represent the generators of the link Floer chain
complex $\operatorname{CFL}^{-}$ as a collection of lattice points: we mark a
lattice point $\mathbf{j}=(j_{1},\dots,j_{\ell})\in\mathbb{H}(L)$ as in
$\operatorname{CFL}^{-}$ if there is an $\mathbb{F}$-generator
$x=U_{1}^{m_{1}}\dots U_{\ell}^{m_{\ell}}\cdot\mathbf{x}$ such that
$j_{i}=\mathcal{F}_{i}(x)=\mathcal{F}_{i}(\mathbf{x})-m_{i}$. In this picture
multiplication by a variable $U_{i}$ acts as

$(j_{1},\dots,j_{\ell})\mapsto(j_{1},\dots,j_{i}-1,\dots,j_{\ell})\ ,$

and the sub-complex $A_{\mathbf{s}}^{-}$ is identified with the set of
generators lying inside the semi-space of lattice points
$\mathbf{j}=(j_{1},\dots,j_{\ell})\in\mathbb{H}(L)$ with coordinates
$j_{i}\leq s_{i}$, for $i=1,\dots,\ell$.

Note that $\operatorname{CFL}^{-}$ can be completely reconstructed by
translating the lattice points corresponding to the
$\mathbb{F}[U_{1},\dots,U_{\ell}]$-generators (intersection points in the
Lagrangian Floer set-up), and that these all lie inside the Floer polytope
$\mathcal{L}$ [34]. It follows that there are no lattice points marked as
generators in the strip
$\\{\mathbf{j}\leq\mathbf{s}\\}\setminus\\{\mathbf{j}\leq\mathbf{s}_{i}\\}\subset\mathbb{H}(L)$,
and we have the equality $A_{\mathbf{s}}^{-}=A_{\mathbf{s}_{i}}^{-}$.

_Case two: $s_{i}<0$._ Observe that $\operatorname{CFL}^{-}(L)$ comes equipped
with a $U_{i}$ filtration for all $1\leq i\leq n$. Let $\\{K_{i}\\}_{1\leq
i\leq\ell}$ be the components of $L$. Observe that the effect of switching the
orientation of the component $K_{i}$ is to switch the $U_{i}$ and
$\mathcal{F}_{i}$ filtrations; see [34]. We introduce some new notation to
keep track of this; if $L$ is a link let $A_{(s_{1},s_{2},,\dots
s_{n}),t}^{-}(L)$ denote the subcomplex of $\operatorname{CFL}^{-}(L)$
generated by elements with each $\mathcal{F}_{j}$ filtration at most $s_{j}$
and $U_{i}$ filtration at most $t$. Now let $L_{i}$ denote the link $L$ with
the orientation of $L$ reversed. It follows that

$H_{*}(A^{-}_{\mathbf{s}}(L))\simeq H_{*}(A^{-}_{(s_{1},s_{2},\dots
s_{i-1},0,s_{i+1}\dots s_{\ell}),s_{i}}(L_{i})).$

Since $s_{i}<-q_{i}$, multiplication by $U_{i}^{s_{i}}$ induces an
isomorphism:

$H_{*}(A^{-}_{(s_{1},s_{2},\dots s_{i-1},-s_{i},s_{i+1}\dots
s_{n}),0}(L_{i}))\to H_{*}(A^{-}_{(s_{1},s_{2},\dots s_{i-1},0,s_{i+1}\dots
s_{n}),s_{i}}(L_{i}))$

As in case one we have that $H_{*}(A^{-}_{(s_{1},s_{2},\dots-s_{i},\dots
s_{n}),0},L_{i})\simeq H_{*}(A^{-}_{(s_{1},s_{2},\dots q_{i},\dots
s_{n}),0},L_{i})$ for all $i$. The result follows. ∎

######  Remark 3.13.

Note that the argument of Lemma 3.12 specifies the constant $b$ in the
statement of [23, Lemma 10.1].

###### Proof of Theorem 1.8.

Suppose that $Y\simeq S^{3}_{\mathbf{n}}(L)$ for some $\ell$ component link
$L\subset S^{3}$ with $\mathbf{n}$ large in the sense of Definition 3.6. Set

$N:=\min\\{k\geq 0:U^{k}\cdot\operatorname{HF}^{-}_{\text{red}}(Y)=0\\}.$

If $N=0$ or $N=1$ there is nothing to prove, thus we can assume that $N>1$.

Given a lattice path $(\mathbf{s}_{1},\dots,\mathbf{s}_{k})$ consisting of
lattice points in $\overline{P(\Lambda)}$, if we can ensure that the sequence
satisfies the following conditions:

  1. (1)

$\mathbf{s}_{i}\in P(\Lambda)$ for $i\in\\{1,\dots,k-1\\}$, so that for each
of these lattice points we can apply Theorem 3.4 to show that
$H_{*}(A^{-}_{\mathbf{s}})\simeq\operatorname{HF}^{-}(S^{3}_{\mathbf{n}}(L),\mathbf{s}_{1})$,

  2. (2)

the start-point $\mathbf{s}_{1}$ is such that
$\operatorname{HF}^{-}(S^{3}_{\mathbf{n}}(L),\mathbf{s}_{1})$ contains a copy
of $T^{-}_{N}$,

  3. (3)

the end-point $\mathbf{s}_{k}$ is such that $H_{*}(A^{-}_{\mathbf{s}_{k}})$ is
torsion free.

then using Lemma 3.8 we can conclude that the summand
$\bigoplus_{i=1}^{k-1}\operatorname{HF}^{-}(S^{3}_{\mathbf{n}}(L),\mathbf{s}_{i})$
contains a copy of $T^{-}_{N}\oplus T^{-}_{N-1}\dots\oplus T_{1}^{-}$.

Choose a lattice point $\mathbf{s}\in P(\Lambda)$ representing a
$\text{spin}^{c}$ structure such that
$\operatorname{HF}^{-}(Y,\mathbf{s})\simeq\operatorname{HF}^{-}(S^{3}_{\mathbf{n}}(L),\mathbf{s})$
contains a $T^{-}_{N}$-summand. Let
$Q=[-q_{1},q_{1}]\times[-q_{2},q_{2}]\dots[-q_{\ell},q_{\ell}]$ be the
smallest hyperbox containing the link Floer polytope, and
$\mathbf{q}=(q_{1},\dots,q_{\ell})$. By Lemma 3.12 there is a lattice point
$\mathbf{s}^{\prime}$ in $Q\subset\overline{P(\Lambda)}$ that also contains a
$T^{-}_{N}$-summand. If $\mathbf{s}^{\prime}\in P(\Lambda)$ then we can find a
path
$(\mathbf{s}_{1}=s^{\prime},\mathbf{s}_{2}\dots,\mathbf{s}_{k}=\mathbf{q})$,
with $\mathbf{s}_{1}\dots,\mathbf{s}_{k-1}\in Q\cap P(\Lambda)$, and we are
done.

Of course it could be that $\mathbf{s}^{\prime}\in
Q\cap\partial\overline{P(\Lambda)}$, where
$\partial\overline{P(\Lambda)}=\overline{P(\Lambda)}\setminus P(\Lambda)$. In
this limiting case we can find an adjacent point $\mathbf{s}^{\prime\prime}\in
Q\setminus\partial\overline{P(\Lambda)}=Q\cap P(\Lambda)$ and start a lattice
path from $\mathbf{s}^{\prime\prime}$ instead. This shows that
$\operatorname{HF}^{-}(S^{3}_{\mathbf{n}}(L))$ contains a copy of
$T^{-}_{N-1}\dots\oplus T_{1}^{-}$ as well as a copy of $T^{-}_{N}$,
concluding the proof. ∎

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