# Sutured instanton homology and Heegaard diagrams

John A. Baldwin  Department of Mathematics
Boston College<EMAIL_ADDRESS>,  Zhenkun Li  Department of Mathematics
Stanford University<EMAIL_ADDRESS>and  Fan Ye  Department of Pure
Mathematics and Mathematical Statistics
University of Cambridge<EMAIL_ADDRESS>
###### Abstract.

Suppose $\mathcal{H}$ is an admissible Heegaard diagram for a balanced sutured
manifold $(M,\gamma)$. We prove that the number of generators of the
associated sutured Heegaard Floer complex is an upper bound on the dimension
of the sutured instanton homology $\mathit{SHI}(M,\gamma)$. It follows, in
particular, that strong L-spaces are instanton L-spaces.

JAB was supported by NSF CAREER Grant DMS-1454865 and NSF FRG Grant
DMS-1952707.

##  1\. Introduction

Let $(M,\gamma)$ be a balanced sutured manifold. Kronheimer and Mrowka
conjectured [KM10] that its sutured instanton homology is isomorphic to its
sutured Heegaard Floer homology,

(1.1) $\mathit{SHI}(M,\gamma)\cong\mathit{SFH}(M,\gamma)\otimes\mathbb{C}.$

Proving this remains a major open problem. In particular, it would imply
isomorphisms

$\displaystyle I^{\\#}(Y)$
$\displaystyle\cong\widehat{\mathit{HF}}(Y)\otimes\mathbb{C}$
$\displaystyle\mathit{KHI}(Y,K)$
$\displaystyle\cong\widehat{\mathit{HFK}}(Y,K)\otimes\mathbb{C}$

between the invariants of closed 3-manifolds and knots in the instanton and
Heegaard Floer settings.

There has been a flood of recent work proving these isomorphisms for various
families of closed 3-manifolds and knots; see [BS20, ABDS20, LPCS20, LY20,
GL19]. In this paper, we initiate a systematic approach to the general
isomorphism (1.1). Before stating our main result, let us establish some
notation.

Given a sutured Heegaard diagram

$\mathcal{H}=\big{(}\Sigma,\alpha=\\{\alpha_{1},\dots,\alpha_{k}\\},\beta=\\{\beta_{1},\dots,\beta_{k}\\}\big{)}$

for a balanced sutured manifold $(M,\gamma)$, let

$\mathbb{T}_{\alpha}:=\alpha_{1}\times\dots\times\alpha_{k}\textrm{ and
}\mathbb{T}_{\beta}:=\beta_{1}\times\dots\times\beta_{k}\subset\operatorname{Sym}^{k}(\Sigma)$

denote the usual tori in the $k$-fold symmetric product of $\Sigma$, and let

$\mathfrak{S}(\mathcal{H}):=\mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta}\subset\operatorname{Sym}^{k}(\Sigma).$

If $\mathcal{H}$ is _admissible_ , then $\mathfrak{S}(\mathcal{H})$ is the set
of generators for the sutured Heegaard Floer complex
$\mathit{SFC}(\mathcal{H})$ as defined by Juhász in [Juh06].111See §3.2 for
the definition of _admissible_ ; every $\mathcal{H}$ is admissible when
$H_{1}(M,\partial M;\mathbb{Q})=0$. Our main theorem is the following.

######  Theorem 1.1.

If $\mathcal{H}$ is an admissible sutured Heegaard diagram for $(M,\gamma)$,
then

$\dim_{\mathbb{C}}\mathit{SHI}(M,\gamma)\leq|\mathfrak{S}(\mathcal{H})|.$

######  Remark 1.2.

Theorem 1.1 does not hold without the assumption that $\mathcal{H}$ is
admissible; see Remark 3.5.

######  Remark 1.3.

Our proof of Theorem 1.1 also works for sutured monopole homology
($\mathit{SHM}$) in place of $\mathit{SHI}$. Of course, the $\mathit{SHM}$
version of our main result follows from the isomorphism

$\mathit{SHM}(M,\gamma)\cong\mathit{SFH}(M,\gamma),$

which is a consequence of the equivalence between monopole and Heegaard Floer
homology; see [Lek13]. Still, it may be of value to know that one can prove
the inequality

$\dim_{\mathbb{Z}}\mathit{SHM}(M,\gamma)\leq|\mathfrak{S}(\mathcal{H})|$

without going through the proof of this equivalence.

Given a balanced sutured manifold $(M,\gamma)$, we define the _simultaneous
trajectory number_ $\mathcal{T}(M,\gamma)$ to be the minimum of
$|\mathfrak{S}(\mathcal{H})|$ over all admissible sutured Heegaard diagrams
$\mathcal{H}$ for $(M,\gamma)$. This is the generalization to balanced sutured
manifolds of a notion originally defined for rational homology 3-spheres by
Ozsváth and Szabó in [OS04]. It admits a purely Morse-theoretic interpretation
when $H_{1}(M,\partial M;\mathbb{Q})=0$, and is a measure of the topological
complexity of $(M,\gamma)$—for example, $\mathcal{T}(M,\gamma)=1$ iff
$(M,\gamma)$ is a product sutured manifold.222This is a fun exercise which we
have not seen written down before; it is a generalization of the well-known
fact that $S^{3}$ is the only closed 3-manifold with simultaneous trajectory
number one [OS04]. Further, it is clear from the definition that

$\textrm{rk}_{\mathbb{Z}}\mathit{SFH}(M,\gamma)\leq\mathcal{T}(M,\gamma).$

We have the following immediate corollary of Theorem 1.1.

######  Corollary 1.4.

If $(M,\gamma)$ is a balanced sutured manifold, then

$\dim_{\mathbb{C}}\mathit{SHI}(M,\gamma)\leq\mathcal{T}(M,\gamma).$

For the corollaries below, we recall the natural sutured manifolds associated
to closed 3-manifolds and knots therein. Given a closed 3-manifold $Y$, let
$(Y(1),\delta)$ denote the sutured manifold obtained by removing a 3-ball from
$Y$, where $\delta$ is a simple closed curve on $\partial Y(1)\cong S^{2}$.
Similarly, given a knot $K\subset Y$, let $(Y(K),m\cup-m)$ be the sutured
manifold obtained by removing a tubular neighborhood of $K$, where $m$ and
$-m$ are oppositely oriented meridional curves on $\partial Y(K)\cong T^{2}$.
The framed instanton and Heegaard Floer homologies of a closed 3-manifold $Y$
are given by

$\displaystyle I^{\\#}(Y)$ $\displaystyle\cong\mathit{SHI}(Y(1),\delta)$
$\displaystyle\widehat{\mathit{HF}}(Y)$
$\displaystyle\cong\mathit{SFH}(Y(1),\delta).$

Likewise the instanton and Heegaard knot Floer homologies of a knot $K\subset
Y$ are given by

$\displaystyle\mathit{KHI}(Y,K)$
$\displaystyle\cong\mathit{SHI}(Y(K),m\cup-m)$
$\displaystyle\widehat{\mathit{HFK}}(Y,K)$
$\displaystyle\cong\mathit{SFH}(Y(K),m\cup-m).$

Recall that the ranks of the Heegaard Floer homology and framed instanton
homology of a rational homology 3-sphere $Y$ are each bounded below by
$|H_{1}(Y)|$. An _L-space_ , respectively _instanton L-space_ , is a rational
homology 3-sphere which achieves these lower bounds

$\displaystyle\textrm{rk}_{\mathbb{Z}}\widehat{\mathit{HF}}(Y)$
$\displaystyle=|H_{1}(Y)|,$ $\displaystyle\dim_{\mathbb{C}}I^{\\#}(Y)$
$\displaystyle=|H_{1}(Y)|,$

respectively. A _strong L-space_ , as defined by Levine and Lewallen in
[LL12], is a rational homology 3-sphere $Y$ which satisfies the stronger
condition

$\mathcal{T}(Y(1),\delta)=|H_{1}(Y)|.$

Indeed, this condition implies that $Y$ is an L-space, since

$|H_{1}(Y)|\leq\textrm{rk}_{\mathbb{Z}}\widehat{\mathit{HF}}(Y)\leq\mathcal{T}(Y(1),\delta).$

The following is then an immediate corollary of Theorem 1.1.

######  Corollary 1.5.

If $Y$ is a strong L-space, then it is an instanton L-space.

More generally, we say that a sutured Heegaard diagram $\mathcal{H}$ for a
sutured manifold $(M,\gamma)$ is _strong_ if $\mathcal{H}$ is admissible and
the sutured Floer complex $\mathit{SFC}(\mathcal{H})$ has trivial
differential. We then have the following.

######  Corollary 1.6.

If $(M,\gamma)$ is a balanced sutured manifold which has a strong sutured
Heegaard diagram, then

$\dim_{\mathbb{C}}\mathit{SHI}(M,\gamma)\leq\textrm{rk}_{\mathbb{Z}}\mathit{SFH}(M,\gamma).$

For example, when $K$ is a $(1,1)$-knot in a lens space $L(p,q)$, the sutured
manifold

$\mathit{SHI}((L(p,q))(K),m\cup-m)$

has a strong Heegaard diagram. We thus reproduce the following result of Li
and Ye [LY20].

######  Corollary 1.7.

If $K\subset L(p,q)$ is a $(1,1)$-knot, then

$\dim_{\mathbb{C}}\mathit{KHI}(L(p,q),K)\leq\textrm{rk}_{\mathbb{Z}}\widehat{\mathit{HFK}}(L(p,q),K).$

###  1.1. On the proof

Given a vertical tangle $T$ in a balanced sutured manifold $(M,\gamma)$, one
forms an associated sutured manifold $(M_{T},\gamma_{T})$ by removing a
neighborhood of $T$ from $M$, and adding meridians of the components of $T$ to
$\gamma$; see §2 for more details. Li and Ye proved the following dimension
inequality in [LY20, Proposition 3.14].

######  Theorem 1.8.

If $T$ is a vertical tangle in $(M,\gamma)$ such that $[T_{i}]=0$ in
$H_{1}(M,\partial M;\mathbb{Q})$ for each component $T_{i}$ of $T$, then

$\dim_{\mathbb{C}}\mathit{SHI}(M,\gamma)\leq\dim_{\mathbb{C}}\mathit{SHI}(M_{T},\gamma_{T}).$

To prove Theorem 1.1, we first establish the same inequality under the weaker
assumption that $T$ (rather than each of its components) is rationally
nullhomologous, in §2.

######  Theorem 1.9.

If $T$ is a vertical tangle in $(M,\gamma)$ such that $[T]=0$ in
$H_{1}(M,\partial M;\mathbb{Q})$, then

$\dim_{\mathbb{C}}\mathit{SHI}(M,\gamma)\leq\dim_{\mathbb{C}}\mathit{SHI}(M_{T},\gamma_{T}).$

Next, given an admissible sutured Heegaard diagram $\mathcal{H}$ for
$(M,\gamma)$, we construct a vertical tangle $T\subset(M,\gamma)$ with $[T]=0$
in $H_{1}(M,\partial M;\mathbb{Q})$, such that

$\dim_{\mathbb{C}}\mathit{SHI}(M_{T},\gamma_{T})=|\mathfrak{S}(\mathcal{H})|.$

This is the content of §3. Theorem 1.1 then follows from Theorem 1.9.

######  Remark 1.10.

Theorem 1.8 suffices to prove Theorem 1.1 in the case where $H_{1}(M,\partial
M;\mathbb{Q})=0$, but we need the stronger Theorem 1.9 in general.

###  1.2. Organization

In §2, we prove the inequality in Theorem 1.9. We then use this in §3 to prove
our main result, Theorem 1.1. Finally, in §4, we discuss further directions,
some of which are in progress. In particular, we discuss the possibility of
using the proof of Theorem 1.1 to construct a grading on $\mathit{SHI}$ by
homotopy classes of 2-plane fields, and the prospects for upgrading Theorem
1.1 to a proof of the isomorphism (1.1).

###  1.3. Acknowledgements

We thank Adam Levine, Tye Lidman, and Steven Sivek for helpful conversations.

##  2\. A dimension inequality

A _vertical tangle_ $T=T_{1}\cup\dots\cup T_{n}$ in a balanced sutured
manifold $(M,\gamma)$ is a properly embedded 1-manifold in $M$, with boundary
in $R(\gamma)$, whose components $T_{i}$ are oriented from $R_{+}(\gamma)$ to
$R_{-}(\gamma)$. One forms an associated balanced sutured manifold
$(M_{T},\gamma_{T})$ by removing tubular neighborhoods of the components
$T_{i}$, and adding positively-oriented meridians of these components to the
suture $\gamma$, as in [LY20, Section 3] and depicted in Figure 7. In this
section, we prove Theorem 1.9, which states that

$\dim_{\mathbb{C}}\mathit{SHI}(M,\gamma)\leq\dim_{\mathbb{C}}\mathit{SHI}(M_{T},\gamma_{T})$

when $[T]=0$ in $H_{1}(M,\partial M;\mathbb{Q})$. The rough idea is to turn
$T$ into a related tangle $T^{\prime}$ whose components are rationally
nullhomologous, and apply Theorem 1.8.

###### Proof of Theorem 1.9.

Let $T_{1},\dots,T_{n}$ be the components of $T$. For $i=1,\dots,n$, let

$\partial T_{i}=q_{i}-p_{i},$

where $p_{i}\in R_{+}(\gamma)$ and $q_{i}\in R_{-}(\gamma)$. We may assume
that $\gamma$ is connected, since we can achieve this by adding contact
1-handles to $(M,\gamma)$, an operation which does not change
$\mathit{SHI}(M,\gamma)$ (equivalently, $\mathit{SHI}$ is invariant under
product disk decomposition [KM10, BS16]). Then we can find a sequence of
pairwise disjoint arcs

$\xi_{1},\dots,\xi_{n}\subset\partial M$

such that, for each $i=1,\dots,n$, we have:

  * •

$\partial\xi_{i}=p_{i+1}-q_{i}$ (where $p_{n+1}:=p_{1}$), and

  * •

$\xi_{i}$ intersects $\gamma$ in exactly one point.

For every $i\in\\{2,\dots,n\\}$, choose an arc $t_{i}\subset\partial M$ in a
neighborhood of the unique intersection point $\xi_{i-1}\cap\gamma$, as
depicted in Figures 1 and 2. Push the interior of $t_{i}$ into the interior of
$M$ to turn this arc into a vertical tangle $T_{i}^{\prime}$, and let

$T^{\prime\prime}=T_{2}^{\prime}\cup\dots\cup T_{n}^{\prime}.$

Then

$(M_{T^{\prime\prime}},\gamma_{T^{\prime\prime}})=(M-N(T^{\prime\prime}),\gamma\cup\mu_{2}^{\prime}\cup\dots\cup\mu_{n}^{\prime}),$

where $\mu_{i}^{\prime}$ is a positively-oriented meridian of
$T_{i}^{\prime}$. Each component $T_{i}^{\prime}$ cobounds a disk in $M$ with
the arc $t_{i}$. These disks then restrict to properly embedded disks

$D_{2},\dots,D_{n}\subset M_{T^{\prime\prime}}$

with $|D_{i}\cap\gamma|=1$ and $|D_{i}\cap\mu_{j}^{\prime}|=\delta_{ij}$, so
that

$|D_{i}\cap\gamma_{T^{\prime\prime}}|=2.$

Thus, each $D_{i}$ is a product disk.

2pt $t_{i}$ at 73 3 $\xi_{i-1}$ at 136 48 $R_{-}(\gamma)$ at 12 65
$R_{+}(\gamma)$ at 50 65 $\partial M$ at 150 70 $\gamma$ at 129 28

Figure 1. The point of view is from the interior of $M$, looking at $\partial
M$.

$\dots$ at 200 59 2pt $M$ at 272 59 $T_{1}$ at 48 70 $T_{2}$ at 98 70 $T_{3}$
at 150.5 70 $T_{n}$ at 238.5 70 $\xi_{1}$ at 68 30 $\xi_{3}$ at 167 30
$\xi_{2}$ at 120 94 $\xi_{n}$ at 215 108 $p_{1}$ at 48 98 $q_{1}$ at 48 24
$p_{n}$ at 251 24 $q_{n}$ at 251 98 $t_{2}$ at 87 35.5 $t_{3}$ at 149 89
$t_{n}$ at 229 35.5

Figure 2. The tangle $T=T_{1}\cup\dots\cup T_{n}$ in $M$ and the arcs
$\xi_{i}$ and $t_{i}$ in $\partial M$. The suture $\gamma$ is shown in red.

Next, consider the arc

$t_{1}=T_{1}\cup\xi_{1}\cup T_{2}\cup\xi_{2}\cup\dots\cup
T_{n-1}\cup\xi_{n-1}\cup T_{n}\subset M_{T^{\prime\prime}}.$

Push its interior into the interior of $M_{T^{\prime\prime}}$ to form a
vertical tangle $T_{1}^{\prime}$ with $\partial T_{1}^{\prime}=q_{n}-p_{1},$
as in Figure 3. Let $T^{\prime}$ be the tangle in $M$ given by

$T^{\prime}=T_{1}^{\prime}\cup T^{\prime\prime}=T_{1}^{\prime}\cup\dots\cup
T_{n}^{\prime}.$

We will refer to a tangle $T^{\prime}$ formed in this way as a _mixed tangle
for $T$_. Note that

$(M_{T^{\prime}},\gamma_{T^{\prime}})=((M_{T^{\prime\prime}})_{T_{1}^{\prime}},(\gamma_{T^{\prime\prime}})_{T_{1}^{\prime}})=(M-N(T^{\prime}),\gamma\cup\mu_{1}^{\prime}\cup\dots\cup\mu_{n}^{\prime}),$

where $\mu_{1}^{\prime}$ is a positively-oriented meridian of
$T_{1}^{\prime}$. Observe that the disks $D_{2},\dots,D_{n}\subset
M_{T^{\prime\prime}}$ restrict to properly embedded annuli

$A_{2},\dots,A_{n}\subset M_{T^{\prime}}$

since $T_{1}^{\prime}$ intersects each disk in exactly one point, as shown in
Figure 4.

$\dots$ at 200 59 2pt $M$ at 272 59 $T_{1}^{\prime}$ at 48 70

$\xi_{n}$ at 215 108 $p_{1}$ at 48 98 $q_{n}$ at 251 98 $T_{2}^{\prime}$ at 87
38 $T_{3}^{\prime}$ at 136 90 $T_{n}^{\prime}$ at 228 38

Figure 3. The tangle $T^{\prime}=T_{1}^{\prime}\cup\dots\cup T_{n}^{\prime}$
in $M$.

2pt $M_{T^{\prime}}$ at 210 105 $\partial N(T_{1}^{\prime})$ at 9 53 $\partial
N(T_{i}^{\prime})$ at 147 30

$\mu_{i}^{\prime}$ at 126 116 $\mu_{1}^{\prime}$ at 52 51 $\gamma$ at 187 83
$A_{i}$ at 94 92 $R_{+}$ at 43 82 $R_{-}$ at 106 8 $R_{+}$ at 26 37.5 $R_{-}$
at 71 57

Figure 4. The annulus $A_{i}$ in $M_{T^{\prime}}$ near the boundaries of the
tubular neighborhoods of the components $T_{1}^{\prime}$ and $T_{i}^{\prime}$,
as seen from inside $M_{T^{\prime}}$.

The endpoints of the arc $\xi_{n}\subset\partial M_{T^{\prime\prime}}$ agree
with $\partial T_{1}^{\prime}$, and $|\xi_{n}\cap\gamma|=1$. We can thus use
$\xi_{n}$ together with $\gamma$ and $\mu_{1}^{\prime}$ to define a sequence
of sutures $\Gamma_{m}\subset\partial M_{T^{\prime}}$ for $m\in\mathbb{N}$, as
in [LY20, Section 3.2], which one should regard as “longitudinal” sutures for
$T_{1}^{\prime}$; see Figure 5. By the construction of $T_{1}^{\prime}$ and
the assumption that $T$ is rationally nullhomologous in $(M,\partial M)$, we
have

$[T_{1}^{\prime}]=[T]=0\in H_{1}(M_{T^{\prime\prime}},\partial
M_{T^{\prime\prime}};\mathbb{Q}).$

Therefore, by [LY20, Lemmas 3.21 and 3.22], we have the following.

######  Lemma 2.1.

There is an exact triangle

$\textstyle{\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{m+1}}$$\textstyle{\mathit{SHI}(-M_{T^{\prime\prime}},-\gamma_{T^{\prime\prime}}),\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G_{m}}$

coming from the surgery exact triangle associated to surgeries on the meridian
$\mu_{1}^{\prime}$ of $T_{1}^{\prime}$. Furthermore, $G_{m}\equiv 0$ for $m$
sufficiently large.

2pt $-M_{T^{\prime}}$ at 383 105 $-\Gamma_{m+1}$ at 27 60
$-\gamma_{T^{\prime}}$ at 210 60 $A_{i}$ at 94 92 $A_{i}^{-}$ at 272 92

Figure 5. Left, the suture $-\Gamma_{m+1}$. The bypass arc $\eta_{-}$ is shown
in green. Right, the suture $-\gamma_{T^{\prime}}$ resulting from the bypass
attachment along $\eta_{-}$, and the negatively stabilized annulus
$A_{i}^{-}\subset(-M_{T^{\prime}},-\gamma_{T^{\prime}})$.

For each $i=2,\dots,n$ and every $m\in\mathbb{N}$, we have that

$|A_{i}\cap\Gamma_{m}|=4.$

Let us orient each $A_{i}$ so that the induced orientation on $\partial A_{i}$
is opposite the orientation of $\partial D_{i}$ coming from that of
$T_{i}^{\prime}$, as in Figures 4 and 5. By [GL19], the disks
$D_{2},\dots,D_{n}$ induce a $\mathbb{Z}^{n-1}$-grading on

$\mathit{SHI}(-M_{T^{\prime\prime}},-\gamma_{T^{\prime\prime}}).$

Similarly, the annuli $A_{2},\dots,A_{n}$ induce a $\mathbb{Z}^{n-1}$-grading
on

$\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m})$

for each $m\in\mathbb{N}$, and we have the following graded version of the
triangle in Lemma 2.2.

######  Lemma 2.2.

The exact triangle of Lemma 2.1 restricts to the exact triangle

$\textstyle{\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m},(A_{2},\dots,A_{n}),(0,\dots,0))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m+1},(A_{2},\dots,A_{n}),(0,\dots,0))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{m+1}}$$\textstyle{\mathit{SHI}(-M_{T^{\prime\prime}},-\gamma_{T^{\prime\prime}},(D_{2},\dots,D_{n}),(0,\dots,0)).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G_{m}}$

######  Remark 2.3.

A more general version of Lemma 2.2 will be proved in [LY].

###### Proof of Lemma 2.2.

We will prove that the map $F_{m+1}$ preserves the gradings. The arguments for
the other two maps are similar.

Let us first recall the definition of $F_{m+1}$ from [LY20, Section 3]. Pick a
_closure_

$(Y_{m+1},R_{m+1},\omega_{m+1})$

for $(-M_{T^{\prime}},-\Gamma_{m+1})$ so that each annulus $A_{i}$ extends to
a closed surface $\bar{A}_{i}\subset Y_{m+1}$, as in [Li19, Section 3]. By the
construction therein,

$g(\bar{A}_{i})=2$

for $i=2,\dots,n$, since each component of $\partial A_{i}$ intersects
$\Gamma_{m+1}$ in two points. The sutured instanton homology of
$(-M_{T^{\prime}},-\Gamma_{m+1})$ is defined as a certain direct summand

$\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m+1})=I_{*}(Y_{m+1}|R_{m+1})_{\omega_{m+1}}$

of the instanton Floer homology $I_{*}(Y_{m+1})_{\omega_{m+1}}$ as in [KM10,
Section 7]. The summand

$\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m+1},(A_{2},\dots,A_{n}),(i_{2},\dots,i_{n}))$

is defined as the simultaneous (generalized)
$(2i_{2},\dots,2i_{n})$-eigenspace of commuting operators

$\mu(\bar{A}_{2}),\dots,\mu(\bar{A}_{n}):\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m+1})\to\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m+1})$

associated to these surfaces.

The meridian $\mu_{1}^{\prime}$ of $T_{1}^{\prime}$ can be thought of as an
embedded circle in $Y_{m+1}$. Let $Y$ be the manifold obtained from $Y_{m+1}$
via $0$-surgery on $\mu_{1}^{\prime}$, with respect to the framing of
$\mu_{1}^{\prime}$ induced by $\partial M_{T^{\prime}}$. Since
$\mu_{1}^{\prime}$ is disjoint from $R_{m+1}$ and the $\bar{A}_{i}$, these
surfaces survive in $Y$. By [BS16, Section 3.3], $(Y,R_{m+1},\omega_{m+1})$ is
a closure of $(-M_{T^{\prime\prime}},-\gamma_{T^{\prime\prime}})$. The map
$F_{m+1}$ is then induced by the cobordism given by the trace of $0$-surgery
on $\mu_{1}^{\prime}$. Since $\bar{A}_{i}\subset Y_{m+1}$ is homologous to
$\bar{A}_{i}\subset Y$ in this cobordism, $F_{m+1}$ respects the eigenspaces
of $\mu(\bar{A}_{i})$. Thus, $F_{m+1}$ maps

$\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m+1},(A_{2},\dots,A_{n}),(i_{2},\dots,i_{n}))$

into

$\mathit{SHI}(-M_{T^{\prime\prime}},-\gamma_{T^{\prime\prime}},(A_{2},\dots,A_{n}),(i_{2},\dots,i_{n})).$

Now, the $0$-surgery on $\mu_{1}^{\prime}$ makes each $\bar{A}_{i}$
compressible in $Y$; in particular, each $\bar{A}_{i}\subset Y$ is homologous
to the disjoint union of two tori

$T_{i}^{1}\cup T_{i}^{2}\subset Y.$

One of these tori, say $T_{i}^{1}$, is the extension $\bar{D}_{i}\subset Y$ of
$D_{i}\subset-M_{T^{\prime\prime}}$ that is used to define the grading on
$\mathit{SHI}(-M_{T^{\prime\prime}},-\gamma_{T^{\prime\prime}})$ associated to
$D_{i}$. Since

$\bar{A}_{i}=\bar{D}_{i}+T_{i}^{2}$

in $H_{2}(Y)$, the $k$-eigenspace of $\mu(\bar{A}_{i})$ agrees with the
$k$-eigenspace of $\mu(\bar{D}_{i})$ for every $k$, by [BS18, Corollary 2.9].
Thus, we have that

$\displaystyle\mathit{SHI}(-M_{T^{\prime\prime}},-\gamma_{T^{\prime\prime}},(A_{2},\dots,A_{n}),(i_{2},\dots,i_{n}))$
$\displaystyle=\,$
$\displaystyle\mathit{SHI}(-M_{T^{\prime\prime}},-\gamma_{T^{\prime\prime}},(D_{2},\dots,D_{n}),(i_{2},\dots,i_{n})).$

Putting these arguments together, we see that $F_{m+1}$ preserves the
$\mathbb{Z}^{n-1}$-gradings as claimed in the lemma. ∎

Note that decomposing $(-M_{T^{\prime\prime}},-\gamma_{T^{\prime\prime}})$
along $D_{2}\cup\dots\cup D_{n}$ yields $(-M,-\gamma)$. By [Li19, Lemma 4.2],
we therefore have

$\mathit{SHI}(-M_{T^{\prime\prime}},-\gamma_{T^{\prime\prime}},(D_{2},\dots,D_{n}),(0,\dots,0))\cong\mathit{SHI}(-M,-\gamma).$

Hence, for $m$ sufficiently large, Lemmas 2.1 and 2.2 imply that that

(2.1) $\displaystyle\dim_{\mathbb{C}}\mathit{SHI}(-M,-\gamma)=$
$\displaystyle\dim_{\mathbb{C}}\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m+1},(A_{2},\dots,A_{n}),(0,\dots,0))$
$\displaystyle-\dim_{\mathbb{C}}\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m},(A_{2},\dots,A_{n}),(0,\dots,0)),$

since $G_{m}\equiv 0$.

Next, we consider attaching a bypass to $(-M_{T^{\prime}},-\Gamma_{m+1})$
along the arc $\eta_{-}$ in Figure 5. By [BS18, Section 4], this attachment
gives rise to a bypass exact triangle. As discussed in [LY20, Section 3], the
other two sutures involved in the triangle are $-\Gamma_{m}$ and
$-\gamma_{T^{\prime}}$. It is straightforward to check that the bypass
attachment along $\eta_{-}$ creates a _negative stabilization_

$A_{i}^{-}\subset(-M_{T^{\prime}},-\gamma_{T^{\prime}})$

of $A_{i}$, for each $i=2,\dots,n$, in the sense of [Li19, Definition 3.1].
Hence, as in the proof of [Li19, Proposition 5.5], we have the following
graded version of the bypass exact triangle of [BS18, Theorem 1.20],

$\textstyle{\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m},(A_{2},\dots,A_{n}),(0,\dots,0))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m+1},(A_{2},\dots,A_{n}),(0,\dots,0))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathit{SHI}(-M_{T^{\prime}},-\gamma_{T^{\prime}},(A_{2}^{-},\dots,A_{n}^{-}),(0,\dots,0)),\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

which implies that

(2.2)
$\displaystyle\dim_{\mathbb{C}}\mathit{SHI}(-M_{T^{\prime}},-\gamma_{T^{\prime}},(A_{2}^{-},\dots,A_{n}^{-}),(0,\dots,0))$
$\displaystyle\geq$
$\displaystyle\dim_{\mathbb{C}}\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m+1},(A_{2},\dots,A_{n}),(0,\dots,0))$
$\displaystyle-\dim_{\mathbb{C}}\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m},(A_{2},\dots,A_{n}),(0,\dots,0)).$

From the grading shifting property [Li19, Theorem 1.12] and [Wan20,
Proposition 4.1], we have

(2.3)
$\displaystyle\mathit{SHI}(-M_{T^{\prime}},-\gamma_{T^{\prime}},(A_{2}^{-},\dots,A_{n}^{-}),(0,\dots,0))$
$\displaystyle=\,\,$
$\displaystyle\mathit{SHI}(-M_{T^{\prime}},-\gamma_{T^{\prime}},(A_{2}^{+},\dots,A_{n}^{+}),(-1,\dots,-1)),$

where $A_{i}^{+}$ is a _positive stabilization_ of $A_{i}$. Moreover, from the
construction of the gradings and stabilizations in [Li19, Section 3], we have

(2.4)
$\displaystyle\mathit{SHI}(-M_{T^{\prime}},-\gamma_{T^{\prime}},(A_{2}^{+},\dots,A_{n}^{+}),(-1,\dots,-1))$
$\displaystyle=\,\,$
$\displaystyle\mathit{SHI}(-M_{T^{\prime}},-\gamma_{T^{\prime}},(-(A_{2}^{+}),\dots,-(A_{n}^{+})),(1,\dots,1))$
$\displaystyle=\,\,$
$\displaystyle\mathit{SHI}(-M_{T^{\prime}},-\gamma_{T^{\prime}},((-A_{2})^{-},\dots,(-A_{n})^{-}),(1,\dots,1)).$

By [Li19, Lemma 4.2], this last group is isomorphic to the sutured instanton
homology of the manifold obtained from
$(-M_{T^{\prime}},-\gamma_{T^{\prime}})$ by decomposing along
$(-A_{2})^{-}\cup\dots\cup(-A_{n})^{-}$. By [Li19, Lemma 3.2], this is the
same as the manifold obtained by decomposing along $-A_{2}\cup\dots\cup-
A_{n}$, which is, after reversing orientation, the manifold obtained from
$(M_{T^{\prime}},\gamma_{T^{\prime}})$ by decomposing along
$A_{2}\cup\dots\cup A_{n}$. It is straightforward to check that the latter
manifold is simply $(M_{T},\gamma_{T})$, as indicated in Figure 6 in the case
$n=2$. Thus,

(2.5)
$\mathit{SHI}(-M_{T^{\prime}},-\gamma_{T^{\prime}},((-A_{2})^{-},\dots,(-A_{n})^{-}),(1,\dots,1))\cong\mathit{SHI}(-M_{T},-\gamma_{T}).$

2pt $M_{T}$ at 210 105

$\mu_{1}$ at 52 51 $\mu_{2}$ at 112 77

$\gamma$ at 187 83 $A_{2}$ at 94 92

Figure 6. The result of decomposing $(M_{T^{\prime}},\gamma_{T^{\prime}})$
along $A_{2}\cup\dots\cup A_{n}$ is simply $(M_{T},\gamma_{T})$. This is
illustrated above in the case $n=2$.

Finally, combining (2.1)–(2.5), we have that

$\displaystyle\dim_{\mathbb{C}}\mathit{SHI}(-M_{T},-\gamma_{T})=$
$\displaystyle\dim_{\mathbb{C}}\mathit{SHI}(-M_{T^{\prime}},-\gamma_{T^{\prime}},((-A_{2})^{-},\dots,(-A_{n})^{-}),(1,\dots,1))$
$\displaystyle=$
$\displaystyle\dim_{\mathbb{C}}\mathit{SHI}(-M_{T^{\prime}},-\gamma_{T^{\prime}},(A_{2}^{-},\dots,A_{n}^{-}),(0,\dots,0))$
$\displaystyle\geq$
$\displaystyle\dim_{\mathbb{C}}\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m+1},(A_{2},\dots,A_{n}),(0,\dots,0))$
$\displaystyle-\dim_{\mathbb{C}}\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m},(A_{2},\dots,A_{n}),(0,\dots,0))$
$\displaystyle=$ $\displaystyle\dim_{\mathbb{C}}\mathit{SHI}(-M,-\gamma).$

Given the symmetry of $\mathit{SHI}$ under orientation reversal, this proves
Theorem 1.9. ∎

##  3\. Proof of Theorem 1.1

###  3.1. Full tangles

Let $(M,\gamma)$ be a balanced sutured manifold. Let

$\mathcal{H}=\big{(}\Sigma,\alpha=\\{\alpha_{1},\dots,\alpha_{k}\\},\beta=\\{\beta_{1},\dots,\beta_{k}\\}\big{)}$

be any (not necessarily admissible) sutured Heegaard diagram for $(M,\gamma)$.
This means that $M$ is obtained from $\Sigma\times[-1,1]$ by attaching
3-dimensional 2-handles

$\displaystyle\mathbb{D}_{\alpha_{i}}$ $\displaystyle=D_{\alpha_{i}}^{2}\times
I$ $\displaystyle\mathbb{D}_{\beta_{i}}$
$\displaystyle=D_{\beta_{i}}^{2}\times I$

along $A_{i}\times\\{-1\\}$ and $B_{i}\times\\{+1\\}$, where $A_{i}$ and
$B_{i}$ are annular neighborhoods of $\alpha_{i}$ and $\beta_{i}$,
respectively, for $i=1,\dots,k$. The suture $\gamma$ is given by

$\gamma=\partial\Sigma\times\\{0\\}.$

We will next define a special class of vertical tangles in $(M,\gamma)$
associated to $\mathcal{H}$.

Let $R_{1},\dots,R_{n}$ be the regions of $\Sigma-\alpha-\beta$ disjoint from
$\partial\Sigma$. For each $i=1,\dots,n$, let $p_{i1},\dots,p_{ia_{i}}$ be
$a_{i}$ distinct points in $R_{i}$, for some integer $a_{i}\geq 1$. Let

$T_{ij}=p_{ij}\times[-1,1]\subset\Sigma\times[-1,1],$

and let

$T=\bigcup_{i=1}^{n}\bigcup_{j=1}^{a_{i}}T_{ij}.$

Then $T$ is a vertical tangle in $(M,\gamma)$, oriented from $R_{+}(\gamma)$
to $R_{-}(\gamma)$. Let $D_{ij}^{2}$ be a tubular neighborhood of the point
$p_{ij}\in R_{i}$, let

$N_{ij}=D^{2}_{ij}\times[-1,1]$

be a tubular neighborhood of the component $T_{ij}$ in $M$, and let

$\gamma_{ij}=\partial D^{2}_{ij}\times\\{0\\}\subset\partial N_{ij}$

be a positively-oriented meridian of this component; see Figure 7. Let
$(M_{T},\gamma_{T})$ be the balanced sutured manifold obtained from $M$ by
removing these tubular neighborhoods,

$M_{T}=M-\bigcup_{i=1}^{n}\bigcup_{j=1}^{a_{i}}N_{ij},$

where $\gamma_{T}$ is the union of $\gamma$ with meridians of the $T_{ij}$,

$\gamma_{T}=\gamma\cup\bigcup_{i=1}^{n}\bigcup_{j=1}^{a_{i}}\gamma_{ij},$

as in §2. We refer to any tangle obtained in this way as a _full tangle for
$\mathcal{H}$_.

2pt $1$ at 10 85 $-1$ at 5 44 $T_{ij}$ at 136 74 $\gamma_{ij}$ at 348 74

Figure 7. Left, the component $T_{ij}=p_{ij}\times[-1,1]\subset M$. Right, the
complement of $N_{ij}$ with the meridian $\gamma_{ij}$ in red.

The main result of this section is the following.

######  Proposition 3.1.

If $T$ is a full tangle for $\mathcal{H}$, then
$\dim_{\mathbb{C}}\mathit{SHI}(M_{T},\gamma_{T})=|\mathfrak{S}(\mathcal{H})|.$

######  Remark 3.2.

The analogue of this proposition for $\mathit{SFH}$ is immediate. One forms a
sutured Heegaard diagram $\mathcal{H}_{T}$ for $(M_{T},\gamma_{T})$ from
$\mathcal{H}$ by removing neighborhoods of the $p_{ij}\in\Sigma$. Since there
is at least one such point in every region of $\Sigma-\alpha-\beta$ not
intersecting $\partial\Sigma$, this ensures that $\mathcal{H}_{T}$ is
admissible and that the differential on $\mathit{SFC}(\mathcal{H}_{T})$ is
zero, so that

$\textrm{rk}_{\mathbb{Z}}\mathit{SFH}(M_{T},\gamma_{T})=\textrm{rk}_{\mathbb{Z}}\mathit{SFC}(\mathcal{H}_{T})=|\mathfrak{S}(\mathcal{H}_{T})|=|\mathfrak{S}(\mathcal{H})|.$

This was the inspiration for our result above.

We will need the following for the proof of Proposition 3.1; see [GL19,
Corollary 4.3].

######  Proposition 3.3.

Suppose $(M,\gamma)$ is a balanced sutured manifold and $D\subset M$ is a
properly embedded disk which intersects $\gamma$ in four points. Then

$\mathit{SHI}(M,\gamma)\cong\mathit{SHI}(M^{\prime},\gamma^{\prime})\oplus\mathit{SHI}(M^{\prime\prime},\gamma^{\prime\prime}),$

where $(M^{\prime},\gamma^{\prime})$ and
$(M^{\prime\prime},\gamma^{\prime\prime})$ are the decompositions of
$(M,\gamma)$ along $D$ and $-D$, respectively.

###### Proof.

Ghosh and Li prove this in [GL19, Corollary 4.3] under the assumption that
$(M,\gamma)$ is taut and at least one of $(M^{\prime},\gamma^{\prime})$ and
$(M^{\prime\prime},\gamma^{\prime\prime})$ is taut. We will show that this
additional assumption is unnecessary, by showing that the proposition still
holds when the assumption is not true.

First, suppose $(M,\gamma)$ is not taut. Then neither
$(M^{\prime},\gamma^{\prime})$ nor $(M^{\prime\prime},\gamma^{\prime\prime})$
is taut, by [Gab87, Lemma 0.4]. In this case, we have

$\mathit{SHI}(M,\gamma)\cong\mathit{SHI}(M^{\prime},\gamma^{\prime})\cong\mathit{SHI}(M^{\prime\prime},\gamma^{\prime\prime})=0,$

and the proposition holds.

2pt $D$ at 6 71 $(M,\gamma)$ at 150 18 $(M,\gamma_{1}^{\prime})$ at 246 18
$(M,\gamma_{1}^{\prime\prime})$ at 340 18

Figure 8. Left, a neighborhood of $D\subset M$; the suture $\gamma$ is shown
in red. Second to left, the arc of attachment for the initial bypass in the
triangle in green.

Next, suppose neither $(M^{\prime},\gamma^{\prime})$ nor
$(M^{\prime\prime},\gamma^{\prime\prime})$ is taut. Consider the bypass exact
triangle of [BS18]:

$\textstyle{\mathit{SHI}(-M,-\gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathit{SHI}(-M,-\gamma^{\prime}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathit{SHI}(-M,-\gamma^{\prime\prime}_{1}),\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

determined by an initial bypass attachment to $(M,\gamma)$ along an arc in
$\partial D$ as shown in Figure 8. The other manifolds
$(M,\gamma^{\prime}_{1})$ and $(M,\gamma^{\prime\prime}_{1})$ in the triangle
product disk decompose (along a copy of $D$ which intersects the new sutures
in two points) to $(M^{\prime},\gamma^{\prime})$ and
$(M^{\prime\prime},\gamma^{\prime\prime})$, respectively. Therefore, since
$\mathit{SHI}$ is invariant under product disk decomposition, we have

$\displaystyle\mathit{SHI}(M,\gamma^{\prime}_{1})$
$\displaystyle\cong\mathit{SHI}(M^{\prime},\gamma^{\prime})\cong 0,$
$\displaystyle\mathit{SHI}(M,\gamma^{\prime\prime}_{1})$
$\displaystyle\cong\mathit{SHI}(M^{\prime\prime},\gamma^{\prime\prime})\cong
0.$

It then follows from the bypass triangle, and the symmetry of $\mathit{SHI}$
under orientation reversal, that $\mathit{SHI}(M,\gamma)=0$ as well, so the
proposition holds. ∎

###### Proof of Proposition 3.1.

Recall that $D_{ij}^{2}$ denotes a tubular neighborhood of $p_{ij}\in R_{i}$.
There exists a (possibly empty) set of disjoint, properly embedded arcs

$d_{1},\dots,d_{m}\subset\Sigma-\bigcup_{i=1}^{n}\bigcup_{j=1}^{a_{i}}D_{ij}$

which satisfy the following three conditions:

  1. (1)

for every $e=1,\dots,m$, the arc $d_{e}$ is contained in some region of
$\Sigma-\alpha-\beta$,

  2. (2)

for every $e$, either both endpoints of $d_{e}$ are on $\partial\Sigma$, or
each is on some $\partial D_{ij}^{2},$ and

  3. (3)

$\Sigma-\bigcup_{i=1}^{n}\bigcup_{j=1}^{a_{i}}D_{ij}-d_{1}-\dots-d_{m}$
deformation retracts onto $\alpha\cup\beta$.

Now consider the disk

$\delta_{e}=d_{e}\times[-1,1]\subset\Big{(}\Sigma-\bigcup_{i=1}^{n}\bigcup_{j=1}^{a_{i}}D_{ij}\Big{)}\times[-1,1]\subset\partial
M_{T},$

for $e=1,\dots,m$. The boundary of each $\delta_{e}$ intersects $\gamma_{T}$
in two points; hence, $\delta_{e}$ is a product disk. Since $\mathit{SHI}$ is
invariant under product disk decomposition, let $(M_{T},\gamma_{T})$
henceforth refer to the balanced sutured manifold obtained after decomposing
along $\delta_{1},\dots,\delta_{m}$.

$M_{T}$ then admits the following description. Let $q_{1},\dots,q_{t}$ denote
the intersection points between the $\alpha$ and $\beta$ curves. If $q_{\ell}$
is an intersection point between $\alpha_{i}$ and $\beta_{j}$, let
$r_{\ell}\subset\Sigma$ denote the rectangular component of $A_{i}\cap B_{j}$
which contains $q_{\ell}$. Then $M_{T}$ is the given by the union

$M_{T}=\mathbb{D}_{\alpha_{1}}\cup\dots\cup\mathbb{D}_{\alpha_{k}}\cup\mathbb{D}_{\beta_{1}}\cup\dots\cup\mathbb{D}_{\beta_{k}}\cup\tau_{1}\cup\dots\cup\tau_{t}$

of the usual 2-handles with the _tubes_

$\tau_{\ell}=r_{\ell}\times[-1,1],$

as shown in Figure 9. Let $c_{\ell}$ denote the union of the four corners of
the rectangle $r_{\ell}$. Then the suture $\gamma_{T}$ is given by

$\gamma_{T}=\bigcup_{i=1}^{k}(\partial D_{\alpha_{i}}^{2}\times\partial
I)\cup\bigcup_{i=1}^{k}(\partial D_{\beta_{i}}^{2}\times\partial
I)\cup\bigcup_{\ell=1}^{t}(c_{\ell}\times[-1,1])-\bigcup_{\ell=1}^{t}(\partial
r_{\ell}\times\\{-1,1\\})$

as shown and oriented in the figure near a tube $\tau_{\ell}$. Let

$m_{\ell}=r_{\ell}\times\\{0\\}\subset\tau_{\ell}$

denote the meridional disk of $\tau_{\ell}$, oriented as in Figure 9, for
$\ell=1,\dots,t$. Note that the boundary of each $m_{\ell}$ intersects the
suture $\gamma_{T}$ in four points.

2pt $\alpha_{i}$ at 85 274 $\beta_{j}$ at 39 234 $\Sigma$ at 85 214
$\mathbb{D}_{\beta_{i}}$ at 308 290 $\mathbb{D}_{\alpha_{i}}$ at 308 198
$m_{\ell}$ at 328 242 $\epsilon_{\ell}=-$ at 115 30 $\epsilon_{\ell}=+$ at 310
30

Figure 9. Top left, an intersection point
$q_{\ell}\in\alpha_{i}\cap\beta_{j}$. The rectangular component $r_{\ell}$ of
$A_{i}\cap B_{j}$ containing $q_{\ell}$ is shown in darker gray. Top right,
the 2-handles $\mathbb{D}_{\alpha_{i}}$ and $\mathbb{D}_{\beta_{j}}$ glued
together by the tube $\tau_{\ell}$. The suture $\gamma_{T}$ is shown in red.
The meridional disk $m_{\ell}$ is shown in gray; its oriented normal points
upwards. Bottom left, the result of decomposing along $-m_{\ell}$. Bottom
right, the result of decomposing along $m_{\ell}$.

For each $t$-tuple of signs

$I=(\epsilon_{1},\dots,\epsilon_{t})\in\\{+,-\\}^{t},$

let $(M_{T}^{I},\gamma_{T}^{I})$ be the sutured manifold obtained by
decomposing $(M_{T},\gamma_{T})$ along the disks

$\epsilon_{1}m_{1}\cup\dots\cup\epsilon_{t}m_{t}.$

Then

$\mathit{SHI}(M_{T},\gamma_{T})\cong\bigoplus_{I\in\\{+,-\\}^{t}}\mathit{SHI}(M_{T}^{I},\gamma_{T}^{I}),$

by Proposition 3.3. Each $(M_{T}^{I},\gamma_{T}^{I})$ is simply a union of
3-balls

$M_{T}^{I}\cong\mathbb{D}_{\alpha_{1}}\cup\dots\cup\mathbb{D}_{\alpha_{k}}\cup\mathbb{D}_{\beta_{1}}\cup\dots\cup\mathbb{D}_{\beta_{k}},$

which means that $\mathit{SHI}(M_{T}^{I},\gamma_{T}^{I})$ is either
$\mathbb{C}$ or trivial, according as whether $\gamma_{T}^{I}$ has exactly one
component on the boundary of each of these 3-balls or not. We claim that the
nonzero summands $\mathit{SHI}(M_{T}^{I},\gamma_{T}^{I})$ are in one-to-one
correspondence with the elements of $\mathfrak{S}(\mathcal{H})$, which will
then complete the proof.

For this claim, we consider the restriction of $\gamma_{T}^{I}$ to the ball
$\mathbb{D}_{\alpha_{i}}$. Let

$q_{\ell_{1}},\dots,q_{\ell_{p}}\in\\{q_{1},\dots,q_{t}\\}$

denote the intersection points between $\alpha_{i}$ and $\beta$. Then
$\gamma_{T}^{I}$ restricts to exactly one component on the boundary of
$\mathbb{D}_{\alpha_{i}}$ iff exactly one of
$\epsilon_{\ell_{1}},\dots\epsilon_{\ell_{p}}$ is $-$ and the rest are $+$.
The analogous statement holds for the restriction of $\gamma_{T}^{I}$ to
$\mathbb{D}_{\beta_{j}}$. Thus, if we let

$q(I)=\\{q_{\ell}\mid\epsilon_{\ell}=-\\},$

then $\gamma_{T}^{I}$ restricts to exactly one component on each 3-ball in
$M_{T}^{I}$ iff $q(I)\in\mathfrak{S}(\mathcal{H})$. ∎

######  Remark 3.4.

Proposition 3.1 also follows from the fact that $\mathit{SHI}$ and
$\mathit{SFH}$ obey the same decomposition laws (Proposition 3.3, and the
invariance under product disk decomposition), agree in rank for sutured
3-balls, and

$\textrm{rk}_{\mathbb{Z}}\mathit{SFH}(M_{T},\gamma_{T})=|\mathfrak{S}(\mathcal{H})|,$

per Remark 3.2. Our original proof has the advantages that it does not rely on
the definition of the differential in $\mathit{SFC}$, and it establishes a
very concrete bijection between the nonzero summands
$\mathit{SHI}(M_{T}^{I},\gamma_{T}^{I})$ and elements of
$\mathfrak{S}(\mathcal{H})$.

###  3.2. The proof

Recall that a sutured Heegaard diagram $\mathcal{H}$ for a balanced sutured
manifold $(M,\gamma)$ is _admissible_ iff every nontrivial periodic domain has
both positive and negative multiplicities [Juh06]. This is automatically true
of any $\mathcal{H}$ when $H_{1}(M,\partial M;\mathbb{Q})=0$ (there are no
nontrivial periodic domains in this case), though every balanced sutured
manifold admits an admissible diagram.

###### Proof of Theorem 1.1.

Let $\mathcal{H}=(\Sigma,\alpha,\beta)$ be an admissible sutured Heegaard
diagram for $(M,\gamma)$. Then we can assign a positive integer area $a_{i}$
to each region $R_{i}$ of $\Sigma-\alpha-\beta$ disjoint from
$\partial\Sigma$, so that the signed area of every periodic domain is zero;
see [OS04, Lemma 4.2].555Ozsváth and Szabó state this for real-valued areas,
but proof shows the same is true for integer areas. Fix $a_{i}$ distinct
points $p_{i1},\dots,p_{ia_{i}}\in R_{i}$ for each $i$, and let

$T=\bigcup_{i=1}^{n}\bigcup_{j=1}^{a_{i}}T_{ij}\subset M$

be the corresponding full tangle for $\mathcal{H}$, as in §3.1.

We claim that $[T]=0$ in $H_{1}(M,\partial M;\mathbb{Q})$. To see this, note
that for every periodic domain $P$ of $\mathcal{H}$, the intersection number
of $T$ with the 2-cycle in $M$ represented by $P$ is negative the signed area
of $P$, which is zero. Since the homology classes represented by periodic
domains span $H_{2}(M)$, the claim follows. Theorem 1.9 therefore implies that

$\dim_{\mathbb{C}}\mathit{SHI}(M,\gamma)\leq\dim_{\mathbb{C}}\mathit{SHI}(M_{T},\gamma_{T}).$

Theorem 1.1 then follows from the fact that

$\dim_{\mathbb{C}}\mathit{SHI}(M_{T},\gamma_{T})=|\mathfrak{S}(\mathcal{H})|,$

by Proposition 3.1. ∎

######  Remark 3.5.

It is not true that the inequality

$\dim_{\mathbb{C}}\mathit{SHI}(M,\gamma)\leq|\mathfrak{S}(\mathcal{H})|$

holds for _any_ sutured Heegaard diagram $\mathcal{H}$ for an arbitrary
balanced sutured $(M,\gamma)$. For example, consider the diagram
$\mathcal{H}=(T^{2}-D^{2},\alpha_{1},\beta_{1})$ for

$(M,\gamma)=((S^{1}\times S^{2})(1),\delta)$

in which $\alpha_{1}$ and $\beta_{1}$ are disjoint curves on the punctured
torus. In this case, we know that

$\dim_{\mathbb{C}}\mathit{SHI}(M,\gamma)=2,$

while $|\mathfrak{S}(\mathcal{H})|=0$. The issue here is that $\mathcal{H}$ is
not admissible.

##  4\. Further directions

Let $\mathcal{H}$ be an admissible sutured Heegaard diagram for a balanced
sutured manifold $(M,\gamma)$. Let $T$ be a full tangle for $\mathcal{H}$, as
defined in §3. In ongoing work, we prove that

$\mathit{SHI}(-M_{T},-\gamma_{T})\cong\mathbb{C}^{|\mathfrak{S}(\mathcal{H})|}$

has a basis given by the contact invariants of the tight contact structures on
$(M_{T},\gamma_{T})$. In particular, this sutured instanton homology group is
naturally graded by homotopy classes of 2-plane fields. We discuss potential
applications of this fact below.

Let $T^{\prime}$ be a mixed tangle for $T$, as defined in §2. Let $V_{T}$ and
$V_{m}$ be the groups

$\displaystyle V_{T}$
$\displaystyle=\mathit{SHI}(-M_{T^{\prime}},-\gamma_{T^{\prime}},(A_{2}^{-},\dots,A_{n}^{-}),(0,\dots,0)),$
$\displaystyle V_{m}$
$\displaystyle=\mathit{SHI}(-M_{T^{\prime}},-\Gamma_{m},(A_{2},\dots,A_{n}),(0,\dots,0))$

from §2, for $m\in\mathbb{N}$. To prove Theorem 1.1, we proved that
$V_{T}\cong\mathit{SHI}(-M_{T},-\gamma_{T})$ and

$\dim_{\mathbb{C}}\mathit{SHI}(-M,-\gamma)=\dim_{\mathbb{C}}V_{m+1}-\dim_{\mathbb{C}}V_{m}\leq\dim_{\mathbb{C}}V_{T}$

for sufficiently large $m$. But, in fact, this inequality can be viewed as
coming from a spectral sequence similar to that in [LY20, Section 4]. This
spectral sequence can also be described as follows. From the bypass exact
triangle used in §2, we have that

$V_{T}\cong H_{*}(\textrm{Cone}(\psi_{-}:V_{m}\to V_{m+1})),$

where $\psi_{-}$ is the map associated to the bypass attachment along the arc
$\eta_{-}$. One can prove, on the other hand, that

$\mathit{SHI}(-M,-\gamma)\cong H_{*}(\textrm{Cone}(\psi_{-}-\psi_{+}:V_{m}\to
V_{m+1}))$

for $m$ large, where $\psi_{+}$ is a related bypass attachment map. The groups
$V_{m}$ and $V_{m+1}$ can be graded using the rational Seifert surface for
$T^{\prime}$, as in [Li19]. After adjusting this grading by an overall shift,
the map $\psi_{-}$ is grading-preserving while $\psi_{+}$ decreases the
grading by $1$, for $m$ large. The complex $\textrm{Cone}(\psi_{-}-\psi_{+})$
is then filtered, and the $E_{1}$ page of the associated spectral sequence is
$H_{*}(\textrm{Cone}(\psi_{-}))$. In sum, we have a spectral sequence

(4.1) $\mathbb{C}^{|\mathfrak{S}(\mathcal{H})|}\cong V_{T}\cong
H_{*}(\textrm{Cone}(\psi_{-}))\implies
H_{*}(\textrm{Cone}(\psi_{-}-\psi_{+}))\cong\mathit{SHI}(-M,-\gamma).$

The first potential application of these ideas involves defining a grading on
$\mathit{SHI}(-M,-\gamma)$ by homotopy classes of plane fields. Indeed,
$\mathit{SHI}(-M_{T},-\gamma_{T})$ has such a grading, as mentioned above, as
it is generated by contact invariants of contact structures. The manifold
$(M_{T^{\prime}},\gamma_{T^{\prime}})$ is obtained by gluing
$(M_{T},\gamma_{T})$ along annuli, as in §2, and we believe that the tight
contact structures on the latter glue to give tight contact structures on the
former whose invariants form a basis for $V_{T}$. So, there should be a
natural grading by homotopy classes of 2-plane fields on $V_{T}$ as well. The
bypass maps $\psi_{-},\psi_{+}$ are natural from a contact-geometric
standpoint, and should therefore shift plane field gradings in a sensible way.
We expect that one can then use the relation between $V_{T}$ and
$\textrm{Cone}(\psi_{-})$ and the structure of the latter to define a plane
field grading on $\textrm{Cone}(\psi_{-})$, and then on
$\textrm{Cone}(\psi_{-}-\psi_{+})$.

A grading by homotopy classes of 2-plane fields on $\mathit{SHI}$ would enable
one to define $\textrm{Spin}^{c}$ decompositions of these groups, as well as
an analogue of the Maslov grading in Heegaard Floer homology (see [LY20,
Section 4] for another approach to such a decomposition). The current lack of
such structure makes it difficult to translate arguments from the Heegaard
Floer setting to the instanton Floer setting.

A related second application is towards proving the isomorphism (1.1). Indeed,
there is some hope that one could understand the spectral sequence (4.1)
purely in terms of contact geometry, and thereby obtain a more axiomatic proof
that

(4.2)
$\mathit{SHI}(M,\gamma)\cong\mathit{SFH}(M,\gamma)\otimes\mathbb{C}\cong\mathit{SHM}(M,\gamma)\otimes\mathbb{C},$

since the analogous spectral sequences can be defined in the Heegaard Floer
and monopole Floer settings by the same contact-geometric means.

A more pedestrian approach to (4.2) is the following: first, prove that one
can understand the spectral sequence

$\mathbb{C}^{|\mathfrak{S}(\mathcal{H})|}\implies\mathit{SHI}(-M,-\gamma)$

as coming from a differential

$\partial=\partial_{1}+\partial_{2}+\dots:\mathbb{C}^{|\mathfrak{S}(\mathcal{H})|}\to\mathbb{C}^{|\mathfrak{S}(\mathcal{H})|},$

where $\partial_{k}$ shifts a grading (coming from homotopy classes of 2-plane
fields) on $\mathbb{C}^{|\mathfrak{S}(\mathcal{H})|}\cong V_{T}$ by $k$, such
that

$\mathit{SHI}(-M,-\gamma)\cong
H_{*}(\mathbb{C}^{|\mathfrak{S}(\mathcal{H})|},\partial).$

Then, for generators $x,y\in\mathfrak{S}(\mathcal{H})$ and the corresponding
basis elements $e_{x},e_{y}\in\mathbb{C}^{|\mathfrak{S}(\mathcal{H})|}$,
perhaps one could use the 2-plane field gradings to show that the coefficient

(4.3) $\langle\partial e_{x},e_{y}\rangle$

is nonzero only if there is a homotopy class of Whitney disks

(4.4) $\varphi\in\pi_{2}(x,y)$

with positive domain in $\mathcal{H}$ and Maslov index one. If even this were
true, then one could prove, for example, that the inequality in Corollary 1.7
is an equality,

$\dim_{\mathbb{C}}\mathit{KHI}(L(p,q),K)\leq\textrm{rk}_{\mathbb{Z}}\widehat{\mathit{HFK}}(L(p,q),K)$

for $(1,1)$-knots $K\subset L(p,q)$. More generally, the hope would be that
for a _nice_ diagram $\mathcal{H}$ (the regions of $\Sigma-\alpha-\beta$
disjoint from $\partial\Sigma$ are bigons or rectangles), one could show that
the coefficient (4.3) is $\pm 1$ iff there is a class as in (4.4) with
positive domain and Maslov index one (the domain of such a class is
necessarily an embedded bigon or rectangle in this case). This would be enough
to prove (1.1) by [SW10], and then (4.2) by the same methods.

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