$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};\overrightarrow{\rho})$
diffeomorphic to $(0,1]$; the count of such ends is equal to
$\\#\mathcal{M}^{B}(\bm{x},\bm{y};-\rho_{1230})$ when
$\overrightarrow{\rho}=(-\rho_{0},-\rho_{123})$, and is equal to
$\\#\mathcal{M}^{B}(\bm{x},\bm{y};-\rho_{3012})$ when
$\overrightarrow{\rho}=(-\rho_{012},-\rho_{3})$.

By Proposition 2.62 and 2.57, boundary degenerations are further divided into
boundary degenerations with or without corners. In particular, different types
of degeneration do not appear simultaneously. When boundary degeneration
without corners appear, the situation is covered in Proposition 2.62. When
boundary degenerations with corners appear, Proposition 2.58 and Proposition
2.60 show the moduli space of degenerate disks are smoothly cut out. In
particular, the standard gluing results can be applied to show each boundary
degeneration in
$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};\overrightarrow{\rho})$ has a
neighborhood diffeomorphic to $(0,1]$. The number of such ends is equal to

$\sum_{\\{(q,B_{1})|\exists B_{2}\in
T(q),B_{1}+B_{2}=B\\}}\\#(\mathcal{M}^{B_{1}}(\bm{x},\bm{y};q)\times_{ev}\mathcal{N}^{B_{2}}(q;\overrightarrow{\rho}))$

(We have suppressed the almost complex structure $J_{s}$, which can be chosen
generically so that the evaluation maps are transversal to each other.) This
quantity is even when $\overrightarrow{\rho}\neq(-\rho_{0},\ldots,-\rho_{3})$
in view of Proposition 2.60. Otherwise, it has the same parity as

$\sum_{\\{(q,B_{1})|\exists B_{2}\in
T(q),B_{1}+B_{2}=B\\}}\\#\mathcal{M}^{B_{1}}(\bm{x},\bm{y};q)$

in view of Proposition 2.58. ∎

###  2.8. Ends of moduli spaces of 1-P curves

This subsection characterizes the ends of one-dimensional moduli spaces of 1-P
holomorphic curves. Given a generator $\bm{x}$, we say
$\iota(\bm{x})=\iota_{1}$ if and only if $\bm{x}$ is in
$\mathbb{T}_{\alpha,1}$; otherwise $\iota(\bm{x})=\iota_{0}$. The main result
is the following.

######  Proposition 2.63.

Let $B\in\tilde{\pi}_{2}(\bm{x},\bm{y})$ such that $\iota(\bm{x})=\iota_{1}$
and $\text{ind}(B;U)=2$. Then fixing a generic almost complex structure, the
compactified moduli space $\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};U)$ is a
compact 1-manifold with boundary. The boundaries are of the following types:

  * (1)

Two-story building

  * (2)

simple holomorphic combs $(u,v)$ with $v$ being an orbit curve

  * (3)

boundary degeneration with corners

  * (4)

boundary degeneration without corners

Moreover,

  * (a)

The number of type (2) ends is
$\\#\mathcal{M}^{B}(\bm{x},\bm{y};-\rho_{1230})+\\#\mathcal{M}^{B}(\bm{x},\bm{y};-\rho_{3012})$.

  * (b)

The number of type (3) ends is mod 2 congruent to

$\sum_{\\{(B_{1},\ q)|B_{2}\in T(q),\
B_{1}+B_{2}=B\\}}\\#\mathcal{M}^{B_{1}}(\bm{x},\bm{y};q)$

  * (c)

The number of type (4) ends is even.

######  Remark 2.64.

A similar proposition holds in the case when $\iota(\bm{x})=\iota_{0}$. One
simply needs to change the Reeb chords in (a) by a cyclic permutation of the
digits in the subscript.

####  2.8.1. Reformulation of the moduli spaces

We reformulate $\mathcal{M}^{B}(\bm{x},\bm{y};U)$ in terms of holomorphic
disks in $Sym^{g}(\Sigma)$. Assume $\iota(\bm{x})=\iota_{1}$ throughout the
rest of the section.

######  Definition 2.65.

${\mathcal{M}}^{B}_{Sym}(\bm{x},\bm{y};U)$ is defined to be the space of
holomorphic maps $u:[0,1]\times\mathbb{R}\backslash\\{(s_{0},0)\\}\rightarrow
Sym^{g}(\Sigma)$ such that:

  * (1)

$(s_{0},0)$ is in the interior of $[0,1]\times\mathbb{R}$ and is allowed to
vary;

  * (2)

$u(\\{0\\}\times\mathbb{R})\subset\mathbb{T}_{\beta}$;

  * (3)

$u(\\{1\\}\times\mathbb{R})\subset\mathbb{T}_{\alpha,1}$. Moreover,
$u|_{\\{1\\}\times\mathbb{R}}$ lifts through
$f_{1}:(0,1)\times\mathbb{T}^{g-2}\times(\amalg S^{1})\rightarrow
Sym^{g}(\Sigma)$;

  * (4)

$\lim_{t\rightarrow\infty}u(s+it)=\bm{y}$, and
$\lim_{t\rightarrow-\infty}u(s+it)=\bm{x}$;

  * (5)

$\lim_{(s,t)\rightarrow(s_{0},0)}u(s+it)$ is a closed
$\overrightarrow{R}$-orbit $\sigma\times\bm{w}$, where $\bm{w}\in
Sym^{g-1}(\Sigma)$ and $\sigma$ stands for a closed Reeb orbit that traverses
$\partial\overline{\Sigma}$ once;

  * (6)

$\frac{du}{ds}+J_{s}\frac{du}{dt}=0$;

  * (7)

$u$ is in the homology class specified by $B$.

Again, we have the tautological correspondence that identifies the moduli
spaces defined here and the ones in Section 2.2. Therefore, we shall no longer
keep the subscript “Sym” in the notation.

We also define the moduli spaces of one-punctured degenerate disks (with or
without corners).

######  Definition 2.66 (One-punctured degenerate disks without corners).

Let $J$ be a nearly-symmetric almost complex structure. Let
$\bm{x}\in\mathbb{T}_{\alpha}$. $\mathcal{N}_{J}(\bm{x};U)$ is the space of
maps $v:\mathbb{H}\backslash\\{i\\}\rightarrow Sym^{g}(\Sigma)$ such that

  * (1)

$v(\mathbb{R})\subset\mathbb{T}_{\alpha,1}$. Moreover, the restriction of
$v|_{\mathbb{R}}$ lifts through
$f_{1}:(0,1)\times\mathbb{T}^{g-2}\times(\amalg S^{1})\rightarrow
Sym^{g}(\Sigma)$;

  * (2)

$\lim_{z\rightarrow\infty}v(z)=\bm{x}$, and the path obtained from
$v|_{\partial\mathbb{H}}$ by continuous extension at $\infty$ lifts through
$\iota_{1}$;

  * (3)

$\lim_{z\rightarrow i}v(z)$ is some closed $\overrightarrow{R}$-orbit
$\sigma\times\bm{w}$, where $\bm{w}\in Sym^{g-1}(\Sigma)$ and $\sigma$ stands
for a closed Reeb orbit that traverses the $\partial\overline{\Sigma}$ once;

  * (4)

$\frac{du}{ds}+J\frac{du}{dt}=0$.

######  Definition 2.67 (One-cornered one-punctured degenerate disks).

Let $J$ be a nearly-symmetric almost complex structure. Let $q$ be a self-
intersection point of $\alpha_{im}$. $\mathcal{N}_{J}(q;U)$ is the space of
maps $v:\mathbb{H}\backslash\\{i\\}\rightarrow Sym^{g}(\Sigma)$ such that

  * (1)

$v(\mathbb{R})\subset\mathbb{T}_{\alpha,1}$. Moreover, the restriction of
$v|_{\mathbb{R}}$ lifts through
$f_{1}:(0,1)\times\mathbb{T}^{g-1}\times(\amalg S^{1})\rightarrow
Sym^{g}(\Sigma)$;

  * (2)

$\lim_{z\rightarrow\infty}v(z)=(q,\bm{p})$ for some
$p\in\alpha^{a}_{1}\times\alpha^{c}_{1}\times\cdots\times\alpha^{c}_{g-2}$,
and and the path obtained from $v|_{\partial\mathbb{H}}$ by continuous
extension at $\infty$ does not lift through $\iota_{1}$;

  * (3)

$\lim_{z\rightarrow i}v(z)$ is some closed $\overrightarrow{R}$-orbit
$\sigma\times\bm{w}$, where $\bm{w}\in Sym^{g-1}(\Sigma)$ and $\sigma$ stands
for a closed Reeb orbit that traverses $\partial\overline{\Sigma}$ once;

  * (4)

$\frac{du}{ds}+J\frac{du}{dt}=0$.

We call $q$ the corner of such a degenerate disk. We also have an evaluation
map
$ev_{J}:\mathcal{N}_{J}(q;U)\rightarrow\alpha^{a}_{i}\times\alpha^{c}_{1}\times\cdots\times\alpha_{g-2}^{c}$
defined by $v\mapsto\bm{p}$ if $\lim_{z\rightarrow\infty}v(z)=(q,\bm{p})$.

####  2.8.2. One-punctured boundary degeneration with corners

######  Proposition 2.68.

If a boundary degeneration with corners appears in the compactification of a
one-dimensional moduli space $\mathcal{M}^{B}(\bm{x},\bm{y};U)$, then the
nodal comb is of simple form, and the domain for the degenerate disk is a
stabilized teardrop with an acute corner.

###### Proof.

The proof is similar to the proof of Proposition 2.57. There is only one
modification needed: We no longer have east boundary punctures when
considering the bubble tree $\mathbb{B}$ of the nodal curve; instead, there is
one and only one interior puncture. With this, the rest of the proof follows
exactly as in Proposition 2.57. ∎

######  Proposition 2.69.

Let $q$ be a self-intersection point of $\alpha_{im}$ and let $B\in T(q)$ be a
stabilized teardrop with acute corner. For a generic nearly symmetric almost
complex structure $J$, the moduli space of degenerate disks
$\mathcal{N}_{J}^{B}(q;U)$ is a $(g-1)$-manifold, and a generic fiber of the
evaluation map
$ev_{J}:\mathcal{N}_{J}^{B}(q;U)\rightarrow\alpha^{a}_{1}\times\alpha^{c}_{1}\times\cdots\times\alpha_{g-2}^{c}$
is a compact 0-dimensional manifold consisting of an odd number of points.

###### Proof.

The regularity of $\mathcal{N}_{J}^{B}(q;U)$ and compactness of a generic
fiber are proved in the same way as in Proposition 2.69.

The parity of the cardinality of a generic fiber follows from a similar neck-
stretching and cobordism argument as in Proposition 2.69, using Lemma 2.70
below instead of Lemma 2.59. ∎

######  Lemma 2.70.

Assume $g(\Sigma)=2$. Fix some point $p\in\alpha^{a}_{1}$. For a sufficiently
stretched almost complex structure $j$ on $\Sigma$, the fiber
$ev_{Sym^{2}(j)}^{-1}(p)$ is transversally cut out and consists of one point.

###### Proof.

View $\Sigma$ be the connected sum $(E_{1},\alpha^{a}_{1},\alpha^{a}_{2})$ and
$(E_{2},\alpha_{im})$, where $E_{1}$ is the punctured Riemann surface of genus
one and $E_{2}$ is a closed Riemann surface of genus one. Let $z^{\prime}$
denote the points on $E_{1}$ and $E_{2}$ where the connected sum is performed.
The domain $B$ gives rise to a teardrop domain $B^{\prime}$ in $E_{2}$ with
$n_{z^{\prime}}(B^{\prime})=1$. The Riemann mapping theorem implies that the
moduli space $\mathcal{N}^{B^{\prime}}(q)$ of holomorphic disks in $E_{2}$
with corner $q$ and domain $B^{\prime}$ is smoothly cut out and has only one
element. The gluing argument in Section 10 of [OS04b] shows for a sufficiently
stretched almost complex structure, maps in $ev_{Sym^{2}(j)}^{-1}(p)$ are
obtained by splicing the one-punctured holomorphic sphere in $Sym^{2}(E_{1})$
passing through $(z^{\prime},p)$ and the holomorphic disk in
$\mathcal{N}^{B^{\prime}}(q)$.666Strictly speaking, Ozsváth and Szabó’s
argument concerns splicing closed holomorphic spheres while in our case the
sphere is punctured, but this does not affect the argument applying to our
case. Alternatively, we may treat one-punctured holomorophic disks or spheres
as the corresponding object without interior punctures, but intersecting
$\\{e\\}\times\Sigma_{\bar{e}}$ once in $Sym^{2}(\Sigma_{\bar{e}})$. In
particular, $ev_{Sym^{2}(j)}^{-1}(p)$ is identified with
$\mathcal{N}^{B^{\prime}}(q)$ and hence consists of only one element. ∎

####  2.8.3. One-punctured boundary degeneration without corners

######  Proposition 2.71.

In the assumption of Proposition 2.63, if a boundary degeneration without
corner occurs, then:

  * (1)

There is only one degenerate disk, and its domain $[B]$ is $[\Sigma]$.

  * (2)

$\bm{x}=\bm{y}$.

  * (3)

Such degenerate disk do not occur simultaneously with other types of
degeneration.

  * (4)

The number of ends corresponding to such boundary degeneration is even.

###### Proof.

The proof of (1), (2), and (3) are straightforward modifications of that of
Proposition 2.62 and are omitted. (4) follows from the standard gluing result
and Proposition 2.72 below, which differs from the counterpart in the 0-P
case. ∎

######  Proposition 2.72.

For a generic almost complex structure $J$,
$\mathcal{N}^{[\Sigma]}_{J}(\bm{x};U)$ is a compact, 0-dimensional manifold
that consists of an even number of points.

###### Proof.

The argument for compactness and transversality is the same as in [OS04b,
Proposition 3.14], which is the counterpart of Proposition 2.72 when the
Heegaard surface is closed; we will omit this part. By a similar cobordism
argument used in Proposition 2.58, we can reduce understanding the parity of
the moduli space to the base case $g(\Sigma)=2$, which is addressed in Lemma
2.73 below. ∎

######  Lemma 2.73.

Assume $g(\Sigma)=2$. View
$(\Sigma,\alpha_{1}^{a},\alpha^{a}_{2},\alpha_{im})=(E_{1},\alpha_{1}^{a},\alpha_{2}^{a})\\#(E_{2},\alpha_{im})$,
where $E_{1}$ is a punctured Riemann surface of genus one and $E_{2}$ is a
Riemann surface of genus one. If $j$ is a sufficiently stretched complex
structure on $\Sigma$, then $\mathcal{N}^{[\Sigma]}_{Sym^{2}(j)}(\bm{x};U)$ is
empty.

###### Proof.

Otherwise, the same neck-stretching procedure as in Lemma 2.61 produces a
limit nodal holomorphic curve $u_{\infty}:\mathbb{B}\rightarrow
Sym^{2}(E_{1}\vee E_{2})$. It consists of a (possibly punctured) holomorphic
disk $v$ that maps to $E_{1}\times E_{2}$ with boundary in
$\mathbb{T}_{\alpha,1}$ and possibly some (possibly punctured) sphere bubbles
in $Sym^{2}(E_{i})$, $i=1,2$. We claim $v$ must be a constant map. It is clear
that $Pr_{E_{1}}\circ v$ is constant, for
$\pi_{2}(E_{1,\bar{e}},\alpha^{a}_{1}\cup\\{e\\})=0$, where $E_{1,\bar{e}}$
denote the Riemann surface obtained by filling in the east puncture. We move
to see $Pr_{E_{2}}\circ v$ is constant. Suppose $Pr_{E_{2}}\circ v$ is not a
constant map. Note the domain of $(Pr_{E_{2}}\circ v)$ is a zero-cornered
$\alpha$-bounded domain $D$ in $E_{2}$. Stabilizing by $E_{1}$, this domain
induces a zero-cornered $\alpha$-bounded domain $D^{\prime}$ in $\Sigma$ with
$n_{z}(D^{\prime})\leq 1$. If $n_{z}(D^{\prime})=0$, then $D^{\prime}$ does
not exist as $\mathcal{H}$ is unobstructed, and hence $D$ does not exist. So
$n_{z}(D^{\prime})=1$, and hence $D^{\prime}=\Sigma$ since $\mathcal{H}$ is
unobstructed. This implies $D=E_{2}$. Therefore, $\partial(Pr_{E_{2}}\circ v)$
is null-homotopic in $\alpha_{im}$. So $Pr_{E_{2}}\circ v$ induces a
nontrivial element in $\pi_{2}(E_{2})$. This, however, contradicts that
$\pi_{2}(E_{2})=0$. Therefore, $Pr_{E_{2}}\circ v$ is also constant, and hence
$v$ is the constant map with image $\bm{x}$. Now $\\{\bm{x}\\}$ intersects
neither $Sym^{2}(E_{i})$, $i=1,2$, and hence there are no sphere bubbles in
$u_{\infty}$. So the Gromov limit $u_{\infty}$ is a constant map. In
particular, $n_{z}(u_{\infty})=0$. However, $n_{z}(u_{\infty})=1$ as it is the
limit of a sequence of holomorphic maps whose multiplicity at $z$ is one. This
is a contradiction. Therefore, $\mathcal{N}^{[\Sigma]}_{Sym^{2}(j)}(\bm{x};U)$
is empty provided $j$ is sufficiently stretched. ∎

####  2.8.4. Proof of Proposition 2.63

###### Proof of Proposition 2.63.

In view of Proposition 2.40, Proposition 2.68, and Proposition 2.71 we know
the degenerations that can appear in the boundary of the compactified moduli
spaces are two-story curves, simple combs with orbit curve ends, or simple
boundary degenerations with or without corners. In all cases, gluing arguments
can be applied to see the compactified moduli space
$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};U)$ is a one-manifold with boundary.

For conclusion (a), note that ends of type (2) correspond to pairs of curves
$(u,v)$ where $u$ is in $\mathcal{M}^{B}(\bm{x},\bm{y};-\rho_{1230})$ or
$\mathcal{M}^{B}(\bm{x},\bm{y};-\rho_{3012})$ and $v$ is an orbit curve, but
the moduli space of orbit curves consists of a single element by the Riemann
mapping theorem so the count of type (2) boundaries agrees with
$\\#\mathcal{M}^{B}(\bm{x},\bm{y};-\rho_{1230})+\\#\mathcal{M}^{B}(\bm{x},\bm{y};-\rho_{3012})$.
For conclusion (b), standard gluing results imply that the number of such ends
is equal to

$\sum_{\\{(q,B_{1})|\exists B_{2}\in
T(q),B_{1}+B_{2}=B\\}}\\#(\mathcal{M}^{B_{1}}(\bm{x},\bm{y};q)\times_{ev}\mathcal{N}^{B_{2}}(q;U))$

This is mod 2 equal to

$\sum_{\\{(q,B_{1})|\exists B_{2}\in
T(q),B_{1}+B_{2}=B\\}}\\#\mathcal{M}^{B_{1}}(\bm{x},\bm{y};q)$

as a generic fiber of $ev$ in $\mathcal{N}^{B_{2}}(q;U)$ is odd by Proposition
2.69. For (c), note by gluing results the number of such ends is equal to
$\\#\mathcal{N}^{[\Sigma]}(\bm{x};U)$. This is even by Proposition 2.72. ∎

###  2.9. Type D structures

We define type D structures from an immersed bordered Heeggard diagram
$\mathcal{H}=(\Sigma,\bm{\beta},\bm{\bar{\alpha}},z)$ in this subsection.

Figure 7. The quiver presentation of the torus algebra (left) and the pointed
match circle of $\mathcal{H}$ with reversed boundary orientation (right).

Let $\mathcal{A}$ denote the torus algebra, which is isomorphic to the quiver
algebra of the quiver in Figure 7 (left). For $I\in\\{1,2,3,12,23,123\\}$,
$\rho_{I}\in\mathcal{A}$ is understood as the product of the $\rho_{i}$’s for
those $i$ appear in $I$. This algebra arises naturally in the context of
bordered Heegaard diagrams, where $\mathcal{A}$ is associated to the pointed
match circle determined by $\mathcal{H}$ with the reversed boundary
orientation (Figure 7 (right)); we refer the readers to [LOT18, Chapter 11.1]
for a detailed definition of the torus algebra in terms of pointed match
circles, and we only point out that the element $\rho_{I}\in\mathcal{A}$ for
$I\in\\{1,2,3,12,23,123\\}$ corresponds to the Reeb chord with the same label
on the pointed match circle. Let
$\mathcal{I}=\langle\iota_{0}\rangle\oplus\langle\iota_{1}\rangle$ denote the
ring of idempotents of $\mathcal{A}$. We recall the definition of a type D
structure.

######  Definition 2.74.

A type D structure over the torus algebra $\mathcal{A}$ is a left
$\mathcal{I}$-module $N$ together with a linear map
$\delta:N\rightarrow\mathcal{A}\otimes N$ such that the map

$\partial\coloneqq(\mu_{\mathcal{A}}\otimes\mathbb{I}_{N})\circ(\mathbb{I}_{\mathcal{A}}\otimes\delta):\mathcal{A}\otimes
N\rightarrow\mathcal{A}\otimes N$

is a differential, i.e., $\partial^{2}=0$. The left differential
$\mathcal{A}$-module $\mathcal{A}\otimes N$ is called the type D module of the
type D structure $(N,\delta)$.

Next, we spell out the construction of a type D structure from an immersed
bordered Heegaard diagram. Recall
$\mathbb{T}_{\beta}=\beta_{1}\times\cdots\times\beta_{g}$ and
$\mathbb{T}_{\alpha,i}=\alpha^{a}_{i}\times\alpha_{1}^{c}\times\cdots\times\alpha_{g-1}^{c}$,
$i=1,2$. Let
$\mathbb{T}_{\alpha}=\mathbb{T}_{\alpha,1}\cup\mathbb{T}_{\alpha,2}$. Let
$\mathcal{G}(\mathcal{H})=\\{\bm{x}|\bm{x}\in\mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta}\\}$.
Denote the local system on $\alpha_{im}$ as a vector bundle
$\mathcal{E}\rightarrow\alpha_{im}$ together with a parallel transport $\Phi$.
Note that this induces a local system on $\mathbb{T}_{\alpha}$, the tensor
product of $\mathcal{E}$ and the trivial local system on the other alpha
curves (or arcs). Abusing notation, we still denote the local system on
$\mathbb{T}_{\alpha}$ by $(\mathcal{E},\Phi)$. Now define an
$\mathcal{I}$-module
$X^{\mathcal{E}}(\mathcal{H})=\oplus_{\bm{x}\in\mathcal{G}(\mathcal{H})}\mathcal{E}|_{\bm{x}}$,
where the $\mathcal{I}$-action on an element $\eta\in\mathcal{E}|_{\bm{x}}$ is
specified by

$\iota_{i}\cdot\eta=\begin{cases}\eta,\ o(\bm{x})\equiv i\pmod{2}\\\ 0,\
\text{otherwise}\end{cases}$

Here $o(\bm{x})=i$ if and only if $\bm{x}\in\mathbb{T}_{\alpha,i}$, $i=1,2$.

Given a sequence of Reeb chords
$\overrightarrow{\sigma}=(\sigma_{1},\ldots,\sigma_{k})$ of a pointed match
circle $\mathcal{Z}$, $a(-\overrightarrow{\sigma})$ is defined to be
$(-\sigma_{1})\cdot(-\sigma_{2})\cdot\ldots(-\sigma_{k})\in\mathcal{A}(-\mathcal{Z})$.
Note given $B\in\pi_{2}(\bm{x},\bm{y})$, the parallel transport restricted to
the arc $\partial_{\alpha_{im}}B\subset\alpha_{im}$ induces an isomorphism
from $\mathcal{E}|_{\bm{x}}$ to $\mathcal{E}|_{\bm{y}}$, which we denote by
$\Phi^{B}_{\bm{x},\bm{y}}$.

######  Definition 2.75.

Let $\mathcal{H}$ be an unobstructed, provincially admissible, immersed
bordered Heegaard diagram. Fix a generic almost complex structure on
$\Sigma\times[0,1]\times\mathbb{R}$. The type D module
$\widehat{CFD}(\mathcal{H})$ is defined to be the $\mathcal{A}$-module

$\mathcal{A}\otimes_{\mathcal{I}}X^{\mathcal{E}}(\mathcal{H})$

together with a differential given by

$\partial(a\otimes\eta)=a\cdot(\sum_{\bm{y}}\sum_{\\{(B,\overrightarrow{\sigma})|\
n_{z}(B)=0,\
\text{ind}(B,\overrightarrow{\sigma})=1\\}}\\#\mathcal{M}^{B}(\bm{x},\bm{y};\overrightarrow{\sigma})a(-\overrightarrow{\sigma})\otimes\Phi^{B}_{\bm{x},\bm{y}}\eta),$

where $a\in\mathcal{A}$, $\eta\in\mathcal{E}|_{\bm{x}}$, and the pairs
$(B,\overrightarrow{\sigma})$ are compatible. The underlying type D structure
is the pair $(X^{\mathcal{E}}(\mathcal{H}),\delta)$ where
$\delta(\eta)\coloneqq\partial(1\otimes\eta)$ for any $\eta\in
X^{\mathcal{E}}(\mathcal{H})$.

Abusing notation, we will also use $\widehat{CFD}(\mathcal{H})$ to denote its
underlying type D structure.

######  Remark 2.76.

Note when the local system is trivial, we can identify $\bm{x}$ with
$\mathcal{E}|_{\bm{x}}$, and the differential defined above can be more
conveniently written as

$\partial(a\otimes\bm{x})=a\cdot(\sum_{\bm{y}}\sum_{\\{(B,\overrightarrow{\sigma})|\
n_{z}(B)=0,\
\text{ind}(B,\overrightarrow{\sigma})=1\\}}\\#\mathcal{M}^{B}(\bm{x},\bm{y};\overrightarrow{\sigma})a(-\overrightarrow{\sigma})\otimes\bm{y}).$

######  Proposition 2.77.

The operator $\partial$ in Definition 2.75 is well-defined and
$\partial^{2}=0$.

###### Proof.

We first point out $\partial$ is well-defined, i.e., the sum defining
$\partial$ is finite. This reduces to the provincial admissibility of
$\mathcal{H}$, which implies there are only finitely many positive domains
with prescribed Reeb chords connecting any given pair of generators. The proof
is standard, and we do not repeat it here.

We move to see $\partial^{2}(\bm{x})=0$. For ease of explanation, we begin
with the case of trivial local systems. Let $a$ be a non-zero element of
$\mathcal{A}$, and let $\langle\partial^{2}\bm{x},a\bm{y}\rangle\in\mathbb{F}$
denote the coefficient of the term $a\bm{y}$ in $\partial^{2}\bm{x}$. Then

(2.7)
$\langle\partial^{2}\bm{x},a\bm{y}\rangle=\sum_{\bm{w}\in\mathcal{G}}\sum\\#\mathcal{M}^{B_{1}}(\bm{x},\bm{w};\overrightarrow{\sigma_{1}})\\#\mathcal{M}^{B_{2}}(\bm{w},\bm{y};\overrightarrow{\sigma_{2}}),$

where the second sum is over all the index-one compatible pairs
$(B_{i},\overrightarrow{\sigma_{i}})$ ($i=1,2$) with
$a(-\overrightarrow{\sigma_{1}})\cdot a(-\overrightarrow{\sigma_{2}})=a$. In
view of Proposition 2.47 and the gluing result, the right-hand side of
Equation (2.7) is

$\sum_{\\{(B,\overrightarrow{\sigma})|\text{ind}(B,\overrightarrow{\sigma})=2,\
\
a(-\overrightarrow{\sigma})=a\\}}\\#\partial\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};\overrightarrow{\sigma})\equiv
0\pmod{2}$

This finishes the proof in the case of trivial local systems. For the case of
non-trivial local systems, the proof is a slight modification of the above
argument. One needs to note that given $B_{1}\in\pi_{2}(\bm{x},\bm{w})$ and
$B_{2}\in\pi_{2}(\bm{w},\bm{y})$, we have
$\Phi_{\bm{x},\bm{w}}^{B_{1}}\circ\Phi_{\bm{w},\bm{y}}^{B_{2}}=\Phi_{\bm{x},\bm{y}}^{B_{1}+B_{2}}$.
Therefore, given an $\eta\in\mathcal{E}|_{\bm{x}}$, the terms in
$\partial^{2}(\eta)$ corresponding to two-story ends of a one-dimensional
moduli space $\mathcal{M}^{B}(\bm{x},\bm{y};\overrightarrow{\sigma})$ are
multiples of the same element in $\mathcal{E}|_{\bm{y}}$, namely
$\Phi_{\bm{x},\bm{y}}^{B}(\eta)$, and hence the coefficient is zero mod $2$. ∎

###  2.10. Weakly extended Type D structures

We define the weakly extended type D structure $\widetilde{CFD}(\mathcal{H})$
in this subsection. The weakly extended torus algebra $\tilde{\mathcal{A}}$
can be represented by the quiver with relations shown in Figure 8.

Figure 8. The weakly extended torus algebra. The subscripts in the relation
are understood mod $4$.

Note as in the torus algebra, we have the idempotent ring
$\mathcal{I}=\langle\iota_{0}\rangle\oplus\langle\iota_{1}\rangle$. Let
$\bm{U}$ be $\rho_{0123}+\rho_{1230}+\rho_{2301}+\rho_{3012}$, which is a
central element of $\tilde{\mathcal{A}}$.

######  Definition 2.78.

A weakly extended type D structure over $\tilde{\mathcal{A}}$ is a left
$\mathcal{I}$-module $N$ together with a linear map
$\tilde{\delta}:N\rightarrow\tilde{\mathcal{A}}\otimes N$ such that the map

$\tilde{\partial}\coloneqq(\mu_{\tilde{\mathcal{A}}}\otimes\mathbb{I}_{N})\circ(\mathbb{I}_{\tilde{\mathcal{A}}}\otimes\tilde{\delta}):\tilde{\mathcal{A}}\otimes
N\rightarrow\tilde{\mathcal{A}}\otimes N$

squares to $\bm{U}$, i.e. $\tilde{\partial}^{2}=\bm{U}$. The curved left
$\tilde{\mathcal{A}}$-module $\tilde{\mathcal{A}}\otimes N$ is called the
weakly extended type D module of the weakly extended type D structure
$(N,\tilde{\delta})$.

Let $X^{\mathcal{E}}(\mathcal{H})$ be the $\mathcal{I}$-module defined the
same way as in Section 2.9.

######  Definition 2.79.

Let $\mathcal{H}$ be an unobstructed, provincially admissible, immersed
bordered Heegaard diagram. Fix a generic admissible almost complex structure
on $\Sigma\times[0,1]\times\mathbb{R}$. The weakly extended type D module
$\widetilde{CFD}(\mathcal{H})$ is defined to be the
$\tilde{\mathcal{A}}$-module

$\tilde{\mathcal{A}}\otimes_{\mathcal{I}}X^{\mathcal{E}}(\mathcal{H})$

together with a differential given by

$\tilde{\partial}(a\otimes\eta)=a\cdot(\sum_{\bm{y}}\sum_{\\{(B,\overrightarrow{\sigma})|\
\text{ind}(B,\overrightarrow{\sigma})=1\\}}\\#\mathcal{M}^{B}(\bm{x},\bm{y};\overrightarrow{\sigma})a(-\overrightarrow{\sigma})\otimes\Phi^{B}_{\bm{x},\bm{y}}\eta),$

where $a\in\tilde{\mathcal{A}}$, $\eta\in\mathcal{E}|_{\bm{x}}$,
$\overrightarrow{\sigma}$ is a sequence of Reeb chords that include the case
of a single closed Reeb orbit $\\{U\\}$ (in which case the corresponding
moduli space consists of 1-P holomorphic curves), and the pairs
$(B,\overrightarrow{\sigma})$ are compatible. When
$\overrightarrow{\sigma}=\\{U\\}$, we define $a(-U)=\bm{U}$. The underlying
weakly extended type D structure is
$(X^{\mathcal{E}}(\mathcal{H}),\tilde{\delta})$ where
$\tilde{\delta}(\eta)\coloneqq\tilde{\partial}(1\otimes\eta)$.

######  Remark 2.80.

Again, by abusing notation, we also use $\widetilde{CFD}(\mathcal{H})$ to
denote the underlying weakly extended type D structure. When the local system
is trivial, we have the following more familiar formula for the differential:

$\tilde{\partial}(a\otimes\bm{x})=a\cdot(\sum_{\bm{y}}\sum_{\\{(B,\overrightarrow{\sigma})|\text{ind}(B,\overrightarrow{\sigma})=1\\}}\\#\mathcal{M}^{B}(\bm{x},\bm{y};\overrightarrow{\sigma})a(-\overrightarrow{\sigma})\otimes\bm{y}).$

######  Proposition 2.81.

The operator $\tilde{\partial}$ in Definition 2.79 is well-defined and
$\tilde{\partial}^{2}=\bm{U}$.

###### Proof.

A standard argument shows that the provincial admissibility of $\mathcal{H}$
implies the sum defining $\tilde{\partial}$ in Definition 2.79 is finite, and
hence $\tilde{\partial}$ is well-defined.

Next, we show $\tilde{\partial}^{2}=\bm{U}$. Once again, we first give the
proof when the local system is trivial for conciseness. Recall the length of
an element $a\in\tilde{\mathcal{A}}$ is the number of factors
$\rho_{i}\in\\{\rho_{0},\rho_{1},\rho_{2},\rho_{3}\\}$ when we write $a$ as a
product of the generators
$\\{\iota_{0},\iota_{1},\rho_{0},\rho_{1},\rho_{2},\rho_{3}\\}$. (For example,
$\rho_{123}$ has length $3$ and $\iota_{0}$ has length $0$.)

For an element $a\in\tilde{\mathcal{A}}$ whose length is less than or equal to
$3$, the proof of Proposition 2.77 carries over to show
$\langle\tilde{\partial}^{2}\bm{x},a\bm{y}\rangle=0$ for any $\bm{x}$ and
$\bm{y}$ (by permuting the region we where we put the base point $z$).

We are left to consider the case where the algebra element is of length $4$.
We claim that for a generator $\bm{x}$ such that
$\iota_{1}\cdot\bm{x}=\bm{x}$, we have

$\langle\tilde{\partial}^{2}\bm{x},\rho_{0123}\bm{y}\rangle=\begin{cases}0,\
if\ \bm{x}\neq\bm{y},\\\ 1,\ if\ \bm{x}=\bm{y}.\end{cases}$

Assuming this claim, by permuting the subscripts we also have that
$\langle\tilde{\partial}^{2}\bm{x},\rho_{2301}\bm{y}\rangle$ is $1$ if
$\bm{x}=\bm{y}$ and $0$ otherwise, and an idempotent consideration shows
$\langle\tilde{\partial}^{2}\bm{x},a\bm{y}\rangle=0$ when
$a\in\\{\rho_{1230},\rho_{3012}\\}$. These together imply
$\tilde{\partial}^{2}\bm{x}=\bm{U}\cdot\bm{x}$ when $\iota(\bm{x})=\iota_{1}$.
A similar consideration shows this is true for $\bm{x}$ with
$\iota(\bm{x})=\iota_{0}$ as well. This finishes the proof of the proposition
modulo the claim.

Next, we prove the claim. Note

(2.8)
$\langle\tilde{\partial}^{2}\bm{x},\rho_{0123}\bm{y}\rangle=\sum_{\bm{w}}\sum_{\begin{subarray}{c}\text{ind}(B_{i},\overrightarrow{\sigma_{i}})=1,\\\
i=1,2\end{subarray}}\\#\mathcal{M}^{B_{1}}(\bm{x},\bm{w};\overrightarrow{\sigma_{1}})\\#\mathcal{M}^{B_{2}}(\bm{w},\bm{y};\overrightarrow{\sigma_{2}}),$

where $(B_{i},\overrightarrow{\sigma_{i}})$ ($i=1,2$) is compatible and
$a(-\overrightarrow{\sigma_{1}})a(-\overrightarrow{\sigma_{1}})=\rho_{0123}$
or $\bm{U}$; the possible pairs of
$(\overrightarrow{\sigma_{1}},\overrightarrow{\sigma_{2}})$ are listed below:

$\displaystyle\bigg{\\{}(\emptyset,\\{-\rho_{0},-\rho_{1},-\rho_{2},-\rho_{3}\\}),(\\{-\rho_{0}\\},\\{-\rho_{1},-\rho_{2},-\rho_{3}\\}),(\\{-\rho_{0},-\rho_{1},-\rho_{2}\\},\\{-\rho_{3}\\}),$
$\displaystyle(\\{-\rho_{0},-\rho_{1}\\},\\{-\rho_{2},-\rho_{3}\\}),(\\{-\rho_{0},-\rho_{1},-\rho_{2},-\rho_{3}\\},\emptyset),(\emptyset,\\{-\rho_{0},-\rho_{123}\\}),$
$\displaystyle(\\{-\rho_{0}\\},\\{-\rho_{123}\\}),(\\{-\rho_{0},-\rho_{123}\\},\emptyset),(\emptyset,\\{-\rho_{012},-\rho_{3}\\}),(\\{-\rho_{012}\\},\\{-\rho_{3}\\}),$
$\displaystyle(\\{-\rho_{012},-\rho_{3}\\},\emptyset),(\emptyset,\\{U\\}),(\\{U\\},\emptyset)\bigg{\\}}.$

Let

$\overline{\mathcal{M}}_{0}\coloneqq\cup_{\text{ind}(B,\overrightarrow{\sigma})=2}\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};\overrightarrow{\sigma}),$

where
$\overrightarrow{\sigma}\in\\{(-\rho_{0},-\rho_{1},-\rho_{2},-\rho_{3}),(-\rho_{0},-\rho_{123}),(-\rho_{012},-\rho_{3})\\}$.
Let

$\overline{\mathcal{M}}_{1}:=\cup_{\text{ind}(B,U)=2}\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};U).$

Equation 2.8 and the gluing result imply that
$\langle\tilde{\partial}^{2}\bm{x},\rho_{0123}\bm{y}\rangle$ is equal to the
number of two-story ends of the moduli space
$\overline{\mathcal{M}}_{0}\cup\overline{\mathcal{M}}_{1}$.

According to Proposition 2.49, the other elements in
$\partial\overline{\mathcal{M}}_{0}$ are:

  * (A-1)

simple holomorphic combs with a single split component;

  * (A-2)

simple boundary degenerations with one corner;

  * (A-3)

simple boundary degenerations without corners.

Proposition 2.63 shows the other boundary points in
$\overline{\mathcal{M}}_{1}$ in addition to two-story ends are:

  * (B-1)

simple holomorphic combs with an orbit curve;

  * (B-2)

simple boundary degenerations with one corner;

  * (B-3)

simple boundary degenerations without corners.

Note by Proposition 2.49 and Proposition 2.63, the number of boundary points
of type (A-1) is equal to that of type (B-1), both of which is

$\sum_{\text{ind}(B,-\rho_{3012})=1}\\#\mathcal{M}^{B}(\bm{x},\bm{y};-\rho_{3012})+\sum_{\text{ind}(B,-\rho_{1230})=1}\\#\mathcal{M}^{B}(\bm{x},\bm{y};-\rho_{1230}).$

The parity of the number of boundary points of type (A-2) is equal to that of
type (B-2), which are both mod $2$ equal to

$\sum_{q}\sum_{\\{(B_{1},B_{2})|B_{2}\in T(q),\
\text{ind}(B_{1}+B_{2};U)=2\\}}\\#\mathcal{M}^{B_{1}}(\bm{x},\bm{y};q),$

where $q$ ranges over self-intersection points of $\alpha_{im}$, and $T(q)$
denotes the set of stabilized teardrops at $q$.

The parity of the number of boundary points of type (B-3) is even according to
Proposition 2.63.

In summary, the parity of the number of boundary points of
$\overline{\mathcal{M}}_{0}\cup\overline{\mathcal{M}}_{1}$ corresponding to
two-story ends is equal to that of type (A-3), which is odd if and only if
$\bm{x}=\bm{y}$ by Proposition 2.49. Therefore,
$\langle\tilde{\partial}^{2}\bm{x},\rho_{0123}\bm{y}\rangle$ is odd if and
only if $\bm{x}=\bm{y}$, finishing the proof of the claim.

In the presence of non-trivial local systems, we simply need to consider the
above argument for each domain. For a domain $B$, let
$\overline{\mathcal{M}}_{0}^{B}$ be the subset of $\overline{\mathcal{M}}_{0}$
consisting of holomorphic curves with domain $B$, and similarly define
$\overline{\mathcal{M}}_{1}^{B}$. The two-story ends in
$\overline{\mathcal{M}}_{0}^{B}\cup\overline{\mathcal{M}}_{1}^{B}$ all
correspond to the same parallel transport. When ends of type (A-3) do not
occur, the two-story ends cancel in pairs by the same argument as above. When
ends of type (A-3) appear, we have $B=[\Sigma]$ and
$\sigma=(-\rho_{0},-\rho_{1},-\rho_{2},-\rho_{3})$. In particular,
$\partial_{\alpha_{im}}B=\emptyset$, which induces the identity endomorphism
of $\mathcal{E}|_{\bm{x}}$. Also, the number of two-story ends is odd as the
number of (A-3) ends is odd. The claim follows from these. ∎

There is a canonical quotient map
$\pi:\tilde{\mathcal{A}}\rightarrow\mathcal{A}$. We say a weakly extended type
D structure $(N,\tilde{\delta})$ extends a type D structure
$(N^{\prime},\delta)$ if $(N^{\prime},\delta)$ is isomorphic to
$(N,(\pi\otimes\mathbb{I}_{N})\circ\tilde{\delta})$. Clearly,
$\widetilde{CFD}(\mathcal{H})$ extends $\widehat{CFD}(\mathcal{H})$ when both
are defined.

###  2.11. Invariance

In this subsection, we address the invariance of the (weakly extended) type D
structures.

######  Proposition 2.82.

The homotopy type of the type D structure defined in Definition 2.75 is
independent of the choice of the almost complex structure and is invariant
under isotopy of the $\alpha$\- or $\beta$-curves.

######  Remark 2.83.

We do not need the invariance under handleslides and stabilizations for our
applications. We only need to prove invariance when perturbing diagrams to
obtain nice diagrams, and this only requires isotopies.

###### Proof of Proposition 2.82.

The standard proof in Section 6.3 of [LOT18] carries over. For instance, to
prove independence of almost complex structures, one first constructs a
continuation map by counting holomorphic curves in
$\Sigma\times[0,1]\times\mathbb{R}$ for a generic almost complex structure $J$
that interpolates two admissible almost complex structures $J_{0}$ and
$J_{1}$. Then, one proves the continuation map is a chain map by analyzing the
ends of one-dimensional moduli spaces. The only possible complication comes
from boundary degenerations since $\alpha_{im}$ is immersed. However, this
does not happen as $\mathcal{H}$ is unobstructed and the holomorphic curves
have $n_{z}=0$. Therefore, no new phenomenon appears in the degeneration of
moduli spaces, and hence the proof stays the same. ∎

######  Proposition 2.84.

The homotopy type of the weakly extended type D structure defined in
Definition 2.79 is independent of the choice of the almost complex structure
and is invariant under isotopy of the $\alpha$\- and $\beta$-curves.

###### Proof.

One could prove this proposition similarly to the previous one. However, such
an approach would require generalizing the analysis of the ends of moduli
spaces in Proposition 6.20 of [LOT18] and hence is slightly tedious to write
down. Here we give a different approach. Let $\mathcal{H}$ denote the immersed
bordered Heegaard diagram. By Proposition 2.82, we know the homotopy type of
$\widehat{CFD}(\mathcal{H})$ is independent of the choice of almost complex
structures and isotopy of the $\alpha$\- or $\beta$-curves. Since
$\widetilde{CFD}(\mathcal{H})$ extends $\widehat{CFD}(\mathcal{H})$ and that
such extension is unique up to homotopy by Proposition 38 of [HRW23], we know
the homotopy type of $\widetilde{CFD}(\mathcal{H})$ is also independent of the
choice of almost complex structures and isotopy of the $\alpha$\- and
$\beta$-curves. ∎

##  3\. Knot Floer homology of immersed Heegaard diagrams

This section defines knot Floer chain complexes of immersed Heegaard diagrams
and proves the homotopy invariance under Heegaard moves.

###  3.1. Immersed doubly-pointed Heeggard diagram

######  Definition 3.1.

An _immersed doubly-pointed Heegaard diagram_ is a 5-tuple
$\mathcal{H}_{w,z}=(\Sigma,\bm{\alpha},\bm{\beta},w,z)$ where

  * (1)

$\Sigma$ is a closed oriented surface of genus $g$.

  * (2)

$\bm{\alpha}=\\{\alpha_{1},\ldots,\alpha_{g-1},\alpha_{g}\\}$, where
$\alpha_{1},\ldots,\alpha_{g-1}$ are embedded disjoint curves in $\Sigma$ and
$\alpha_{g}=\\{\alpha_{g}^{1},\ldots,\alpha_{g}^{n}\\}$ is a collection of
immersed curves decorated with local systems. Moreover, $\alpha_{i}$
($i=1,\ldots,g-1$) are disjoint from $\alpha_{g}$, $\alpha_{g}^{1}$ has the
trivial local system, and
$\\{\alpha_{1},\ldots,\alpha_{g-1},\alpha_{g}^{1}\\}$ induce linearly
independent elements in $H_{1}(\Sigma,\mathbb{Z})$. We also assume that
$\alpha_{g}^{i}$ is trivial in
$H_{1}(\Sigma,\mathbb{Z})/\langle\alpha_{1},\ldots,\alpha_{g-1}\rangle$ for
$i>1$. For convenience, we also denote $\alpha_{g}$ by $\alpha_{im}$.

  * (3)

$\bm{\beta}=\\{\beta_{1},\ldots,\beta_{g}\\}$ are embedded disjoint curves in
$\Sigma$ which induce linearly independent elements in
$H_{1}(\Sigma,\mathbb{Z})$.

  * (4)

$w$ and $z$ are base points such that they both lie in a single connected
region in the complement of $\alpha$-curves as well as a single region in the
complement of $\beta$-curves.

Domains, periodic domains, and $\alpha$-bounded domains are defined similarly
in this setting as for bordered Heegaard diagrams (by ignoring $\alpha$-arcs
and surface boundary). We make a similar but slightly different definition of
unobstructedness and admissibility below.

######  Definition 3.2.

Given an immersed doubly-pointed Heegaard diagram, $\bm{\alpha}$ is called
_unobstructed_ if there are no nontrivial zero- or one-cornered
$\alpha$-bounded domains $B$ with $n_{z}(B)=0$ (or equivalently $n_{w}(B)=0$).
An immersed doubly-pointed Heegaard diagram is called unobstructed if
$\bm{\alpha}$ is unobstructed.

######  Definition 3.3.

An immersed doubly-pointed Heegaard diagram is _bi-admissible_ if any
nontrivial periodic domain $B$ with $n_{z}(B)=0$ or $n_{w}(B)=0$ has both
positive and negative coefficients.

We remark that the restriction to having only one immersed multicurve in the
definition of immersed doubly-pointed Heegaard diagrams is not essential.

###  3.2. The knot Floer chain complex

We define the knot Floer chain complex of an immersed Heegaard diagram similar
to that in the ordinary setup. The only modification is that we only count
stay-on-track holomorphic curves. The definition and analysis of moduli spaces
in this setup is a straightforward modification of that in the previous
section; it is even simpler as we do not need to care about east punctures. We
hence do not repeat the moduli space theory but only mention the key
properties when we need them. We will let $\mathcal{G}(\mathcal{H}_{w,z})$
denote the set of generators, which are $g$-tuples $(x_{1},\ldots,x_{g})$ such
that $x_{i}\in\alpha_{i}\cap\beta_{\sigma(i)}$ $(i=1,\ldots,g)$ where $\sigma$
is a permutation of $\\{1,\ldots,g\\}$. Let
$\mathcal{R}=\mathbb{F}[U,V]/(UV)$. Implicit in the definition below is that
we choose a generic admissible almost complex structure $J$ on
$\Sigma\times[0,1]\times\mathbb{R}$.

######  Definition 3.4.

Let $\mathcal{H}_{w,z}$ be an unobstructed and bi-admissible immersed doubly-
pointed Heegaard diagram. $CFK_{\mathcal{R}}(\mathcal{H}_{w,z})$ is the free
$\mathcal{R}$-module generated over $\mathcal{G}(\mathcal{H}_{w,z})$ with
differential $\partial$ defined as

$\partial\bm{x}=\sum_{y}\sum_{B\in\pi_{2}(\bm{x},\bm{y}),\
\text{ind}(B)=1}\\#\mathcal{M}^{B}(\bm{x},\bm{y})U^{n_{w}(B)}V^{n_{z}(B)}\bm{y},$

where $\bm{x},\bm{y}\in\mathcal{G}$.

######  Remark 3.5.

Here we only give the definition assuming the local system on $\alpha_{im}$ is
trivial. The case in which the local system is non-trivial is only
notationally more complicated, and we leave it for the interested readers to
work out. See Definition 2.75 for an example.

######  Proposition 3.6.

$(CFK_{\mathcal{R}}(\mathcal{H}_{w,z}),\partial)$ is a chain complex, i.e.,
$\partial^{2}=0$.

###### Proof.

The same proof for Proposition 2.77 works here. Note we will only use moduli
spaces with domains $B$ such that $n_{w}(B)=0$ or $n_{z}(B)=0$, and the
unobstructedness of $\mathcal{H}_{w,z}$ excludes the possibility of boundary
degeneration in the compactified 1-dimensional moduli space supported in such
domains. Hence, an analogue version of Proposition 2.47 holds. With this
observation, the proof of Proposition 2.77 carries over. ∎

###  3.3. Bi-grading

We would like to consider gradings on knot Floer chain complexes.

######  Definition 3.7.

A (possibly immersed) doubly-pointed Heegaard diagram is gradable if all non-
trivial periodic domain $P$ satisfies $\text{ind}(P)-2n_{z}(P)=0$ and
$\text{ind}(P)-2n_{w}(P)=0$, where $\text{ind}(-)$ is defined in Definition
2.43.

If $\mathcal{H}_{w,z}$ is gradable then the knot Floer chain complex
$(CFK_{\mathcal{R}}(\mathcal{H}_{w,z}),\partial)$ admits a relative
$\mathbb{Z}\oplus\mathbb{Z}$-grading, as described below. We will be
interested in diagrams $\mathcal{H}_{w,z}$ for which
$\widehat{HF}(\mathcal{H}_{w})\cong\widehat{HF}(\mathcal{H}_{z})\cong\mathbb{F}$,
where $\widehat{HF}(\mathcal{H}_{w})$ and $\widehat{HF}(\mathcal{H}_{z})$ are
homology groups of the chain complexes obtained from
$CFK_{\mathcal{R}}(\mathcal{H}_{w,z})$ by setting $V=0$ and $U=1$ or $U=0$ and
$V=1$, respectively. In this case we say that the horizontal and vertical
homology has rank one. Gradable diagrams with this property can be given an
absolute grading, as follows.

######  Definition 3.8.

Let $\bm{x},\bm{y}\in\mathcal{G}(\mathcal{H}_{w,z})$ be two generators. Let
$B\in\tilde{\pi}_{2}(\bm{x},\bm{y})$ be a domain. Then the $w$-grading
difference between $\bm{x}$ and $\bm{y}$ is given by

$gr_{w}(\bm{x})-gr_{w}(\bm{y})=\text{ind}(B)-2n_{w}(B),$

and the $z$-grading difference between $\bm{x}$ and $\bm{y}$ is given by

$gr_{z}(\bm{x})-gr_{z}(\bm{y})=\text{ind}(B)-2n_{z}(B).$

If the horizontal and vertical homology of $\mathcal{H}_{w,z}$ is rank one,
then the absolute $w$-grading is normalized so that
$\widehat{HF}(\mathcal{H}_{w})$ is supported in $w$-grading $0$, and absolute
$z$-grading is normalized so that $\widehat{HF}(\mathcal{H}_{z})$ is supported
in $z$-grading $0$.

Equivalently, one can equip $CFK_{\mathcal{R}}(\mathcal{H}_{w,z}(\alpha_{K}))$
with the Maslov grading and the Alexander grading. These two gradings can be
expressed in terms of the $w$-grading and $z$-grading: The Maslov grading is
equal to the $z$-grading, and the Alexander grading is given by
$\frac{1}{2}(gr_{w}-gr_{z})$.

######  Remark 3.9.

The normalization conditions for the absolute gradings are chosen so that the
bi-graded chain complexes model those associated to knots in the 3-sphere.

###  3.4. Invariance

We will show knot Floer chain complexes defined over immersed Heegaard
diagrams satisfy similar invariance properties when varying the almost complex
structure or modifying the Heegaard diagram by isotopy, handleslides, and
stabilizations. While the meaning of isotopy and stabilization are obvious for
immersed Heegaard diagrams, we give a remark on handleslides.

######  Remark 3.10.

When speaking of handleslides of an immersed Heegaard diagram
$\mathcal{H}_{w,z}$, we only allow an $\alpha$-curve to slide over another
_embedded_ $\alpha$-curve, not over an immersed $\alpha$-curve. Furthermore,
we point out that handle-slides do not change the unobstructedness, bi-
admissibility, and gradability of the diagram. To see this, note periodic
domains of two Heegaard diagrams before and after a handleslide are related. A
periodic domain in the old Heegaard diagram with boundary on the arc that
moves in the handleslide give rise to a periodic domain in the new Heegaard
diagram by boundary summing a thin annulus (whose multiplicity can be one or
negative one). In particular, if we started from a somewhere negative domain
$B$, then the new domain $B^{\prime}$ after this procedure is still somewhere
negative; it is also easy to see $\text{ind}(B)=\text{ind}(B^{\prime})$,
$n_{z}(B)=n_{z}(B^{\prime})$, and $n_{w}(B)=n_{w}(B^{\prime})$, which implies
the gradability of two diagrams are the same as well.

######  Proposition 3.11.

Let $\mathcal{H}_{w,z}$ be an unobstructed, bi-admissible, and gradable
immersed doubly-pointed Heegaard diagram. The bigraded chain homotopy type of
$CFK_{\mathcal{R}}(\mathcal{H}_{w,z})$ is invariant under varying the almost
complex structure, isotopy of the $\alpha$\- and $\beta$-curves, handleslides,
and stabilization/destabilization.

###### Proof.

The proof of the bigraded homotopy invariance under the variation of the
almost complex structure, isotopy, and stabilization is the same as the
corresponding results in the embedded-$\alpha$-curve set-up in [Lip06]. In
fact, changing the $\alpha$-curves from embedded to immersed can only
complicate the arguments in that boundary degeneration might appear as ends of
the moduli spaces involved, yet the unobstructedness dispels such worries.

Figure 9. The $\alpha$-curves in proving the handleslide invariance on a
genus-two surface, which is represented as a torus obtained by identifying the
edges of a square together with a handle attached to the two circles inside
the square. The labels $\theta_{im}^{\pm}$ are used interchangeably with
$\theta_{2}^{\pm}$. Similarly, $\theta_{im}^{H,\pm}$ and
$\theta_{im}^{{}^{\prime}\pm}$ are the same as $\theta_{2}^{H,\pm}$ and
$\theta_{2}^{{}^{\prime}\pm}$, respectively. (a) shows any self-intersection
point $q_{i}$ of $\alpha_{im}$ induces two intersection points between
$\alpha_{im}$ and its perturbation $\alpha^{\prime}_{im}$. (b) shows the small
triangles showing
$F(\Theta_{\alpha^{\prime},\alpha^{H}}\otimes\Theta_{\alpha^{H},\alpha})=\Theta_{\alpha^{\prime},\alpha}$

.

The handleslide invariance can also be proved using the same strategy as in
the embedded-$\alpha$-curve case with slightly more caution. The main
difference is that in the embedded-$\alpha$-curve case, there is a unique
maximal graded generator in the Heegaard Floer homology of a Heegaard diagram
where the set of $\alpha$-curves is a small Hamiltonian perturbation of the
$\beta$-curves. In contrast, such a generator needs to be specified more
carefully in our case. We spell this out in more detail.

Denote $\mathcal{H}=(\Sigma,\bm{\alpha},\bm{\beta},w,z)$. For clarity of
exposition, assume $\alpha_{im}$ consists of a single component with a trivial
local system and $n$ self-intersection points. We also restrict to the
interesting case, in which the handleslide is sliding $\alpha_{im}$ over an
embedded $\alpha$-curve. Let $\bm{\alpha^{\prime}}$ denote a small Hamiltonian
perturbation of $\bm{\alpha}$ so that $\alpha_{i}\cap\alpha_{j}=\emptyset$ for
$i\neq j$; for $i=1,\ldots,g-1$, the embedded curves $\alpha_{i}$ and
$\alpha_{i}^{\prime}$ intersects exactly at two points
$\\{\theta_{i}^{+},\theta_{i}^{-}\\}$; $\alpha_{im}$ intersects
$\alpha_{im}^{\prime}$ at $2+2n$ points
$\\{\theta_{g}^{+},\theta_{g}^{-},\xi_{1}^{+},\xi_{1}^{-},\ldots,\xi_{n}^{+},\xi_{n}^{-}\\}$,
where $\xi_{i}^{\pm}$ are intersection points corresponding to the self-
intersection points of $\alpha_{im}$. We label the $\theta$-intersection
points using the convention so that $(\theta_{i}^{+},*)$ is of higher grading
than $(\theta_{i}^{-},*)$ in
$CFK_{\mathcal{R}}(\Sigma,\bm{\alpha^{\prime}},\bm{\alpha},w,z)$,
$(i=1,\ldots,g)$ (see Figure 9 (a)). Let $\alpha_{im}^{H}$ denote the curve
obtained by sliding $\alpha_{im}$ over, say, $\alpha_{g-1}$, so that
$\alpha_{im}^{H}$ intersects each of $\alpha_{im}$ and $\alpha_{im}^{\prime}$
in $2+2n$ points; denote the $\theta$-intersection points by
$\\{\theta_{g}^{H,+},\theta_{g}^{H,-}\\}$ and
$\\{\theta_{g}^{{}^{\prime}+},\theta_{g}^{{}^{\prime}-}\\}$, respectively. Let
$\alpha_{i}^{H}$ ($i=1,\ldots,g-1$) be small Hamiltonian perturbations of
$\alpha_{i}^{\prime}$ so that $\alpha_{i}^{H}$ intersects each of $\alpha_{i}$
and $\alpha_{i}^{\prime}$ at exactly two points, denoted by
$\\{\theta_{i}^{H,+},\theta_{i}^{H,-}\\}$ and
$\\{\theta_{i}^{{}^{\prime}+},\theta_{i}^{{}^{\prime}-}\\}$, respectively. Let
$\Theta_{\alpha^{\prime},\alpha}=(\theta_{1}^{+},\ldots,\theta_{g}^{+})$,
$\Theta_{\alpha^{H},\alpha}=(\theta_{1}^{H,+},\ldots,\theta_{g}^{H,+})$, and
$\Theta_{\alpha^{\prime},\alpha^{H}}=(\theta_{1}^{{}^{\prime}+},\ldots,\theta_{g}^{{}^{\prime}+})$.
These correspond to the maximal graded intersection points used in the
embedded case.777A straightforward computation would show
$\Theta_{\alpha,\alpha^{\prime}}$ are indeed cycles in the Floer chain complex
associated to the immersed Heegaard diagram $(\Sigma,\bm{\alpha}$,
$\bm{\alpha^{\prime}},w,z)$; similar statements hold for
$\Theta_{\alpha^{H},\alpha}$ and $\Theta_{\alpha^{\prime},\alpha^{H}}$.

The rest of the proof is similar to the embedded case. We provide a sketch.
Let $\mathcal{H}^{H}=(\Sigma,\bm{\alpha}^{H},\bm{\beta},w,z)$ and
$\mathcal{H}^{\prime}=(\Sigma,\bm{\alpha^{\prime}},\bm{\beta},w,z)$. By
counting holomorphic triangles (with stay-on-track boundaries), one can define
chain maps

$F(\Theta_{\alpha^{H},\alpha}\otimes-):CFK_{\mathcal{R}}(\mathcal{H})\rightarrow
CFK_{\mathcal{R}}(\mathcal{H}^{H})$

and

$F(\Theta_{\alpha^{\prime},\alpha^{H}}\otimes-):CFK_{\mathcal{R}}(\mathcal{H}^{H})\rightarrow
CFK_{\mathcal{R}}(\mathcal{H}^{\prime})$

Again, the usual proof which shows the above maps are chain maps carries
through, as the unobstructedness excludes boundary degeneration when analyzing
the ends of one-dimensional moduli spaces of holomorphic triangles. Similarly,
by analyzing ends one-dimensional moduli spaces of holomorphic quadrilaterals,
one can show the composition of these two maps is chain homotopic equivalent
to
$F(F(\Theta_{\alpha^{\prime},\alpha^{H}}\otimes\Theta_{\alpha^{H},\alpha})\otimes-)$.
One can show this map is homotopic equivalent to

$F(\Theta_{\alpha^{\prime},\alpha}\otimes-):CFK_{\mathcal{R}}(\mathcal{H})\rightarrow
CFK_{\mathcal{R}}(\mathcal{H}^{\prime})$

by a standard computation which shows
$F(\Theta_{\alpha^{\prime},\alpha^{H}}\otimes\Theta_{\alpha^{H},\alpha})=\Theta_{\alpha^{\prime},\alpha}$
(see Figure 9 (b)). One can show that the map
$F(\Theta_{\alpha^{\prime},\alpha}\otimes-)$ is a chain isomorphism (using the
area-filtration technique in [OS04b], Proposition 9.8). ∎

##  4\. Paring theorems

In Section 4.1–4.2, we introduce a pairing construction which merges a (non-
immersed) bordered Heegaard diagram and an immersed multicurve to produce an
immersed Heegaard diagram. After that, we establish the unobstructedness and
admissibility of these pairing diagrams in Section 4.3–4.5, and then we prove
the bordered invariant of such pairing diagrams admits a box-tensor product
interpretation in Section 4.6. Finally, in Section 4.7 we prove a pairing
theorem for gluing a particular type of doubly-pointed bordered Heegaard
diagram and an immersed bordered Heegaard diagram; this theorem will be useful
in Section 5.

###  4.1. Immersed curves in the marked torus

######  Definition 4.1.

The _marked torus_ $T^{2}$ is the oriented surface
$\mathbb{R}^{2}/\mathbb{Z}^{2}$ together with a base point $z$ located at
$(1-\epsilon,1-\epsilon)$ for some sufficiently small $\epsilon>0$. The images
of the positively oriented $x$-axis and $y$-axis are called the _preferred
longitude_ and _preferred meridian_ respectively.

We will consider immersed multicurves with local systems in the marked torus.
Two immersed multicurves are equivalent if they are regularly homotopic in
$T^{2}\backslash{z}$ and the local systems are isomorphic. Throughout this
paper, we restrict to immersed multicurves $\alpha_{im}$ satisfying the
following assumptions:

  * (C-1)

No component of $\alpha_{im}$ is a circle enclosing the base point $z$ once.

  * (C-2)

No component of the immersed multicurve is null-homotopic in
$T^{2}\backslash\\{z\\}$, and the immersed multicurve is _unobstructed_ in the
sense that it does not bound any teardrops in $T^{2}\backslash\\{z\\}$.

  * (C-3)

The immersed multicurve is _reduced_ , i.e., if we let $[0,1]\times[0,1]$ be
the square obtained by cutting the marked torus open along the preferred
meridian and longitude, then no sub-arcs of $\alpha_{im}$ contained in
$[0,1]\times[0,1]$ have both ends on the same edge of the square.

  * (C-4)

Let $\pi$ denote the projection map from $\mathbb{R}^{2}$ to $T^{2}$. Using
regular homotopy, we assume all immersed curves in the marked torus are
contained in the complement of
$\pi([-\frac{1}{4},\frac{1}{4}]\times[-\frac{1}{4},\frac{1}{4}])$ in $T^{2}$,
the strands contained in
$\pi([-\frac{1}{4},\frac{1}{4}]\times[\frac{1}{4},\frac{3}{4}])$ are
horizontal, and the strands contained in the image of
$\pi([\frac{1}{4},\frac{3}{4}]\times[-\frac{1}{4},\frac{1}{4}])$ are vertical.

An immersed multicurve in the marked torus determines a type D structure over
the torus algebra as follows. First, we introduce some terminology.

Figure 10. Six types of elementary arcs. The orientations are the so-called
correct orientations.

######  Definition 4.2.

An _elementary arc_ is an embedded arc in the marked torus ${T}^{2}$ such that
it only intersects the preferred meridian or longitude at the endpoints. There
are six types of elementary arc based on the position of the endpoints, each
of which is labeled by a Reeb chord in
$\\{\rho_{1},\rho_{2},\rho_{3},\rho_{12},\rho_{23},\rho_{123}\\}$ as shown in
Figure 10.

If we ignore the local systems, then any immersed multicurve is comprised of a
collection of elementary arcs; one can see this by cutting $T^{2}$ open along
the preferred longitude and meridian. Sometimes we also need to consider
oriented elementary arcs.

######  Definition 4.3.

An orientation of an elementary arc is called the correct orientation if it is
the one shown in Figure 10.

Next, we describe how to obtain a type D structure from an immersed multicurve
in terms of elementary arcs. Denote the local system on $\alpha_{im}$ by
$(\mathcal{E},\Phi)$, where $\mathcal{E}$ is a vector bundle over
$\alpha_{im}$ and $\Phi$ is a parallel transport. Let
$\mathcal{G}(\alpha_{im})=\mathcal{G}_{m}\cup\mathcal{G}_{l}$, where
$\mathcal{G}_{m}$ (respectively, $\mathcal{G}_{l}$) is the set of intersection
points of $\alpha_{im}$ and the preferred meridian (respectively, longitude).
Let $\mathcal{X}$ be the vector space
$\oplus_{x\in\mathcal{G}(\alpha_{im})}\mathcal{E}|_{x}$. Next, we define an
$\mathcal{I}$-action on $\mathcal{X}$, where $\mathcal{I}$ is the ring of
idempotent of the torus algebra. If $x\in\mathcal{G}_{m}$, for any
$\tilde{x}\in\mathcal{E}|x$, $\iota_{0}\cdot\tilde{x}=\tilde{x}$ and
$\iota_{1}\cdot\tilde{x}=0$; if $x\in\mathcal{G}_{l}$, for any
$\tilde{x}\in\mathcal{E}|x$, $\iota_{0}\cdot\tilde{x}=0$ and
$\iota_{1}\cdot\tilde{x}=\tilde{x}$. The underlying $\mathcal{A}$-module for
$\widehat{CFD}(\alpha_{im})$ is $\mathcal{A}\otimes_{\mathcal{I}}\mathcal{X}$.
Finally, the differential on $\widehat{CFD}(\alpha_{im})$ decomposes linearly
as maps between $\mathcal{E}|_{x}$ for $x\in\mathcal{G}(\alpha_{im})$. Given
$x,y\in\mathcal{G}(\alpha_{im})$ and $\rho_{I}$ a Reeb element, there is a
differential map $\mathcal{E}|_{x}\rightarrow\rho_{I}\otimes\mathcal{E}|_{y}$
if and only if $x$ and $y$ are connected by a $\rho_{I}$-elementary arc whose
correct orientation goes from $x$ to $y$, in which case the differential is
given by $\partial(\tilde{x})=\rho_{I}\otimes\Phi(\tilde{x})$ for
$\tilde{x}\in\mathcal{E}|_{x}$. In particular, when the local system of
$\alpha_{im}$ is trivial, then the generators of $\widehat{CFD}(\alpha_{im})$
are in one-to-one correspondence with the intersection points of $\alpha_{im}$
with the preferred longitude/meridian, and the differentials are in one-to-one
correspondence with the elementary sub-arcs of $\alpha_{im}$.

The immersed-curve presentation of type D structures is empowered by the
following result.

######  Theorem 4.4 ([HRW23]).

Each Type D structure of a bordered 3-manifold with torus boundary is
homotopic to a type D structure determined by some immersed multicurve (with
local systems) in the marked torus.

######  Remark 4.5.

All immersed multicurves arising from 3-manifolds with torus boundary
satisfies the assumptions (C-1)-(C-4): (C-4) is straightforward, (C-2) and
(C-3) follows from the algorithm of converting type D structures to immersed
multicurves in [HRW23], and for (C-4) see the discussion around Figure 31 and
32 in [HRW22].

We will mainly be interested in the immersed multicurves corresponding to type
D structures of knot complements for knots in the 3-sphere; these immersed
multicurves satisfy some further properties that we specify in Definition 4.6
below, and the proofs of these properties can be found in [HRW22, Section 4].

######  Definition 4.6.

An immersed multicurve
$\alpha_{im}=\\{\alpha_{im}^{0},\ldots,\alpha_{im}^{n-1}\\}$ of $n$ components
(for some $n\geq 1$) with a local system is called knot-like if the local
system restricted to ${\alpha_{im}^{0}}$ is trivial, $\alpha_{im}^{0}$ (with
some orientation) is homologous to the preferred longitude in $T^{2}$, and
$[\alpha_{im}^{i}]$ for $i\geq 1$ is trivial in $H_{1}(T^{2},\mathbb{Z})$.

From now on, we assume all immersed multicurves are knot-like.

###  4.2. Pairing diagrams

We introduce a class of immersed bordered Heegaard diagrams and doubly pointed
Heegaard diagrams. They are respectively obtained from two types of pairing
constructions that we will define:

  * (1)

Pairing an immersed multicurve in the marked torus and an _arced bordered
Heegaard diagram with two boundary components_ to construct an immersed
bordered Heegaard diagram.

  * (2)

Paring an immersed multicurve in the marked torus with a doubly pointed
bordered Heegaard diagram to construct a closed immersed doubly pointed
Heegaard diagram.

We begin with the first type. For convenience, we first recall the definition
of arced bordered Heegaard diagrams below (in the special case where both
boundaries of the corresponding bordered manifold are tori).

######  Definition 4.7.

An arced bordered Heegaard diagram with two boundary components is a quadruple
$\mathcal{H}^{a}=(\bar{\Sigma},\bar{\bm{\alpha}},\bm{\beta},\bm{z})$ where

  * (1)

$\bar{\Sigma}$ is a compact, oriented surface of genus $g$ with two boundary
components
$\partial\bar{\Sigma}=\partial_{L}\bar{\Sigma}\cup\partial_{R}\bar{\Sigma}$;

  * (2)

$\bar{\bm{\alpha}}$ is a collection of pairwise disjoint properly embedded
arcs and curves
$\\{\alpha^{a,L}_{1},\alpha^{a,L}_{2},\alpha^{a,R}_{1},\alpha^{a,R}_{2},\alpha^{c}_{1},\ldots,\alpha^{c}_{g-2}\\}$.
Here, $\alpha^{a,L}_{1}$ and $\alpha^{a,L}_{2}$ are two arcs with endpoints on
$\partial_{L}\bar{\Sigma}$, $\alpha^{a,R}_{1}$ and $\alpha^{a,R}_{2}$ are two
arcs with endpoints on $\partial_{R}\bar{\Sigma}$, and the $\alpha^{c}_{i}$’s
($i=1,\ldots,g-2$) are embedded circles. Moreover, elements in
$\bar{\bm{\alpha}}$ induce linearly independent elements in
$H_{1}(\bar{\Sigma},\partial\bar{\Sigma};\mathbb{Z})$;

  * (3)

$\bm{\beta}$ is a set of $g$ pairwise disjoint embedded circles
$\\{\beta_{1},\ldots,\beta_{g}\\}$ in the interior of $\bar{\Sigma}$ that are
linearly independent as elements in
$H_{1}(\bar{\Sigma},\partial\bar{\Sigma};\mathbb{Z})$;

  * (4)

$\bm{z}$ is a properly embedded arc in
$\bar{\Sigma}\backslash(\bar{\bm{\alpha}}\cup\bm{\beta})$ with one endpoint
$z_{L}$ on $\partial_{L}\bar{\Sigma}$ and the other endpoint $z_{R}$ on
$\partial_{R}\bar{\Sigma}$.

Periodic and provincially period domains for arced bordered Heegaard diagrams
with two boundary components are defined similarly to the case of a single
boundary component. In the two boundary case we will also consider periodic
domains that are adjacent to only one of the boundaries.

######  Definition 4.8.

A domian is _left provincial_ if the multiplicity in the regions adjacent to
$\partial_{L}\bar{\Sigma}$ are zero. We say an arced bordered Heegaard
diagrams with two boundary components is _left provincially admissible_ if all
left provincial periodic domains have both positive and negative
multiplicities.

The pairing construction is illustrated in Figure 11, and is spelled out in
Definition 4.9.

Figure 11. Left: an arced bordered Heegaard diagram. Middle: an immersed
multicurve in the marked torus. The dashed lines are the boundary of
$\pi([-\frac{1}{4},\frac{1}{4}]\times[-\frac{1}{4},\frac{1}{4}])$. Right: a
bordered Heegaard diagram obtained by the pairing construction.

######  Definition 4.9.

Let $\mathcal{H}^{a}=(\bar{\Sigma},\bar{\bm{\alpha}},\bm{\beta},\bm{z})$ be an
arced bordered Heegaard diagram with two boundary components and let
$\alpha_{im}$ be an immersed multicurve in the marked torus $T^{2}$. The
_pairing diagram of $\mathcal{H}^{a}$ and $\alpha_{im}$_, denoted by
$\mathcal{H}^{a}(\alpha_{im})$, is a bordered Heegaard diagram obtained
through the following steps.

  * (1)

Form $\bar{\Sigma}^{\prime}$ from $\bar{\Sigma}$ by collapsing
$\partial_{R}\bar{\Sigma}$. Let $\alpha^{\prime a}_{i}$ be the image of
$\alpha^{a,L}_{i}$ ($i=1,2$), $\alpha^{\prime c}_{i}$ be the image of
$\alpha^{c}_{i}$ ($i=1,\ldots,g-2$), $\bm{\beta}^{\prime}$ be the image of
$\bm{\beta}$, and $z^{\prime}_{L}$ be the image of $z_{L}$. The images of
$\alpha^{a,R}_{i}$ ($i=1,2$), denoted by $\tilde{\alpha}_{i}$, are two circles
intersecting at a single point ${z}^{\prime}_{R}$, the image of $z_{R}$.

  * (2)

Take a neighborhood $U$ of $\tilde{\alpha}_{1}\cup\tilde{\alpha}_{2}$ which
admits a homeomorphism $h:U\rightarrow
T^{2}\backslash\pi([-\frac{1}{4},\frac{1}{4}]\times[-\frac{1}{4},\frac{1}{4}])$
such that $h(\tilde{\alpha}_{1})=\pi(\\{\frac{1}{2}\\}\times[0,1])$,
$h(\tilde{\alpha}_{2})=\pi([0,1]\times\\{\frac{1}{2}\\})$, and each connected
component of $h(\bm{\beta}^{\prime}\cap U)$ is an arc of the form
$\pi(\\{x\\}\times[\frac{1}{4},\frac{3}{4}])$ or
$\pi([\frac{1}{4},\frac{3}{4}]\times\\{y\\})$ for some $x$ or $y$ in
$(2\epsilon,\frac{1}{4})$.

  * (3)

Let $\alpha_{im}^{\prime}=h^{-1}(\alpha_{im})$. Let
$\bar{\bm{\alpha}}^{\prime}=\\{\alpha^{\prime a}_{1},\alpha^{\prime
a}_{2},\alpha^{\prime c}_{1},\ldots,\alpha^{\prime
c}_{g-1},\alpha_{im}^{\prime}\\}$.

  * (4)

Let
$\mathcal{H}^{a}(\alpha_{im})=(\bar{\Sigma}^{\prime},\bar{\bm{\alpha}}^{\prime},\bm{\beta}^{\prime},z^{\prime}_{L})$.

Recall a _doubly pointed bordered Heegaard diagram_ is a bordered Heegaard
diagram with an extra basepoint in the complement of the $\alpha$\- and
$\beta$-curves. It encodes a knot in a bordered 3-manifold. There is an
entirely similar pairing construction for a doubly-pointed bordered Heegaard
diagram and an immersed multicurve in the marked torus. We do not spell out
the wordy definition and simply refer the readers to Figure 12 for an example.

Figure 12. Pairing construction that gives rise to a doubly pointed Heegaard
diagram.

We want to establish the unobstructedness and admissibility of the immersed
Heegaard diagrams obtained from pairing constructions. For that we need two
tools, namely _z-adjacency_ and _the collapsing map_ introduced in the next
two subsections.

###  4.3. z-adjacency

We will consider a diagrammatic condition for immersed multicurves that
guarantees the unobstructedness of the paring diagram; this condition can be
achieved easily by finger moves. We begin by introducing some terminology for
convenience.

In the definition below, we orient the curves in $\alpha_{im}$ arbitrarily and
orient the four edges of the cut-open torus using the boundary orientation.
For each edge of the cut-pen torus, let $k_{+}$ and $k_{-}$ denote the number
of elementary arcs intersecting a given edge positively and negatively,
respectively.

Figure 13. The disks $U^{R}_{-}$ and $U^{L}_{-}$. The superscript is chosen to
suggest whether $z$ is one the left or on the right of the strands when we
traverse an arc in the indicated direction.

######  Definition 4.10.

Let $\alpha_{im}$ be an immersed multicurve in the marked torus. Then
$\alpha_{im}$ is _$z$ -adjacent_ if, for each of the four edges of the cut-
open torus, there exist four open disks $U_{\pm}^{R}$ and $U_{\pm}^{L}$ in
$T^{2}$ such that

  * (1)

$(U_{-}^{L},U_{-}^{L}\cap(\alpha_{im}\cup\\{{z}\\})$,
$(U_{-}^{R},U_{-}^{R}\cap(\alpha_{im}\cup\\{{z}\\})$,
$(U_{+}^{L},U_{+}^{L}\cap(\alpha_{im}\cup\\{{z}\\})$ and
$(U_{+}^{R},U_{+}^{R}\cap(\alpha_{im}\cup\\{{z}\\})$ are homeomorphic to the
corresponding disks in Figure 13, where the arcs in the disks are sub-arcs on
the $k_{-}$ distinct elementary arcs intersecting the given edge negatively
for discs with subscript $-$ or sub-arcs on the $k_{+}$ distinct elementary
arcs intersecting the given edge positively for discs with subscript $+$;

  * (2)

if the given edge is the top edge, then $U^{L}_{-}$ and $U^{R}_{+}$ are
contained in $[0,1]\times[0,1]$;

  * (3)

if the given edge is the right edge, then $U^{R}_{-}$ and $U^{L}_{+}$ are
contained in $[0,1]\times[0,1]$.

######  Proposition 4.11.

Every immersed multicurve in the marked torus is regularly homotopic to a
$z$-adjacent multicurve.

###### Proof.

Orient $\alpha_{im}$ arbitrarily. We first define an operation on a collection
of oriented parallel arcs. Assume there are $k_{+}+k_{-}$ arcs, where
$k_{+}$-many of the arcs are oriented in one direction, and the rest are
oriented in the opposite direction.

Figure 14. Finger moves on parallel strands.

The operation is shown in Figure 14: First, by performing the finger moves in
Figure 14 (a) repeatedly, we can arrive at a collection of arcs as shown in
the left of Figure 14 (b): the $P$\- and $P^{-1}$-boxes indicate a pair of
mutually inverse permutations, and between the $P$\- and $P^{-1}$-boxes the
arcs are arranged so that all $k_{-}$ arcs with parallel orientations are
grouped on the left and all the other $k_{+}$ arcs with the opposite
orientations are grouped on the right. Next, do a sequence of finger moves to
the diagram on the left of Figure 14 (b) to arrive at the right-hand-side
diagram of Figure 14 (b). Now perform this operation to the arcs of
$\alpha_{im}$ near all four edges in the cut-open marked torus, then we have a
z-adjacent immersed multicurve; see Figure 15 for the desired open disks. Note
that conditions $(2)$ and $(3)$ are obviously satisfied because $z$ is in the
top right corner of the cut open torus. ∎

Figure 15. A $z$-adjacent immersed curve.

We shall need a technical lemma. Let $l$ be a one-cornered sub-loop of
$\alpha_{im}$ with a corner $q$. If we traverse $l$ in either direction, we
see it begins with an arc starting from $q$ to the meridian or longitude, then
a sequence of elementary arcs, and finally, an arc starting from the meridian
or longitude and ending at $q$. We call the starting and ending arcs the _non-
elementary sub-arcs of $l$_, and the other sub-arcs _the elementary sub-arcs
of $l$_.

######  Lemma 4.12.

Let $\alpha_{im}$ be a $z$-adjacent immersed curve. Let $D$ be a positive
domain in ${T}^{2}$ bounded by a $k$-cornered (sub)loop of $\alpha_{im}$.

  * (1)

If $n_{z}(D)=n$ for some $n\geq 0$ and $k=0$ or $1$, then for any side of the
cut-open marked torus $[0,1]\times[0,1]$ and any sign, the number of
elementary sub-arcs in $\partial D$ intersecting the given side with the given
sign is less than or equal to $n$.

  * (2)

If $n_{z}(D)=0$, then for arbitrary $k\geq 0$, there are no elementary subarcs
contained in $\partial D$.

###### Proof.

We prove (1) first. We will only consider the case in which the elementary
sub-arcs intersect the given edge negatively and remark that the other case is
similar.

We prove by contradiction. Suppose there are $k_{-}>n$ elementary sub-arcs
contained in $\partial D$ intersecting the given edge negatively. Since
$\partial D$ is $0$\- or $1$-cornered it has an orientation induced by the
orientation on $\alpha_{im}$. Examining the local diagram
$(U_{-}^{L},U_{-}^{L}\cap({\partial D}\cup\\{\bm{z}\\}))$ in Figure 13 one
sees $D$ has negative multiplicity $n-k_{-}$ in the left-most region, which
contradicts our assumption that $D$ is a positive domain. Therefore,
$k_{-}\leq n$.

Next, we prove (2). Assume there is an elementary sub-arc in $\partial D$.
Then no matter how this sub-arc is oriented, $z$ is on both the left and right
of it. As $n_{z}(D)=0$, there is a region with $-1$ multiplicity, which
contradicts that $D$ is positive. ∎

###  4.4. The collapsing operation

To relate the domains of the pairing diagram $\mathcal{H}^{a}(\alpha_{im})$
and the arced bordered diagram $\mathcal{H}^{a}$, we define the so-called
_collapsing operation_. This operation was previously defined in the case of
paring genus one bordered Heegaard diagrams with immersed curves [Che23], and
we give the general case here. The operation is pictorially shown in Figure
16, and the definition is given below.

######  Definition 4.13.

The collapsing operation on $\mathcal{H}^{a}(\alpha_{im})$ is defined to be
the composition of the following modifications of the diagram:

  * (Step 1)

Extend the map $h$ in Definition 4.9 to identify
$T^{2}-\pi([-\frac{1}{4}+\epsilon,\frac{1}{4}-\epsilon]\times[-\frac{1}{4}+\epsilon,\frac{1}{4}-\epsilon])$
with a slightly larger neighborhood of
$U=h^{-1}(T^{2}-\pi([-\frac{1}{4},\frac{1}{4}]\times[-\frac{1}{4},\frac{1}{4}]))$.
Here $\epsilon$ is a sufficiently small positive number.

  * (Step 2)

Puncture $h^{-1}((\frac{3}{4},\frac{3}{4}))$, and enlarge it to a hole so that
under the identification map $h$, the boundary of the hole is a square of side
length $\frac{1}{2}+2\epsilon$ and with rounded corner modeled on a quarter of
a circle of radius $\epsilon$. While enlarging the hole, we push immersed
curves it encountered along the way so that part of the immersed curves are
squeezed to the boundary of the hole.

  * (Step 3)

Collapse
$h^{-1}(\pi([-\frac{1}{4}+\epsilon,\frac{1}{4}-\epsilon]\times[\frac{1}{4},\frac{3}{4}]))$
to the core
$h^{-1}(\pi([-\frac{1}{4}+\epsilon,\frac{1}{4}-\epsilon]\times\\{\frac{1}{2}\\}))$,
which is denoted $a^{a,R}_{1}$. Collapse
$h^{-1}(\pi([\frac{1}{4},\frac{3}{4}]\times[-\frac{1}{4}+\epsilon,\frac{1}{4}-\epsilon]))$
to the core
$h^{-1}(\pi(\\{\frac{1}{2}\\}\times[-\frac{1}{4}+\epsilon,\frac{1}{4}-\epsilon]))$,
which is denoted $a^{a,R}_{2}$.

######  Remark 4.14.

  * (1)

Clearly, the outcome of the collapsing operation on
$\mathcal{H}^{a}(\alpha_{im})$ can be identified with $\mathcal{H}^{a}$.

  * (2)

Each elementary arc in $\alpha_{im}$ standing for
$\rho_{I}\in\\{\rho_{1},\rho_{2},\rho_{3},\rho_{12},\rho_{23},\rho_{123}\\}$
is mapped under the collapsing map to an arc that passes the Reeb chord
$\rho_{I}$ in $\mathcal{Z}^{R}$ of $\mathcal{H}^{a}$. Note that an oriented
elementary sub-arc is correctly oriented if it induces a Reeb chord
$\mathcal{Z}^{R}$ under the collapsing map, i.e., the orientations coincide.

  * (3)

The intersection points in $\mathcal{G}({\mathcal{H}^{a}(\alpha_{im})})$ are
of the form $\bm{x}\otimes a$ are in one-to-one correspondence with
$\mathcal{G}({\mathcal{H}^{a}})\otimes_{\mathcal{I}_{R}}\mathcal{G}({\alpha_{im}})$,
where the tensor product is taken over
$\mathcal{I}_{R}\subset\mathcal{A}(\mathcal{Z}_{R})$. Indeed, given an
intersection point $\xi\in\mathcal{G}({\mathcal{H}^{a}(\alpha_{im})})$, its
image under the collapsing map yields an intersection point $\bm{x}$ in
$\mathcal{H}^{a}$. Also, the component of $\xi$ on $\alpha_{im}$ uniquely
gives rise to an intersection point $a$ of $\alpha_{im}$ as follows. By the
definition of the pairing operation, when we pull back the intersection point
on $\alpha_{im}$ to the marked torus, it lies in a horizontal or vertical arc
as described in assumption (C-4) on immersed multicurves, which uniquely
corresponds to an intersection point of $\alpha_{im}$ with the longitude or
meridian. Therefore, every intersection point $\xi$ in
$\mathcal{H}^{a}(\alpha_{im})$ can be written as $\bm{x}\otimes y$. It is easy
to see this induces a one-to-one correspondence between
$\mathcal{G}({\mathcal{H}^{a}(\alpha_{im})})$ and
$\mathcal{G}({\mathcal{H}^{a}})\otimes_{\mathcal{I}_{R}}\mathcal{G}({\alpha_{im}})$.

Figure 16. The collapsing operation.

We will give a proposition relating the domains of
$\mathcal{H}^{a}(\alpha_{im})$ and $\mathcal{H}^{a}$. Let $l$ be an oriented
arc $l$ on $\alpha_{im}$ such that all the elementary sub-arcs are oriented
correctly. We use $\overrightarrow{\rho}(l)$ to denote the sequence of Reeb
chords determined by $l$.

######  Proposition 4.15.

Assume the immersed multicurve $\alpha_{im}$ is $z$-adjacent. Let $B$ be a
positive domain in $\mathcal{H}^{a}(\alpha_{im})$ corresponding to a homology
class in $\pi_{2}(\bm{x}\otimes a,\bm{y}\otimes b,\overrightarrow{\sigma})$
with $n_{z}(B)=0$. Then the image of $B$ under the collapsing map is a
positive domain $B^{\prime}$ in $\mathcal{H}^{a}$ corresponding to a homology
class
$\pi_{2}(\bm{x},\bm{y},\overrightarrow{\rho}(\partial_{\alpha_{im}}B),\overrightarrow{\sigma})$
with $n_{z}(B^{\prime})=0$. Here, $\partial_{\alpha_{im}}B$ refers to the arc
on $\alpha_{im}$ connecting the corresponding components of $\bm{x}\otimes a$
and $\bm{y}\otimes b$. Moreover,

$e(B^{\prime})=e(B)-\frac{|\overrightarrow{\rho}(\partial_{\alpha_{im}}B)|}{2}.$

###### Proof.

It is clear that $B^{\prime}$ is positive and $n_{z}(B^{\prime})=0$. It is
also clear that $B^{\prime}$ give rise to a domain connecting $\bm{x}$ and
$\bm{y}$. We need to show that $B^{\prime}$ has the Reeb chords
$\overrightarrow{\rho}(\partial_{\alpha_{im}}B)$ at the east infinity. We
claim all the elementary arcs appear in $\partial_{\alpha_{im}}B$ are
correctly oriented, and hence $\partial_{\alpha_{im}}B$ gives rise to a
monotonic arc (in the sense that all the Reeb chords appearing on the arc
respect the boundary orientation) connecting (the components on the $\alpha$
arc of) $\bm{x}$ to $\bm{y}$ under the collapsing map. The sequence of Reeb
chords appearing in this arc are exactly
$\overrightarrow{\rho}(\partial_{\alpha_{im}}B)$ in view of Remark 4.14 (2).
To see the claim, note $\alpha_{im}$ is $z$-adjacent and $B$ is positive with
$n_{z}(B)=0$. Therefore, if an elementary arc on $\partial_{im}B$ intersects
the top edge or the right edge, then its orientation is forced by the
positivity of domains and condition (2) and (3) in Definition 4.10, and the
orientation is the correct orientation. The only type of elementary arcs that
intersects neither the top edge nor the right edge corresponds to $\rho_{2}$.
If an elementary arc corresponding to $\rho_{2}$ on $\partial_{\alpha_{im}}B$
has a successor or precursor, then correct orientation on the successor or the
precursor would induce the correct orientation on it. Otherwise,
$\partial_{\alpha_{im}}B$ has only one elementary arc corresponding to
$\rho_{2}$, in which case it is clear that the elementary arc is correctly
oriented.

Next, we compare the Euler measures. Divide the domain $B$ into two parts
$B_{1}$ and $B_{2}$, along the square with rounded corners, which is the
boundary of the hole in Step 2 of the collapsing operation. (See Figure 17.)
This time, we do not puncture the interior of the square. Let $B_{1}$ denote
the part of $B$ outside of the square, and let $B_{2}$ denote the part inside
the square (which is pushed onto the boundary circle under the collapsing
map). Then $e(B_{1})=e(B^{\prime})$ since these two domains differ by a bunch
of rectangles whose Euler measure are zero; these rectangles are collapsed in
Step 3 of the collapsing operation. As $\alpha_{im}$ is $z$-adjacent, $B_{2}$
is positive, and $n_{z}(B)=0$, we see $B_{2}$ can be further expressed as a
sum of of simple domains determined by the elementary arcs appearing in
$\partial_{\alpha_{im}}B$ (counted with multiplicity). (See Figure 18.) Each
simple domain of multiplicity one has Euler measure $\frac{1}{2}$, and there
are $|\overrightarrow{\rho}(\partial_{\alpha_{im}}B)|$ many of them being
collapsed (in Step 2 of the collapsing operation) in order to obtain
$B^{\prime}$. Therefore,
$e(B^{\prime})=e(B)-\frac{|\overrightarrow{\rho}(\partial_{\alpha_{im}}B)|}{2}.$
∎

Figure 17. Left: $B=B_{1}+B_{2}$. Right: $B^{\prime}$. Figure 18. Simple
domains corresponding to Reeb elements in $\mathcal{A}$.

###  4.5. Unobstructedness and admissibility of paring diagrams

######  Proposition 4.16.

Let $\alpha_{im}\subset T^{2}$ be a $z$-adjacent immersed multicurve. Then the
pairing diagram $\mathcal{H}^{a}(\alpha_{im})$ of an arced bordered Heegaard
diagram $\mathcal{H}^{a}$ and $\alpha_{im}$ is unobstructed. Furthermore,
$\mathcal{H}^{a}(\alpha_{im})$ is provincially admissible provided
$\mathcal{H}^{a}$ is left provincially admissible. (See Definition 2.8 and
Definition 2.9.)

###### Proof of Proposition 4.16.

Consider the bordered Heegaard diagram
$\mathcal{H}^{a}(\alpha_{im})=(\bar{\Sigma}^{\prime},\bar{\bm{\alpha}}^{\prime},\bm{\beta}^{\prime},z^{\prime})$
obtained from pairing an arced bordered Heegaard diagram
$\mathcal{H}^{a}=(\bar{\Sigma},\bar{\bm{\alpha}},\bm{\beta},\bm{z})$ and a
$z$-adjacent immersed multicurve $\alpha_{im}$.

We begin by showing $\bar{\bm{\alpha}}^{\prime}$ is unobstructed in the sense
of Definition 2.8. Let $B$ be a zero- or one-cornered $\alpha$-bounded domain.
Since the curves
$\\{\bar{\alpha_{1}^{a}},\bar{\alpha_{2}^{a}},\alpha_{1},\ldots,\alpha_{g-2},\alpha_{im}^{0}\\}$
are pairwise disjoint and homologically independent in
$H_{1}(\bar{\Sigma},\partial)$, $[\partial B]$ (as a homology class) is equal
to a linear combination of at most one copy of $\partial\bar{\Sigma}$ and some
homologically trivial zero- or one-cornered loop contained in a single
connected component of $\alpha_{im}$.

We first show there are no positive zero- or one-cornered $\alpha$-bounded
domains $B$ with $n_{z^{\prime}}(B)=0$. In this case, $\partial B$ is a
homologically trivial zero- or one-cornered loop contained in a single
connected component of $\alpha_{im}$, i.e., $\partial\bar{\Sigma}$ does not
appear in $\partial B$. As the $\bm{\beta}$-curves are irrelevant to our
consideration, we may assume the $\alpha$-curves of $\mathcal{H}^{a}$ are in
standard position. Therefore, there is an obvious circle
$C\subset\bar{\Sigma}$ that splits $\bar{\Sigma}$ into a genus-$(g-1)$ surface
$E_{1}$ containing
$\\{\alpha^{a,L}_{1},\alpha^{a,L}_{2},\alpha^{c}_{1},\ldots,\alpha^{c}_{g-2}\\}$
and a genus-one surface $E_{2}$ containing $\alpha^{a,R}_{1}$ and
$\alpha^{a,R}_{2}$. Let $C^{\prime}$ be the corresponding curve on
$\bar{\Sigma}^{\prime}$. Then after surgery along $C^{\prime}$, $B$ induces a
positive domain $D$ in the marked torus $T^{2}$ (obtained from $E_{2}$ in an
obvious way), and $D$ is bounded by a zero- or one-cornered (sub)loop of
$\alpha_{im}$. According to Lemma 4.12, $\partial D$ contains no elementary
sub-arcs, so $D$ cannot exist.

Next we show that if $n_{z^{\prime}}(B)=1$, $B$ is a stabilized teardrop or
$[\Sigma^{\prime}]$ depending on whether $\partial B$ is one-cornered or zero-
cornered. In this case, after performing surgery along the same $C^{\prime}$
as in the previous paragraph, $B$ gives rise to two domains: one is
$[E_{1}^{\prime}]$, where $E^{\prime}_{1}$ is the genus-$(g-1)$ surface, and
the other is a positive domain $D$ contained in the marked torus $T^{2}$ with
$n_{z}(D)=1$. We first consider the case in which $\partial D$ is zero-
cornered. If $\partial D=\emptyset$, then $D=E_{2}$ and hence
$B=[\Sigma^{\prime}]$. If $\partial D\neq\emptyset$, then according to Lemma
4.12, it consists of at most (and at least) $4$ elementary sub-arcs, and hence
is a circle enclosing the $z$-basepoint once. However, such circles are
assumed not to exist. When $\partial D$ is one-cornered, we claim $D$ is a
teardrop. To see this, note that Lemma 4.12 implies that $\partial D$ crosses
the meridian at most three times and the longitude at most three times since
each time the meridian or the longitude is crossed (except possibly the last
time) the intersection is the beginning of an elementary sub-arc and there are
at most two elementary sub-arcs starting on each. Because $\partial D$ is
homologically trivial in $H_{1}({T}^{2})$ it crosses both of the meridian and
the longitude and even number of times, so it crosses each at most twice. It
follows that $\partial D$ must circle once around $z$ and $D$ is a teardrop
with $n_{z}(D)=1$.

Now we show any two-cornered positive $\alpha$-bounded domain $B$ with
$n_{z}(B)=0$ is a bigon. To see, we may split $\Sigma^{\prime}$ as
$E_{1}\\#E_{2}$ as before and regard $B$ as a domain in $E_{2}$ with
$n_{z^{\prime}}=0$. Note by Lemma 4.12 (2), we know $\partial B$ consists of
no elementary subarcs, and hence $B$ must be of the form shown in Figure 19
(up to rotation), which is a bigon. (Note we do not require the corners of the
bigon $B$ to be convex.)

Figure 19. Two-cornered positive $\alpha$-bounded domains.

So far, we have proved $\bar{\bm{\alpha}}^{\prime}$ is unobstructed. We now
show there are no non-trivial positive provincial periodic domains. If not,
let $B$ be a positive provincial periodic domain for
$\mathcal{H}^{a}(\alpha_{im})$. Then by Proposition 4.15, $\Psi(B)$ is a
positive periodic domain for $\mathcal{H}^{a}$, where $\Psi$ denotes the
collapsing map. Note $\Psi(B)$ is left provincial. As $\mathcal{H}^{a}$ is
left provincially admissible, we have $\Psi(B)=0$, and hence $\partial B$ has
no $\beta$-curves. So, $B$ is a positive zero-cornered $\alpha$-bounded domain
with $n_{z}(B)=0$, but such domains are already excluded by unobstructedness.
∎

###  4.6. The first paring theorem

Recall a bordered Heegaard diagram is _nice_ if every connected region in the
complement of the $\alpha$\- and $\beta$-curves is a disk with at most four
corners except for the region containing $z$. Any bordered Heegaard diagram
can be turned into a nice diagram via isotopy and handleslides of the
$\beta$-curves (Proposition 8.2 of [LOT18]). The key property of nice Heegaard
diagrams is that the Euler measure of any region away from the base point is
non-negative. This property imposes great constraints on domains supporting
holomorphic representatives via the index formula. Hence it opens up the
combinatorial vein for proving the pairing theorem.

See 1.4

###### Proof.

In view of the homotopy equivalence of the relevant invariants under isotopy
of $\beta$ curves (Proposition 2.82), we may assume $\mathcal{H}^{a}$ is a
nice arced bordered Heegaard diagram. Note nice arced bordered Heegaard
diagrams are automatically left provincially admissible. Therefore,
$\widehat{CFDA}(\mathcal{H}^{a})$ and
$\widehat{CFD}(\mathcal{H}^{a}({\alpha_{im}}))$ are defined. In fact, a
stronger admissibility condition holds for $\mathcal{H}^{a}$: any periodic
domains with $n_{z}=0$ has both positive and negative local multiplicities.
This implies $\widehat{CFDA}(\mathcal{H}^{a})$ is bounded, and hence the box-
tensor product is expressed as a finite sum.

Implicit in the proof is that we will be using split almost complex structures
for defining $\widehat{CFD}(\mathcal{H}^{a}({\alpha_{im}}))$ and
$\widehat{CFDA}(\mathcal{H}^{a})$. A split almost complex structure is
sufficient for defining $\widehat{CFD}(\mathcal{H}^{a}({\alpha_{im}}))$, since
all the domains involved will be bigons and rectangles. In this setting, up to
a generic perturbation of the $\alpha$ and $\beta$ curves, moduli spaces
defined using a split almost complex structure are transverse (c.f. [Lip06,
Proposition 3.9]).

We will call the two punctures in $\mathcal{H}^{a}$ the $\sigma$-puncture and
the $\rho$-puncture, where the $\rho$-puncture is the one that gets capped off
in the pairing diagram. For now, we assume the local systems on $\alpha_{im}$
are trivial, and we will indicate the modifications needed for dealing with
nontrivial local system later on. First, the generators
$\mathcal{G}(\mathcal{H}^{a}(\alpha_{im}))$ and
$\mathcal{G}(\mathcal{H}^{a})\otimes_{\mathcal{I}_{R}}\mathcal{G}(\alpha_{im})$
are identified as pointed out in Remark 4.14 (3). Next, we prove the
differentials have a one-to-one correspondence.

We first show any differential incurred by the box-tensor product has a
corresponding differential in $\widehat{CFD}(\mathcal{H}^{a}(\alpha_{im}))$. A
differential arising from the box tensor product comes in two types, depending
on whether it involves nontrivial differentials in
$\widehat{CFD}(\alpha_{im})$. If it does not involve non-trivial differential
in $\widehat{CFD}(\alpha_{im})$, then the input from
$\widehat{CFDA}(\mathcal{H}^{a})$ counts curves with the domain being a
provincial bigon, a provincial rectangle, or a bigon with a single Reeb chord
on the $\sigma$-puncture; see [LOT18, Proposition 8.4]. Such bigons or
rectangles clearly have their counterparts in $\mathcal{H}^{a}(\alpha_{im})$,
giving the corresponding differentials in
$\widehat{CFD}(\mathcal{H}^{a}(\alpha_{im}))$. If the box-tensor differential
involves differentials in $\widehat{CFD}(\alpha_{im})$, then the corresponding
input from $\widehat{CFDA}(\mathcal{H}^{a})$ counts curves with the domain
being a bigon with a single Reeb chord on the $\rho$-puncture [LOT18,
Proposition 8.4]. As it pairs with a differential in
$\widehat{CFD}(\alpha_{im})$, this bigon gives rise to a bigon in
$\mathcal{H}^{a}(\alpha_{im})$ (which is a pre-image of the collapsing map),
giving the corresponding differential in
$\widehat{CFD}(\mathcal{H}^{a}(\alpha_{im}))$.

Next, we show that every differential in
$\widehat{CFD}(\mathcal{H}^{a}(\alpha_{im}))$ corresponds to a differential
incurred by the box-tensor product. Suppose $u\in\pi_{2}(\bm{x}\otimes
a,\bm{y}\otimes b)$ admits a holomorphic representative contributing to a
differential for $\widehat{CFD}(\mathcal{H}^{a}(\alpha_{im}))$. Let $B$ be the
domain of $u$, and let $B^{\prime}$ denote the image of $B$ under the
collapsing operation. By Proposition 4.15,
$e(B)=e(B^{\prime})+\frac{|\overrightarrow{\rho}(\partial_{\alpha_{im}}B)|}{2}$.
As $B^{\prime}$ is a positive domain with $n_{z}(B^{\prime})=0$ and
$\mathcal{H}^{a}$ is a nice Heegaard diagram, we have $e(B)\geq
e(B^{\prime})\geq 0$. By the index formula, denoting the source surface of $u$
by $S$, we have

$\text{Ind}(u)=g-\chi(S)+2e(B)+|\overrightarrow{\sigma}|.$

As $\text{Ind}(u)=1$ and $2e(B)+|\overrightarrow{\sigma}|\geq 0$, we have
$\chi(S)=g$ or $g-1$.

When $\chi(S)=g$, $S$ consists of $g$ topological disks; each disk has a $+$
and a $-$ puncture, and there is at most one $\sigma$-puncture overall since
$2e(B)+|\overrightarrow{\sigma}|=1$. We separate the discussion according to
the number of $\sigma$-punctures. First, if there is a $\sigma$-puncture, then
the corresponding domain $B$ in $\mathcal{H}^{a}(\alpha_{im})$ is a bigon with
a single Reeb chord on the $\sigma$-puncture and does not involve
$\alpha_{im}$. This domain clearly has its counterpart in $\mathcal{H}^{a}$
under the collapsing map, giving rise to an operation in
$\widehat{CFDA}(\mathcal{H}^{a})$; the corresponding differential in the box-
tensor product is obtained by pairing this $DA$-operation with an element in
$\widehat{CFD}(\alpha_{im})$. Secondly, if there is no $\sigma$-puncture, then
the domain $B$ is a provincial bigon in $\mathcal{H}^{a}(\alpha_{im})$. There
are two sub-cases to consider depending on whether the $\alpha$-boundary of
$B$ overlaps with $\alpha_{im}$. If the $\alpha$ boundary of $B$ is not on
$\alpha_{im}$, then we argue as in the first case to see that $B$ gives a
corresponding differential in the box-tensor product. If, on the other hand,
the boundary of $B$ is on $\alpha_{im}$, since
$1/2=e(B)=e(B^{\prime})+|\overrightarrow{\rho}(\partial_{\alpha_{im}}B)|/{2}$
we have $|\overrightarrow{\rho}(\partial_{\alpha_{im}}B)|$ is either $0$ or
$1$. If $|\overrightarrow{\rho}(\partial_{\alpha_{im}}B)|=0$, then
$B^{\prime}$ is a provincial domain, giving the type-DA operation for the
corresponding differential obtained by the box-tensor product. If
$|\overrightarrow{\rho}(\partial_{\alpha_{im}}B)|=1$, then $B^{\prime}$ is
obtained from $B$ subtracting a simple region (as in the proof of Proposition
4.15) and then applying the collapsing map. We can see $B^{\prime}$ is a bigon
with a single Reeb chord corresponding to the Reeb chord specified by
$\partial_{\alpha_{im}}B$ on the $\rho$-puncture. The DA-operation given by
$B^{\prime}$ and the type D operation given by $\partial_{\alpha_{im}}B$ pair
up to give the corresponding differential in the box-tensor product.

When $\chi(S)=g-1$, then $S$ consists of $g-1$ topological disks; $g-2$ of the
disks are bigon, while the remaining one is a rectangle. As $e(B)=0$, the
bigons are mapped trivially to $\Sigma$. Therefore, the domain $B$ is a
rectangle. Again, since
$e(B)=e(B^{\prime})+|\overrightarrow{\rho}(\partial_{\alpha_{im}}B)|/{2}$ and
$e(B^{\prime})\geq 0$, we have
$|\overrightarrow{\rho}(\partial_{\alpha_{im}}B)|=0$. Then $B^{\prime}$ is a
provincial rectangular domain in $\mathcal{H}^{a}$, giving rise to a DA-
operation that pairs with a trivial type-D operation to give the corresponding
differential in the box-tensor product. We have finished the proof when the
local system is trivial.

Next, we consider the case where $\alpha_{im}$ admits a non-trivial local
system $(\mathcal{E},\Phi)$. The local system induces a local system on the
$\alpha$ curves in the pairing diagram $H^{a}(\alpha_{im})$. First, the
discussion above identifies the generators at the vector space level: let
$\bm{x}\otimes y$ be an intersection point in
$\mathcal{G}(\mathcal{H}^{a}(\alpha_{im}))$, where
$\bm{x}\in\mathcal{G}(H^{a})$ and $y\in\mathcal{G}(\alpha_{im})$; then
$\bm{x}\otimes y$ corresponds to a direct summand $\mathcal{E}|_{\bm{x}\otimes
y}$ of $\widehat{CFD}(\mathcal{H}^{a}(\alpha_{im}))$ as a vector space, and
$\mathcal{E}|_{\bm{x}\otimes y}$ can be naturally identified with
$\bm{x}\otimes\mathcal{E}|_{y}$, a summand of
$\widehat{CFDA}(\mathcal{H}^{a})\boxtimes\widehat{CFD}(\alpha_{im})$.
Secondly, the discussion in the trivial-local-system case shows that
$\widehat{CFD}(\mathcal{H}^{a}(\alpha_{im}))$ has a differential map between
the summands $\mathcal{E}|_{\bm{x}\otimes
y}\rightarrow\sigma_{I}\otimes\mathcal{E}|_{\bm{x^{\prime}}\otimes
y^{\prime}}$ if and only if the box-tensor product has a differential map
between the corresponding summands
$\bm{x}\otimes\mathcal{E}|_{y}\rightarrow\sigma_{I}\otimes(\bm{x^{\prime}}\otimes\mathcal{E}|_{y^{\prime}})$
in the box-tensor product, and under the natural identification between these
summands both differential maps are induced by the same parallel transport
from $\mathcal{E}|_{y}$ to $\mathcal{E}|_{y^{\prime}}$.

∎

###  4.7. The second pairing theorem

We are interested in computing knot Floer chain complexes over
$\mathcal{R}=\mathbb{F}[U,V]/(UV)$ using bordered Floer homology. We have
already defined an extended type-D structure, and we want to pair it with an
extended type-A structure to get a bi-graded chain complex over
$\mathcal{R}=\mathbb{F}[U,V]/(UV)$. Here we will only restrict to a specific
extended type-A structure associated to the doubly-pointed bordered Heegaard
diagram $\mathcal{H}_{id}$ given in Figure 20. The diagram $\mathcal{H}_{id}$
corresponds to the pattern knot given by the core of a solid torus, which is
the identity pattern.

Figure 20. The bordered diagram $\mathcal{H}_{id}$.

Recall $\tilde{\mathcal{A}}$ denotes the weakly extended torus algebra, and
$\mathcal{I}\subset\tilde{\mathcal{A}}$ is the ring of idempotents (see Figure
8).

######  Definition 4.17.

The extended type-A structure $\widetilde{CFA}(\mathcal{H}_{id})$ is a free
$\mathcal{R}$-module generated by the single intersection point $x$ in
$\mathcal{H}_{id}$. It is equipped with an $\mathcal{I}$-action given by
$x\cdot\iota_{0}=x$ and $x\cdot\iota_{1}=0$, together with a family of
$\mathcal{R}$-linear maps
$m_{i+1}:\widetilde{CFA}(\mathcal{H}_{id})\otimes_{\mathcal{I}}\tilde{A}^{\otimes
i}\rightarrow\widetilde{CFA}(\mathcal{H}_{id})$ ($i\in\mathbb{N}$), where up
to $\mathcal{R}$-linearity the only non-zero relations are:

$\displaystyle m_{2}(x,1)=x,$ $\displaystyle
m_{3+i}(x,\rho_{3},\overbrace{\rho_{23},\ldots,\rho_{23}}^{i},\rho_{2})=U^{i}x,\quad
i\in\mathbb{N},$ $\displaystyle
m_{3+i}(x,\rho_{1},\overbrace{\rho_{01},\ldots,\rho_{01}}^{i},\rho_{0})=V^{i}x,\quad
i\in\mathbb{N}.$

######  Remark 4.18.

This extends the hat-version type A-structure
$\widehat{CFA}(\mathcal{H}_{id})$ by allowing Reeb chords crossing the base
point $z$.

Straightforwardly, the hat-version box-tensor product can be generalized to be
an operation between the extended type A structure
$\widetilde{\mathcal{M}}:=\widetilde{CFA}(\mathcal{H}_{id})$ and a weakly
extended type D structure $(\widetilde{\mathcal{N}},\delta^{i})$: It is the
$\mathcal{R}$-module
$\widetilde{\mathcal{M}}\otimes_{\mathcal{I}}\widetilde{\mathcal{N}}$ together
with a differential $\partial_{\boxtimes}:=\sum_{i\geq
0}(m_{i+1}\otimes\mathbb{I}_{\widetilde{\mathcal{N}}})\circ(\mathbb{I}_{\widetilde{\mathcal{M}}}\otimes\delta^{i})$;
the finiteness of the sum can be guaranteed for type D structures defined
using bi-admissible diagrams (see the proof of Theorem 1.6 below). One may
verify $\partial_{\boxtimes}^{2}=0$ algebraically using the structure
equations defining the (weakly) extended type D and type A structures. We omit
such computation and instead content ourselves with Theorem 1.6 below, which
implies that the $\partial_{\boxtimes}$ induced by gluing bordered Heegaard
diagrams is indeed a differential. We further remark that
$\widetilde{\mathcal{M}}\boxtimes_{\mathcal{I}}\widetilde{\mathcal{N}_{1}}$ is
chain homotopic to
$\widetilde{\mathcal{M}}\boxtimes_{\mathcal{I}}\widetilde{\mathcal{N}_{2}}$
provided $\widetilde{\mathcal{N}_{1}}$ is homotopic to
$\widetilde{\mathcal{N}_{2}}$. The proof of this is similar to that in the hat
version and is omitted.

See 1.6 (See Definition 2.8 and 2.10 for the unobstructedness and bi-
admissibility for $\mathcal{H}$.)

###### Proof of Theorem 1.6.

Note periodic domains of $\mathcal{H}_{id}\cup\mathcal{H}_{im}$ with $n_{z}=0$
(respectively $n_{w}=0$) corresponds to periodic domains of $\mathcal{H}_{im}$
with $n_{\rho_{0}}=n_{\rho_{1}}=0$ (respectively
$n_{\rho_{2}}=n_{\rho_{3}}=0$). Therefore, since $\mathcal{H}_{im}$ is bi-
admissible, $\mathcal{H}_{id}\cup\mathcal{H}_{im}$ is bi-admissible in the
sense of Definition 3.3. Also, zero- or one-cornered $\alpha$-bounded domains
in $\mathcal{H}_{id}\cup\mathcal{H}_{im}$ with $n_{z}=n_{w}=0$ must lie in
$\mathcal{H}_{im}$. So, unobstructedness of $\mathcal{H}_{im}$ implies the
unobstructedness of $\mathcal{H}_{id}\cup\mathcal{H}_{im}$. In summary,
$\mathcal{H}_{id}\cup\mathcal{H}_{im}$ is bi-admissible and unobstructed, and
hence $CFK_{\mathcal{R}}(\mathcal{H}_{id}\cup\mathcal{H}_{im})$ is defined.
The bi-admissibility of $\mathcal{H}_{im}$ also implies $\partial_{\boxtimes}$
is expressed as a finite sum and hence is well-defined. To see this, note for
any $\bm{x},\bm{y}\in\mathcal{G}(\mathcal{H}_{im})$, bi-admissibility implies
there are only finitely many positive domains connecting $\bm{x}$ and $\bm{y}$
with a prescribed Reeb-chord sequence of the form
$\rho_{1},\rho_{01},\ldots\rho_{01},\rho_{0}$ and
$\rho_{3},\rho_{23},\ldots\rho_{23},\rho_{2}$.

Recall that the differential in
$CFK_{\mathcal{R}}(\mathcal{H}_{id}\cup\mathcal{H}_{im})$ counts holomorphic
curves that can only cross at most one of the $w$\- and $z$-base points. Note
also that both of the base points in $\mathcal{H}_{id}$ are adjacent to the
east boundary. Therefore, by the symmetry of the base points $w$ and $z$, it
suffices to prove the theorem for the hat version knot Floer homology, i.e.,

$\widehat{CFK}(\mathcal{H}_{id}\cup\mathcal{H}_{im})\cong\widehat{CFA}(\mathcal{H}_{id})\boxtimes\widehat{CFD}(\mathcal{H}_{im}).$

Though our Heegaard diagrams have immersed $\alpha$-multicurves, given that no
boundary degeneration can occur, the proof for the embedded-$\alpha$-curves
case, which uses neck stretching and time dilation, carries over without
changes; see Chapter 9 of [LOT18] for detail or Section 3 of [LOT14a] for an
exposition. ∎

##  5\. Knot Floer homology of satellite knots

We apply the machinery developed in the previous sections to study the knot
Floer homology of satellite knots. First, we introduce a gentler condition on
the immersed curves than the z-adjacency condition.

######  Definition 5.1.

An immersed multicurve $\alpha_{im}$ in the marked torus $(T^{2},z)$ is
admissible if there are no nontrivial zero- and one-cornered $\alpha$-bounded
positive domains $B$ with $n_{z}(B)=0$.

Note any $z$-adjacent immersed multicurve is admissible in view of Lemma 4.12.

Let $\mathcal{H}_{w,z}$ be a doubly pointed bordered Heegaard diagram for a
pattern knot $(S^{1}\times D^{2},P)$. Recall that we can construct a doubly
pointed immersed diagram $\mathcal{H}_{w,z}(\alpha_{im})$. The admissibility
condition guarantees that $CFK_{\mathcal{R}}(\mathcal{H}_{w,z}(\alpha_{im}))$
is defined in view of the following proposition.

######  Proposition 5.2.

If $\alpha_{im}$ is an admissible immersed multicurve, then
$\mathcal{H}_{w,z}(\alpha_{im})$ is bi-admissible and unobstructed.

###### Proof.

First, we show the diagram $\mathcal{H}_{w,z}(\alpha_{im})$ is unobstructed
(in the sense of Definition 3.2). Let $B$ be a zero- or one-cornered
$\alpha$-bounded domain for $\mathcal{H}_{w,z}(\alpha_{im})$. Note
$n_{w}(B)=n_{z}(B)$ since the base points $w$ and $z$ are in the same region
in the complement of the $\alpha$-curves. We restrict to the domains with
$n_{z}(B)=n_{w}(B)=0$. Recall we want to prove $B$ must be somewhere negative.
Splitting the Heegaard surface as in the proof of Proposition 4.16, we see $B$
corresponds to a zero- or one-cornered $\alpha$-bounded domain $B^{\prime}$ in
the marked torus with $n_{z}(B^{\prime})=0$. Since $\alpha_{im}$ is
admissible, $B^{\prime}$ is somewhere negative. Therefore, $B$ is somewhere
negative.

Next we show the pairing diagram $\mathcal{H}_{w,z}(\alpha_{im})$ is bi-
admissible. Recall that bi-admissibility means any nontrivial periodic domains
$B$ with $n_{w}(B)=0$ or $n_{z}(B)=0$ must have both positive and negative
coefficients. To see this, we first claim any given periodic domain $B$ is
bounded by some multiple of a homologically trivial component of
$\alpha_{im}$. (We warn the reader that the claim is no longer true if one
further performs a handleslide of such a component over an embedded alpha
curve.) To see the claim, note that as homology classes, the curves
$[\alpha_{i}]$ ($i=1,\ldots,g-1$), $[\alpha^{1}_{im}]$, and $[\beta_{i}]$
($i=1,\ldots,g$) are linearly independent, just as the attaching curves in a
Heegaard diagram for $S^{3}$. Now, the claim implies $B$ is a zero-cornered
$\alpha$-bounded domain. In view of the unobstructedness established above,
$B$ is somewhere negative. ∎

######  Definition 5.3.

Let $\alpha_{im}$ and $\alpha^{\prime}_{im}$ be two admissible immersed
multicurves. They are said to be admissibly equivalent if there exists a
finite sequence of admissible immersed curves $\alpha^{i}_{im}$,
$i=1,\ldots,n$ such that

  * (1)

$\alpha^{1}_{im}=\alpha_{im}$ and $\alpha^{n}_{im}=\alpha^{\prime}_{im}$,

  * (2)

For $i=1,\ldots,n-1$, $\alpha^{i}_{im}$ and $\alpha^{i+1}_{im}$ are related by
a finger move that creates/cancels a pair of self-intersection points of the
immersed curves.

######  Proposition 5.4.

Let $\alpha_{im}$ and $\alpha_{im}^{\prime}$ be two admissibly equivalent
immersed multicurves, and let $\mathcal{H}_{w,z}$ be a doubly pointed bordered
Heegaard diagram. Then

$CFK_{\mathcal{R}}(\mathcal{H}_{w,z}(\alpha_{im}))\cong
CFK_{\mathcal{R}}(\mathcal{H}_{w,z}(\alpha^{\prime}_{im})).$

###### Proof.

The proof follows the same strategy as the usual proof of isotopy invariance.
Let $\bm{\alpha_{0}}$ and $\bm{\alpha_{1}}$ be the two sets of
$\alpha$-curves. For simplicity, assume they are related by a single finger
move. We model the finger move using a locally supported exact Hamiltonian
isotopy on $\Sigma$. The isotopy induces a family of $\alpha$-curves,
$\bm{\alpha}_{t}$, on $\Sigma$ ($t\in\mathbb{R}$); for $t\ll 0$ (resp. $t\gg
0$), $\bm{\alpha}_{t}$ is constant with respect to $t$ and is identified with
$\bm{\alpha}_{0}$ (resp. $\bm{\alpha}_{1}$). $\bm{\alpha}_{t}$ induces an
immersed totally real submanifold
$C_{\alpha}=\bm{\alpha}_{t}\times\\{1\\}\times\\{t\\}$ in
$\Sigma\times[0,1]\times\mathbb{R}$. $C_{\alpha}$ can be realized as an
immersion
$\Psi_{t}:(\amalg_{i=1}^{g}S^{1})\times\mathbb{R}\rightarrow\Sigma\times\\{1\\}\times\mathbb{R}.$
Let $C_{\beta}$ be the Lagrangian induced by the $\beta$ curves. For
$\bm{x}\in\mathbb{T}_{\bm{\alpha}_{0}}\cap\mathbb{T}_{\bm{\beta}}$ and
$\bm{y}\in\mathbb{T}_{\bm{\alpha}_{1}}\cap\mathbb{T}_{\bm{\beta}}$, one then
define $\mathcal{M}_{\Psi_{t}}(\bm{x},\bm{y})$ to be the moduli space of
holomorphic curves in $\Sigma\times[0,1]\times\mathbb{R}$ with boundary on
$C_{\alpha}\cup C_{\beta}$ such that the $\alpha$-boundary can be lifted
through $\Psi_{t}$. With this, one can define a map
$\Phi_{0}:CFK_{\mathcal{R}}(\mathcal{H}_{w,z}(\alpha_{im}))\rightarrow
CFK_{\mathcal{R}}(\mathcal{H}_{w,z}(\alpha^{\prime}_{im}))$ by

$\bm{x}\mapsto\sum_{\bm{y}}\sum_{\phi\in\pi_{2}(\bm{x},\bm{y})}\\#\mathcal{M}_{\Psi_{t}}(\bm{x},\bm{y})U^{n_{w}(\phi)}V^{n_{z}(\phi)}\bm{y},$

where $\mathcal{M}_{\Psi_{t}}(\bm{x},\bm{y})$ has dimension zero. Define
$\Phi_{1}:CFK_{\mathcal{R}}(\mathcal{H}_{w,z}(\alpha^{\prime}_{im}))\rightarrow
CFK_{\mathcal{R}}(\mathcal{H}_{w,z}(\alpha_{im}))$ similarly. We remark that
the compactness and gluing results still apply to this setup. The bi-
admissibility of the diagrams obstructs the appearance of boundary
degeneration in the compactification of one-dimensional moduli spaces, and
hence we can still apply the usual argument to show (1) $\Phi_{0}$ and
$\Phi_{1}$ are chain maps, and (2) $\Phi_{0}\circ\Phi_{1}$ and
$\Phi_{1}\circ\Phi_{0}$ are homotopy equivalent to the identity map.
Therefore, $\Phi_{0}$ and $\Phi_{1}$ are chain homotopy equivalences. ∎

######  Definition 5.5.

An immersed multicurve is called $z$-passable if it is admissibly equivalent
to a $z$-adjacent multicurve.

######  Remark 5.6.

We can easily arrange $\alpha_{K}$ to be a $z$-passable multicurve; see
Example 5.7 below. Moreover, when the pattern knot admits a genus-one doubly
pointed Heegaard diagram, we can even drop the admissibility condition; see
Section 6.2.

######  Example 5.7.

We give a simple way to arrange an immersed multicurve $\alpha_{K}$ to be
$z$-passable. Without loss of generality, we consider a single component
$\gamma$ of $\alpha_{K}$ each time, and we orient $\gamma$ arbitrarily. We
view the torus $T^{2}$ as a square as usual and position $\gamma$ such that
the elementary arcs hitting the top edge are separated into two groups of arcs
where the arcs in a single group intersect the top edge in the same direction;
see Figure 21 (1). Next, we perform a Reidemeister-II-like move to the two
groups as in Figure 21 (2). Perform the above modification for every component
of $\alpha_{K}$. We claim the resulting multicurve, which we denote
$\alpha_{K}^{\prime}$, is a $z$-passable multicurve.

Figure 21. (2) is an $z$-passable immersed curve obtained from (1).

We justify the claim when $\gamma$ is homologically trivial; the case where
$\gamma$ is homologically essential is similar. We first check that
$\alpha_{K}^{\prime}$ is admissible by checking that there are no zero- or
one-cornered $\alpha$ bounded domains $B$ with $n_{z}(B)=0$. First note that
for any zero- or one-cornered $\alpha$ bounded domains $B$, $\partial B$ must
include an elementary arc meeting the top edge of the square. To see this,
note that $\partial B$ is a nullhomologous curve in the torus and thus lifts
to a closed path in the universal cover. Cutting along (lifts of) the meridian
(i.e., $\mathbb{Z}\times\mathbb{R}$) breaks $\partial B$ into pieces, with at
least two of these (the leftmost and rightmost piece) forming bigons with the
meridian. At least one of those two pieces has no corners (since $B$ is zero-
or one-cornered). The cornerless piece must intersect the longitude because
$\alpha_{K}^{\prime}$ is reduced, and the subarc of $\partial B$ directly
below this intersection with the longitude gives an elementary arc meeting the
top edge of the square. Next we observe that the elementary arcs near the top
edge of the square are arranged such that each arc has the base point $z$ both
on its left and on its right, in each case without oppositely oriented arcs in
between the arc and $z$, and this implies that no domain whose boundary
includes one of these elementary arcs can have $n_{z}(B)=0$. Having shown the
immersed curve $\alpha_{K}^{\prime}$ is admissible, it remains to check that
it is $z$-passable. Recall from Proposition 4.11 that we can perform a
sequence of finger moves to achieve a $z$-adjacent position. Note that all the
intermediate diagrams are admissible by exactly the same argument above.

###  5.1. Proof of the main theorem, ungraded version

This subsection is devoted to proving the ungraded version of Theorem 1.1.

A satellite knot is constructed via the so-called satellite operation that
requires a pattern knot and a companion knot as input. A pattern knot is an
oriented knot $P$ in an oriented solid torus $S^{1}\times D^{2}$, where an
oriented meridian $\mu$ and an oriented longitude $\lambda$ are chosen for
$\partial(S^{1}\times D^{2})$ so that the orientation determined by
$(\mu,\lambda)$ coincides with the induced boundary orientation. A companion
knot is an oriented knot $K$ in the 3-sphere. We orient any Seifert longitude
of $K$ using the parallel orientation, and orient any meridian $m$ of $K$ so
that $lk(m,K)=1$. The satellite knot $P(K)$ is obtained by gluing
$(S^{1}\times D^{2},P)$ to the companion knot complement
$S^{3}\backslash\nu(K)$ so that the chosen meridian $\mu$ is identified with a
meridian of $K$ and that the chosen longitude $\lambda$ is identified with the
Seifert longitude of $K$; $P(K)$ is given by viewing $P$ as a knot in the
glued-up 3-sphere $(S^{1}\times D^{2})\cup(S^{3}\backslash\nu(K))$.

We state the main theorem again below for the readers’ convenience. Recall
that any pattern knot can be represented by a doubly-pointed bordered Heegaard
diagram [LOT18, Section 11.4]. See 1.1

Given a doubly pointed bordered Heegaard diagram $\mathcal{H}_{w,z}$ for the
pattern knot, we will construct an arced bordered Heegaard diagram
$\mathcal{H}_{X(P)}$; the Heegaard diagram $\mathcal{H}_{X(P)}$ specifies a
bordered 3-manifold $X(P)$ with two boundary components888Strictly speaking,
an arced bordered Heegaard diagram specifies a strongly bordered 3-manifold in
the sense of Definition 5.1 in [LOT15], where there is also a framed arc in
addition to the underlying bordered 3-manifold. This extra structure will not
be relevant to us, so we will not specify it. , where (1) the underlying
3-manifold is $S^{1}\times D^{2}\backslash\nu(P)$, (2) the parametrization of
$\partial(S^{1}\times D^{2})$ is the standard meridian-longitude
parametrization, and (3) the parametrization of interior boundary
$\partial(\nu(P))$ is given by a meridian of $P$ and some longitude of $P$.
(The choice of the longitude of $P$ does not matter).

We describe how to obtain $\mathcal{H}_{X(P)}$ from the doubly pointed
bordered Heegaard diagram $\mathcal{H}_{w,z}$. This is a standard
construction, similar to the one appearing in [LOT18] Section 11.7; the reader
familiar with it may skip this paragraph and consult Figure 22 for an
overview.

Figure 22. An example of obtaining $\mathcal{H}_{X(P)}$ from
$\mathcal{H}_{w,z}$. Here, $\mathcal{H}_{w,z}$ is showed on the top row; it is
a genus-one Heegaard diagram for the $(3,1)$-cable pattern.
$\mathcal{H}_{X(P)}$ is the rightmost diagram on the second row.

Assume $\mathcal{H}_{w,z}$ is of genus $g$. First, we stabilize
$\mathcal{H}_{w,z}=(\bar{\Sigma},\bm{\bar{\alpha}},\bm{\beta},w,z)$ to get a
new doubly pointed bordered Heegaard diagram
$\mathcal{H}^{\prime}_{w,z}=(\bar{\Sigma}^{\prime},\bm{\bar{\alpha}}\cup\\{\alpha^{c}_{g}\\},\bm{\beta}\cup\\{\beta_{g+1}\\},w,z)$.
More concretely, $\bar{\Sigma}^{\prime}$ is obtained from $\bar{\Sigma}$ by
attaching a two-dimensional one-handle, with feet near the base points $w$ and
$z$. Parametrize the new one-handle by $S^{1}\times[0,1]$, where
$S^{1}\times\\{0\\}$ is the feet circle near $z$, and $S^{1}\times\\{1\\}$ is
the feet circle near $w$. We also parametrize $S^{1}$ by $[0,2\pi]/(0\sim
2\pi)$. The new $\alpha$-circle $\alpha^{c}_{g}$ is the belt circle
$S^{1}\times\\{1/2\\}$ of the new one-handle. Let $p_{1}=(0,0)$ and
$p_{2}=(0,1)$ be two points on the two feet circles of the one-handle. The new
$\beta$ circle $\beta_{g+1}$ is the union of two arcs $l_{1}$ and $l_{2}$
connecting $p_{1}$ and $p_{2}$, where $l_{1}$ is an arc in
$\bar{\Sigma}\backslash\bm{\beta}$ and $l_{2}$ is the arc
$\\{(0,t)|t\in[0,1]\\}$ in new one-handle. Next, introduce a new curve
$\bar{\alpha}_{1}^{a,L}$ as follows. Let $l_{z}$ be an arc from $z$ to the
point $(-1,0)\in S^{1}\times\\{0\\}$ does not intersect any of the $\alpha$\-
and $\beta$-curves. Let $l_{2}^{\prime}$ be the arc $\\{(1,t)|t\in[0,1]\\}$ in
the one-handle; denote the endpoints of $l_{2}^{\prime}$ by $p_{1}^{\prime}$
and $p_{2}^{\prime}$. Let $l_{1}^{\prime}$ be an arc connecting
$p_{1}^{\prime}$ and $p_{2}^{\prime}$ in
$\bar{\Sigma}\backslash\\{\bm{\bar{\alpha}}\cup l_{z}\\}$. Let
$\bar{\alpha}_{1}^{a,L}=l_{1}^{\prime}\cup l_{2}^{\prime}$. Then
$\bar{\alpha}_{1}^{a,L}$ intersects $\alpha^{c}_{g}$ geometrically once at a
point $p$. Note $\alpha_{g}^{c}$ is the meridian of $P$, and
$\bar{\alpha}_{1}^{a,L}$ is a longitude of $P$. Let
$\bar{\Sigma}^{\prime\prime}$ be the circle compactification of
$\bar{\Sigma}^{\prime}\backslash\\{p\\}$. Denote the new boundary circle by
$\partial_{L}\bar{\Sigma}^{\prime\prime}$, and denote the boundary circle
inherited from $\partial\bar{\Sigma}$ by
$\partial_{R}\bar{\Sigma}^{\prime\prime}$. Let
${\alpha}_{1}^{a,L}=\bar{\alpha}_{1}^{a,L}\backslash\\{p\\}$, and let
${\alpha}_{2}^{a,L}=\alpha^{c}_{g}\backslash\\{p\\}$. Let
$\alpha_{1}^{a,R}=\alpha_{1}^{a}$, and let $\alpha_{2}^{a,R}=\alpha_{2}^{a}$.
Let
$\bm{\bar{\alpha}}^{\prime\prime}=\\{\alpha^{a,L}_{1},\alpha^{a,L}_{2},\alpha^{a,R}_{1},\alpha^{a,R}_{2},\alpha^{c}_{1},\ldots,\alpha^{c}_{g-1}\\}$.
Let $\bm{\beta}^{\prime\prime}=\bm{\beta}\cup\\{\beta_{g+1}\\}$. Label the
Reeb chords corresponding to the new boundary circle
$\partial_{L}\bar{\Sigma}^{\prime\prime}$ by $\sigma_{i}$ ($i=0,1,2,3$) so
that $\sigma_{2}$ and $\sigma_{3}$ lie on the side attached to the feet near
$w$, and $\sigma_{0}$ and $\sigma_{1}$ lie on the side attached to the feet
near $z$. Let $z_{R}=z$, and let $z_{L}$ be a point on $\sigma_{0}$. Let
$\bm{z}$ be an arc connecting $z_{R}$ and $z_{L}$ in the complement of
$\bm{\bar{\alpha}}^{\prime\prime}\cup\bm{\beta}^{\prime\prime}$; $\bm{z}$
exists since we can obtain such an arc by extending $l_{z}$. Finally, we let
$\mathcal{H}_{X(P)}=(\bar{\Sigma}^{\prime\prime},\bm{\bar{\alpha}}^{\prime\prime},\bm{\beta}^{\prime\prime},\bm{z})$.
See Figure 22.

######  Lemma 5.8.

Let $\mathcal{H}_{X(P)}$ be the arced bordered Heegaard diagram obtained from
$\mathcal{H}_{w,z}$ via the above procedure. Let $\alpha_{im}$ be a
$z$-adjacent multicurve. Then $\mathcal{H}_{X(P)}(\alpha_{im})$ is
unobstructed and bi-admissible.

###### Proof.

The unobstructedness follows from Proposition 4.16. We move to see bi-
admissibility in the sense of Definition 2.10. Note that periodic domains $B$
for $\mathcal{H}_{X(P)}(\alpha_{im})$ with
$n_{\sigma_{0}}(B)=n_{\sigma_{1}}(B)=0$ (respectively
$n_{\sigma_{2}}(B)=n_{\sigma_{3}}(B)=0$) correspond to periodic domains
$B^{\prime}$ for $\mathcal{H}_{w,z}(\alpha_{im})$ with $n_{z}(B^{\prime})=0$
(respectively $n_{w}(B^{\prime})=0$). Therefore, the bi-admissibility of
$\mathcal{H}_{w,z}(\alpha_{im})$, which was shown in Proposition 5.2, implies
the bi-admissibility of $\mathcal{H}_{X(P)}$. ∎

Recall $\mathcal{H}_{id}$ is the standard doubly pointed bordered Heegaard
diagram for the identity pattern knot.

######  Lemma 5.9.

$CFK_{\mathcal{R}}(\mathcal{H}_{w,z}(\alpha_{im}))$ is chain homotopy
equivalent to
$CFK_{\mathcal{R}}(\mathcal{H}_{id}\cup\mathcal{H}_{X(P)}(\alpha_{im}))$.

###### Proof.

Note the doubly pointed Heegaard diagram
$\mathcal{H}_{id}\cup\mathcal{H}_{X(P)}(\alpha_{im})$ is obtained from
$\mathcal{H}_{w,z}(\alpha_{im})$ by two stabilizations; see Figure 23.

Figure 23. An example of $\mathcal{H}_{id}\cup\mathcal{H}_{X(P)}(\alpha_{im})$
(left) and $\mathcal{H}_{w,z}(\alpha_{im})$ (lower right). Here, $P$ is the
$(3,1)$-cable. These two diagrams are related via handleslides and
destabilizations, where the handleslides do not involve sliding over the
immersed $\alpha$-curve.

In particular, it is also bi-admissible, and hence one can define
$CFK_{\mathcal{R}}(\mathcal{H}_{id}\cup\mathcal{H}_{X(P)}(\alpha_{im}))$. We
claim there is a sequence of Heegaard moves relating
$\mathcal{H}_{id}\cup\mathcal{H}_{X(P)}(\alpha_{im})$ and
$\mathcal{H}_{w,z}(\alpha_{im})$ which do not involve sliding $\alpha$ curves
over $\alpha_{im}$. To see this, note that on
$\mathcal{H}_{id}\cup\mathcal{H}_{X(P)}(\alpha_{im})$ there is a
$\beta$-circle between the $w$ and $z$ base points that intersects an
$\alpha$-circle geometrically once; denote these curves by $\beta_{g+2}$ and
$\alpha_{g+2}$ respectively. After sliding other beta curves over
$\beta_{g+2}$ if necessary, we may assume $\alpha_{g+2}$ does not intersect
other beta curves, and hence we can destabilize
$\mathcal{H}_{id}\cup\mathcal{H}_{X(P)}(\alpha_{im})$ along $\alpha_{g+2}$ and
$\beta_{g+2}$. Now we arrive at an intermediate Heegaard diagram; see Figure
23 (upper right). It is a stabilization of $\mathcal{H}_{w,z}(\alpha_{im})$.
On this intermediate Heegaard diagram, there is an $\alpha$-circle
$\alpha_{g+1}$ that intersects only one $\beta$-circle $\beta_{g+1}$, and the
geometric intersection number is one. So, we may slide other $\alpha$-curves
over $\alpha_{g+1}$ if necessary so that $\beta_{g+1}$ do not intersect other
$\alpha$-curves. After this, we destabilize the Heegaard diagram along
$\alpha_{g+1}$ and $\beta_{g+1}$, and the resulting Heegaard diagram is
$\mathcal{H}_{w,z}(\alpha_{im})$. The homotopy equivalence between
$CFK_{\mathcal{R}}(\mathcal{H}_{w,z}(\alpha_{im}))$ and
$CFK_{\mathcal{R}}(\mathcal{H}_{id}\cup\mathcal{H}_{X(P)}(\alpha_{im}))$
follows from the homotopy invariance of knot Floer chain complexes established
in Proposition 3.11. ∎

With these lemmas at hand, we now prove the ungraded version of Theorem 1.1.

###### Proof of Theorem 1.1, ungraded version.

In view of Proposition 5.4, we may assume the immersed multicurve $\alpha_{K}$
for the knot complement of $K$ is z-adjacent. Let $\mathcal{H}_{X(P)}$ be the
arced bordered Heegaard diagram obtained from $\mathcal{H}_{w,z}$ via the
“punctured-stabilization procedure”. Throughout, when referring to the type D
structure of a knot complement, we use the meridian and Seifert longitude to
parametrize the boundary. By standard arguments, we can arrange that
$\mathcal{H}_{X(P)}$ is left provincially admissible at the cost of isotopy of
the $\beta$ curves. By Theorem 1.4, we have

$\displaystyle\widehat{CFD}(\mathcal{H}_{X(P)}(\alpha_{K}))$
$\displaystyle\cong\widehat{CFDA}(\mathcal{H}_{X(P)})\boxtimes\widehat{CFD}(\alpha_{K})$
$\displaystyle\cong\widehat{CFDA}(\mathcal{H}_{X(P)})\boxtimes\widehat{CFD}(S^{3}\backslash\nu(K))$
$\displaystyle\cong\widehat{CFD}(S^{3}\backslash\nu(P(K)))$

Therefore, up to homotopy equivalence, the extended type D structure
$\widetilde{CFD}(\mathcal{H}_{X(P)}(\alpha_{K}))$ extends
$\widehat{CFD}(S^{3}\backslash\nu(P(K)))$. Consequently, we have the
following:

$\displaystyle CFK_{\mathcal{R}}(P(K))$
$\displaystyle\cong\widetilde{CFA}(\mathcal{H}_{id}){\boxtimes}\widetilde{CFD}(S^{3}\backslash\nu(P(K)))$
$\displaystyle\cong\widetilde{CFA}(\mathcal{H}_{id}){\boxtimes}\widetilde{CFD}(\mathcal{H}_{X(P)}(\alpha_{K}))$
$\displaystyle\cong
CFK_{\mathcal{R}}(\mathcal{H}_{id}\cup\mathcal{H}_{X(P)}(\alpha_{K}))$

Here, the last equality follows from applying Theorem 1.6. Note ${\boxtimes}$
in the above equation is well-defined since $\mathcal{H}_{X(P)}(\alpha_{K})$
is bi-admissible by Lemma 5.8. Now, by Lemma 5.9, $CFK_{\mathcal{R}}(P(K))$ is
chain homotopy equivalent to
$CFK_{\mathcal{R}}(\mathcal{H}_{w,z}(\alpha_{K}))$. ∎

###  5.2. $\mathcal{H}_{w,z}(\alpha_{K})$ is gradable

We want to show that the chain homotopy equivalence established in the
previous subsection preserves the $w$-grading and $z$-grading of knot Floer
chain complexes. As the first step, we need to show that
$\mathcal{H}_{w,z}(\alpha_{K})$ is gradable (in the sense of Definition 3.7).

######  Proposition 5.10.

The diagram $\mathcal{H}_{w,z}(\alpha_{K})$ is gradable.

In addition to being gradable, note that the results in the previous
subsection also imply that
$\widehat{HF}(\mathcal{H}_{w}(\alpha_{K}))\cong\widehat{HF}(\mathcal{H}_{z}(\alpha_{K}))\cong\mathbb{F}$.
Therefore we can define an absolute bigrading on
$CFK_{\mathcal{R}}(\mathcal{H}_{w,z}(\alpha_{K}))$.

We will reduce the proof of Proposition 5.10 to the case where
$\mathcal{H}_{w,z}$ is of genus one. If $\mathcal{H}_{w,z}$ is a genus-one
bordered Heegaard diagram, then one can define a Maslov grading $m(-)$ on
$CFK_{\mathcal{R}}(\mathcal{H}_{w,z}(\alpha_{K}))$ as follows. Given any two
generators $x$ and $y$, let $p_{0}$ and $p_{1}$ be two paths from $x$ to $y$
in $\alpha_{K}$ and $\beta$ respectively such that $p_{0}-p_{1}$ lifts to a
closed path $\gamma$ in the universal cover $\mathbb{R}^{2}$ of the genus-one
Heegaard surface. Up to perturbing the curves, we may assume that $p_{0}$ and
$p_{1}$ intersect in right angles at $x$ and $y$. Then $m(x)-m(y)$ is equal to
$\frac{1}{\pi}$ times the total counterclockwise rotation along the smooth
segments of $\gamma$ minus twice the number of the (lifts of) base point $z$
enclosed by $\gamma$; see [HRW22, Definition 35]. This Maslov grading is also
defined (by the same definition) when the $\beta$ curve is only immersed. In
[HRW22], it is shown that the Maslov grading thus defined on a pairing diagram
of two immersed curves agrees with the Maslov grading computed using the
grading package of bordered Heegaard Floer homology. Next, we show this Maslov
grading can be equivalently defined in terms of the index of domains.

######  Proposition 5.11.

Let $\mathcal{H}_{w,z}$ be a genus-one bordered Heegaard diagram and let
$m(-)$ be the Maslov grading on $\mathcal{G}(\mathcal{H}(\alpha_{K}))$
mentioned above. Let $B\in\pi_{2}(x,y)$ be a domain connecting $x$ and $y$
with $\partial B=p_{0}-p_{1}$. Then $m(x)-m(y)=\text{ind}(B)-2n_{z}(B)$.
Moreover, this result extends to the case where the $\beta$ is immersed, in
which we define the index of $B$ by

$\text{ind}(B)=e(B)+n_{x}(B)+n_{y}(B)-s(\partial_{\alpha_{K}}B)-s(\partial_{\beta}B).$

(Here $s(-)$ denotes the self-intersection number of an oriented immersed arc
as defined in Section 2.6)

Before proving Proposition 5.11 we introduce some terminology. It will be
clear later that we can assume $p_{0}-p_{1}$ is immersed and only has discrete
double points.

######  Definition 5.12.

A cornered immersed loop in $T^{2}$ is the union of two oriented immersed arcs
$p_{0}$ and $p_{1}$ with at most discrete double points such that

  * (1)

$p_{0}$ and $p_{1}$ share common endpoints,

  * (2)

the interior of $p_{0}$ and $p_{1}$ intersect transversally,

  * (3)

$p_{0}-p_{1}$ is an oriented loop which is null-homologous,

  * (4)

$p_{0}$ and $p_{1}$ intersect transversally at the endpoints if $p_{0}$ and
$p_{1}$ are non-degenerate (i.e., not a point), and

  * (5)

if one of $p_{0}$ and $p_{1}$ is degenerate, the remaining arc forms a smooth
loop after identifying the endpoints.

The endpoints of $p_{0}$ (or equivalently, $p_{1}$) are called corners of the
cornered immersed loop.

######  Definition 5.13.

Two cornered immersed loops $p_{0}-p_{1}$ and $p_{0}^{\prime}-p_{1}^{\prime}$
in $T^{2}$ are called cornered identical if they share the same set of corners
$\\{x,y\\}$ (or $\\{x\\}$ if the loops have degenerate arcs) and there are
arbitrarily small neighborhoods $N_{x}$ and $N_{y}$ of $x$ and $y$
respectively such that
$(p_{0}-p_{1})|_{N_{x}}=(p^{\prime}_{0}-p^{\prime}_{1})|_{N_{x}}$ and
$(p_{0}-p_{1})|_{N_{y}}=(p^{\prime}_{0}-p^{\prime}_{1})|_{N_{y}}$.

Figure 24. Upper row from left to right: Reidemeister I, II, and III move.
Lower row from left to right: an isotopy that crosses a non-degenerate corner
and a degenerate corner.

######  Lemma 5.14.

If two cornered immersed loops $p_{0}-p_{1}$ and
$p_{0}^{\prime}-p_{1}^{\prime}$ are cornered identical, then they are related
by a finite sequence of moves of the following types:

  * (1)

Reidemeister moves that do not involve the corners and

  * (2)

isotopy that possibly cross the corners.

(See Figure 24) Here, we require $(p_{0}-p_{1})|_{N_{x}}$ and
$(p_{0}-p_{1})|_{N_{y}}$ are fixed throughout the modification for some
sufficiently small neighborhoods ${N_{x}}$ and ${N_{y}}$ of the corners.

###### Proof.

One can prove this applying the usual Reidemeister-move equivalence of knot
diagrams (by treating both immersed loops as diagrams for the unknot via
imposing proper crossing information); note that any Reidemeister move
involving a corner can be traded by an isotopy crossing the corner and a
Reidemeister move that does not involve the corner. ∎

######  Definition 5.15.

Given a cornered immersed loop $p_{0}-p_{1}$ in $T^{2}$. Let
$\tilde{p}_{0}-\tilde{p}_{1}$ be a lift of $p_{0}-p_{1}$ in $\mathbb{R}^{2}$
and let $\tilde{B}$ be the bounded domain in $\mathbb{R}^{2}$ such that
$\partial\tilde{B}=\tilde{p}_{0}-\tilde{p}_{1}$. Let $B$ be the domain in
$T^{2}$ obtained from $\tilde{B}$ by applying the covering projection. Define
the index of the cornered immersed loop as

$\text{ind}(p_{0}-p_{1})=e(B)+n_{x}(B)+n_{y}(B)-s(\partial_{p_{0}}B)-s(\partial_{p_{1}}B),$

where $x$ and $y$ are the corners.

Define the net rotation number $nr(p_{0}-p_{1})$ to be $\frac{1}{\pi}$ times
the counterclockwise net rotation along the smooth segments $p_{0}$ and
$p_{1}$.

######  Lemma 5.16.

Suppose $p_{0}-p_{1}$ and $p_{0}^{\prime}-p_{1}^{\prime}$ are cornered
immersed loops differ by an isotopy or a Reidemeister move. Then

$\text{ind}(p_{0}-p_{1})-\text{ind}(p^{\prime}_{0}-p^{\prime}_{1})=nr(p_{0}-p_{1})-nr(p^{\prime}_{0}-p^{\prime}_{1}).$

Figure 25. Local diagrams for isotopies that cross a corner. The numbers $a$,
$a-1$, and $a+1$ indicate the multiplicities of the regions.

###### Proof.

First, we examine the effect of an isotopy on both quantities. Clearly, the
net rotation number is unchanged. We separate the discussion of the index into
two cases according to whether the isotopy crosses corners or not. If the
isotopy does not cross the corners, it clearly does not change the index as
well whence we are done. If the isotopy crosses a corner, then we claim the
local multiplicity and the self-intersection numbers change in a way that
cancel each other, leaving the index unchanged. This claim can be seen by
examining local diagrams, which are further divided into two cases according
to whether the corner is degenerate or not. When the corner is non-degenerate,
the local diagram of one case is shown in Figure 25 (i); all the other cases
can be obtained from this case by swapping the labels and orientations of the
arcs, and the analysis of all cases are similar. In the case shown in Figure
25 (i), only $n_{x}(B)$ and $s(\partial_{p_{0}}B)$ change: the diagram on the
left has $n_{x}(B)=\frac{a+(a-1)+(a-1)+(a-1)}{4}$ and the local self-
intersection of $p_{0}$ contributes $s_{p_{0}}=-1$; the diagram on the right
has $n_{x}(B)=\frac{(a+1)+a+a+a}{4}$ and there are no self-intersections of
the arcs in the local diagram so the local contribution $s_{p_{0}}=0$. In both
diagrams we have $n_{x}(B)-s_{p_{0}}=\frac{4a+1}{4}$, and hence the index is
unchanged. When the corner is degenerate, one of the cases is shown in Figure
25 (ii). In this case, only $n_{x}$ and the self-intersection of $p_{0}$
change: the diagram on the left has $n_{x}=\frac{a+a+(a-1)+(a-1)}{4}$ and a
local contribution of the self-intersection of $p_{0}$ given by
$s_{p_{0}}=-1$; the diagram on the right has $n_{x}=\frac{(a+1)+(a+1)+a+a}{4}$
and a local contribution of the self-intersection of $p_{0}$ given by
$s_{p_{0}}=1$. In both local diagrams we have
$n_{x}(B)+n_{x}(B)-s_{p_{0}}=2a$, and hence the index is unchanged. All other
cases can be obtained from this case by swapping the labels and orientations
of the arcs, and the analysis of all cases are similar.

Figure 26. The local diagram for Reidemeister I move. The numbers $a$, $a-1$,
and $a-2$ indicate the multiplicities of the regions.

Next, we examine the effect of Reidemeister I move. Up to swapping
orientations and the labels, we may assume the local diagram is as shown in
Figure 26 The net rotation number of the diagram on the right is 2 less than
that of the diagram on the left. For the index comparison, the Euler measure
of the local domain on the right is $1$ less than that of the left diagram and
the self-intersection number $s(\partial_{p_{0}}B)$ of the right diagram is
$1$ more than that of the left diagram; in total the index of the diagram on
the right is 2 less than that of the diagram on the left. Therefore, the
changes in the net rotation and in the index are the same after doing a
Reidemeister I move.

Next, we examine the effect of Reidemeister II moves. It does not change the
net rotation number. Also, it does not affect the Euler measure and the local
multiplicities at the corners. A Reidemeister II move creates/annihilates a
pair of self-intersection points whose signs cancel each other if both arcs
involved are on $p_{0}$ or $p_{1}$, and otherwise does not involve self-
intersections; in both cases the self-intersection numbers are unchanged. So,
the index does not change as well.

Finally, it is easy to see that a Reidemeister III move does not change the
net rotation number. It is also easy to see a Reidemeister III move does not
change the Euler measure, local multiplicities at the corners, or self-
intersections, and hence it does not change the index either. ∎

######  Proposition 5.17.

Let $p_{0}-p_{1}$ be a cornered immersed loop. Then
$\text{ind}(p_{0}-p_{1})=nr(p_{0}-p_{1})$.

###### Proof.

By Lemma 5.16 and Lemma 5.14, it suffices to show that $p_{0}-p_{1}$ is
cornered identical with some cornered immersed loop whose index coincides with
the net rotation number.

If at least one of $p_{0}$ and $p_{1}$ is degenerate, $p_{0}-p_{1}$ is
cornered identical with an embedded circle that passes the corner, and it is
easy to see the index and the net rotation number coincide on an embedded
circle with a degenerate corner.

Figure 27. Deforming $p_{0}-p_{1}$.

Next, we discuss the case where $p_{0}-p_{1}$ is non-degenerate. We first
construct a cornered immersed loop $p_{0}^{\prime}-p_{1}^{\prime}$ that is
cornered identical to $p_{0}-p_{1}$ as follows. Let $p_{0}^{\prime}=p_{0}$. We
shall construct $p_{1}^{\prime}$ to be a path which is almost a parallel push-
off of $p_{0}$. (See Figure 27 for examples.) To spell out the construction,
let $f_{0}:[0,1]\rightarrow T^{2}$ be an immersion such that
$f_{0}([0,1])=p_{0}$. Let $\hat{N}$ be a sufficiently small tubular
neighborhood of $p_{0}$ such that it can realized as the image of an extension
of $f_{0}$, i.e., there exits an immersion
$\tilde{f_{0}}:[0,1]\times[-\epsilon,\epsilon]\rightarrow T^{2}$ such that
$\tilde{f_{0}}|_{[0,1]\times\\{0\\}}=f_{0}$ and
$\tilde{f_{0}}([0,1]\times\\{pt\\})$ is a parallel push-off of $p_{0}$ for any
$pt\in[-\epsilon,0)\cup(0,\epsilon]$. We can further assume near the two
corners $x=f_{0}(0)$ and $y=f_{0}(1)$, the other arc $p_{1}$ is contained in
$\tilde{f_{0}}(\\{0,1\\}\times[-\epsilon,\epsilon])$; denote these two arcs on
$p_{1}$ near $x$ and $y$ by $p_{x}$ and $p_{y}$ respectively. We construct