# Satellite knots and immersed Heegaard Floer homology

Wenzhao Chen  Department of Mathematics, University of British Columbia.
E-mail address<EMAIL_ADDRESS>and  Jonathan Hanselman  Department of
Mathematics, Princeton University.
E-mail address<EMAIL_ADDRESS>
###### Abstract.

We describe a new method for computing the $UV=0$ knot Floer complex of a
satellite knot given the $UV=0$ knot Floer complex for the companion and a
doubly pointed bordered Heegaard diagram for the pattern, showing that the
complex for the satellite can be computed from an immersed doubly pointed
Heegaard diagram obtained from the Heegaard diagram for the pattern by
overlaying the immersed curve representing the complex for the companion. This
method streamlines the usual bordered Floer method of tensoring with a
bimodule associated to the pattern by giving an immersed curve interpretation
of that pairing, and computing the module from the immersed diagram is often
easier than computing the relevant bordered bimodule. In particular, for (1,1)
patterns the resulting immersed diagram is genus one, and thus the computation
is combinatorial. For (1,1) patterns this generalizes previous work of the
first author which showed that such immersed Heegaard diagram computes the
$V=0$ knot Floer complex of the satellite. As a key technical step, which is
of independent interest, we extend the construction of a bigraded complex from
a doubly pointed Heegaard diagram and of an extended type D structure from a
torus-boundary bordered Heegaard diagram to allow Heegaard diagrams containing
an immersed alpha curve.

###### Contents

  1. 1 Introduction
    1. 1.1 Satellite knots and immersed curves
    2. 1.2 Immersed Heegaard Floer theory
    3. 1.3 Strategy to prove the main theorem
    4. 1.4 Further discussions
      1. 1.4.1 Immersed Heegaard diagrams
      2. 1.4.2 Pattern detection
    5. 1.5 Organization
  2. 2 Bordered Floer invariants of immersed Heegaard diagrams
    1. 2.1 Immersed bordered Heegaard diagrams
    2. 2.2 Moduli spaces of stay-on-track holomorphic curves
      1. 2.2.1 Definition of moduli spaces of holomorphic curves
      2. 2.2.2 Regularity and the expected dimension
    3. 2.3 Compactification
      1. 2.3.1 Holomorphic curves in the end at east-infinity
      2. 2.3.2 Compactification by Holomorphic combs
    4. 2.4 Gluing results
    5. 2.5 Degeneration of moduli spaces
    6. 2.6 Embedded holomorphic curves
    7. 2.7 Ends of moduli spaces of 0-P curves
      1. 2.7.1 Ends of $\mathcal{M}^{B}$ when $n_{z}(B)=0$
      2. 2.7.2 Ends of $\mathcal{M}^{B}$ when $n_{z}(B)=1$
      3. 2.7.3 Reformulation of the moduli spaces
      4. 2.7.4 Boundary degeneration with corners
      5. 2.7.5 Boundary degeneration without corners
      6. 2.7.6 Proof of Proposition 2.49
    8. 2.8 Ends of moduli spaces of 1-P curves
      1. 2.8.1 Reformulation of the moduli spaces
      2. 2.8.2 One-punctured boundary degeneration with corners
      3. 2.8.3 One-punctured boundary degeneration without corners
      4. 2.8.4 Proof of Proposition 2.63
    9. 2.9 Type D structures
    10. 2.10 Weakly extended Type D structures
    11. 2.11 Invariance
  3. 3 Knot Floer homology of immersed Heegaard diagrams
    1. 3.1 Immersed doubly-pointed Heeggard diagram
    2. 3.2 The knot Floer chain complex
    3. 3.3 Bi-grading
    4. 3.4 Invariance
  4. 4 Paring theorems
    1. 4.1 Immersed curves in the marked torus
    2. 4.2 Pairing diagrams
    3. 4.3 z-adjacency
    4. 4.4 The collapsing operation
    5. 4.5 Unobstructedness and admissibility of paring diagrams
    6. 4.6 The first paring theorem
    7. 4.7 The second pairing theorem
  5. 5 Knot Floer homology of satellite knots
    1. 5.1 Proof of the main theorem, ungraded version
    2. 5.2 $\mathcal{H}_{w,z}(\alpha_{K})$ is gradable
    3. 5.3 Gradings in the main theorem
  6. 6 (1,1) Patterns
    1. 6.1 $(1,1)$ diagrams
    2. 6.2 Removing the $z$-passable assumption
    3. 6.3 One-bridge braids
    4. 6.4 Immersed curves for 1-bridge braid satellites
    5. 6.5 L-space slopes, $\tau$, and $\epsilon$ for one-bridge braid satellites
    6. 6.6 Mazur satellites

##  1\. Introduction

The study of knot Floer chain complexes of satellite knots has many
applications. For instance, computation of knot-Floer concordance invariants
of satellite knots is instrumental in establishing a host of results in knot
concordance, like [CFHH13, Hom15, Lev16, HKL16, OSS17, HKP21, DHST21, HKPS22].
To further understand the behavior of knot Floer chain complexes under
satellite operations, the current paper introduces an immersed-curve technique
to compute knot Floer chain complexes of satellite knots. This method subsumes
most of the previous results in this direction, including [Hed05, Hed07,
Hed09, VC10, Pet13, Hom14a, HLV14, Lev16, HW23, Che23]. This technique is
derived from an immersed Heegaard Floer theory that is developed in this
paper, which is built on the work by the second author, Rasmussen, and Watson
[HRW23].

###  1.1. Satellite knots and immersed curves

Knot Floer homology was introduced by Ozsváth and Szabó and independently by
Rasmussen [OS04a, Ras03]. Recall that any knot can be encoded by a doubly-
pointed Heegaard diagram, which is a closed oriented surface with two sets of
embedded circles and two base points. Knot Floer theory, using the machinery
of Lagrangian Floer theory, associates a bigraded chain complex over
$\mathbb{F}[U,V]$ to such a double pointed Heegaard diagram, and the bigraded
chain homotopy type of this chain complex is an invariant of the isotopy type
of the knot. The literature studies various versions of the knot Floer chain
complex obtained by setting the ground ring to be a suitable quotient ring of
$\mathbb{F}[U,V]$; throughout this paper we will consider the complex defined
over the ground ring $\mathcal{R}=\mathbb{F}[U,V]/UV$. The knot Floer chain
complex over $\mathcal{R}$ of a knot $K$ in the 3-sphere is equivalent to the
bordered Floer invariant of the knot complement $S^{3}\backslash\nu(K)$, and
it was shown in [HRW23] that this is equivalent to an immersed multicurve in
the punctured torus decorated with local systems. The punctured torus we refer
to here is a torus with a single puncture and a parametrization allowing us to
identify it with the boundary of the knot complement with a chosen basepoint.

A satellite knot is obtained by gluing a solid torus that contains a knot $P$
(called the pattern knot) to complement of a knot $K$ in the 3-sphere (called
the companion knot) in a compatible way, after which the glued-up manifold is
a 3-sphere and the pattern knot $P$ gives rise to the satellite knot $P(K)$ in
the 3-sphere. Just as knots in closed 3-manifolds are encoded by doubly-
pointed Heegaard diagrams, a pattern knot in the solid torus can be
represented by a doubly-pointed bordered Heegaard diagram, which is an
oriented surface of some genus $g$ with one boundary component, together with
two base points and a suitable collection of $g$ $\beta$-curves, $g-1$
$\alpha$-curves, and two $\alpha$ arcs.

Our technique involves constucting an immersed doubly pointed Heegaard diagram
by combining a doubly pointed bordered Heegaard diagram for the pattern $P$
with the immersed curve associated with the companion $K$. More precisely, we
fill in the boundary of the bordered Heegaard diagram for $P$ and remove the
two $\alpha$ arcs and then add the immersed curve for $K$ to the diagram by
identifying the punctured torus containing $K$ with a neighborhood of the now
filled in boundary and the $\alpha$ arcs in a way dictated by the given
parametrizations. The resulting diagram is just like a standard genus $g$
doubly pointed Heegaard diagram except that one of the $\alpha$ curves, which
are usually embedded, is now replaced with a decorated immersed multicurve.
See the top row of Figure 1 and 2 for examples of immersed doubly pointed
diagrams constructed in this way.

Our main theorem asserts that this diagram can be used to compute the knot
Floer complex over $\mathcal{R}$ of $P(K)$. We state the main theorem below,
with technical inputs referenced in the remark afterwards.

######  Theorem 1.1.

Let $\mathcal{H}_{w,z}$ be a doubly-pointed bordered Heegaard diagram for a
pattern knot $P$, and let $\alpha_{K}$ be the immersed multicurve associated
to a companion knot $K$. Let $\mathcal{H}_{w,z}(\alpha_{K})$ be the immersed
doubly-pointed Heegaard diagram obtained by pairing $\mathcal{H}_{w,z}$ and
$\alpha_{K}$, in which $\alpha_{K}$ is put in a z-passable position. Then the
knot Floer chain complex
$CFK_{\mathcal{R}}(\mathcal{H}_{w,z}(\alpha_{K}),\mathfrak{d})$ defined using
$\mathcal{H}_{w,z}(\alpha_{K})$ and a generic choice of auxiliary data
$\mathfrak{d}$ is bi-graded homotopy equivalent to the knot Floer chain
complex of the satellite knot $P(K)$ over $\mathcal{R}$, where
$\mathcal{R}=\mathbb{F}[U,V]/UV$.

######  Remark 1.2.

The paring operation for constructing $\mathcal{H}_{w,z}(\alpha_{K})$ is
defined in Section 4.2. The knot Floer chain complex of an immersed doubly-
pointed Heegaard diagram is defined in Section 3. While the definition of the
Heegaard Floer theory with immersed Heegaard diagrams is similar to that in
the usual setup, it is complicated by the appearance of boundary
degenerations. The $z$-passable condition on $\alpha_{K}$ is a diagrammatic
condition used to handle boundary degenerations; it is specified in Definition
5.5 and can be arranged easily via finger moves as in Example 5.7. Moreover,
the $z$-passable condition is not required when $\mathcal{H}_{w,z}$ is a
genus-one diagram; see Theorem 6.1. The proof of Theorem 1.1 is separated into
two stages: we first prove the ungraded version in Section 5.1 and then the
gradings are addressed in Section 5.3. The gradings can be combinatorially
computed using an index formula established in Section 2.6; also see
Definition 3.8.

Figure 1. Obtaining the immersed curve for the $(2,1)$-cable of the trefoil
$T_{2,3}$. The top row shows the pairing diagram
$\mathcal{H}_{w,z}(\alpha_{K})$ obtained by merging a doubly-pointed bordered
Heegaard diagram $\mathcal{H}_{w,z}$ for the $(2,1)$-cable pattern and the
immersed curve $\alpha_{K}$ for the trefoil $T_{2,3}$. The bottom row shows a
lift of this diagram to a suitable covering space and a planar transform that
sends the lift of the immersed curve for $T_{2,3}$ to that of
$(T_{2,3})_{2,1}$.

Theorem 1.1 is especially useful when the pattern knot is a (1,1) pattern,
meaning that it admits a genus one doubly pointed bordered Heegaard diagram
$\mathcal{H}_{w,z}$. This is because in this setting the immersed doubly
pointed Heegaard diagram $\mathcal{H}_{w,z}(\alpha_{K})$ is genus one, and the
complex $CFK_{\mathcal{R}}(\mathcal{H}_{w,z}(\alpha_{K}),\mathfrak{d})$ for
such a diagram is straightforward to compute even in the presence of immersed
curves; it only requires counting bigons which can be done combinatorially. An
example is shown in Figure 1, where we compute the knot Floer complex
$CFK_{\mathcal{R}}$ of the $(2,1)$ cable of the trefoil $T_{2,3}$. The top row
of the figure gives the pairing diagram, formed from a doubly pointed bordered
Heegaard diagram $\mathcal{H}_{w,z}$ for the cable knot and the immersed curve
$\alpha_{K}$ associated with $T_{2,3}$. The bottom left shows the curves
lifted to an appropriate covering space, after a homotopy putting them in
minimal position. There are seven generators, labeled as in the figure, and it
is straightforward to count the bigons that cover only $w$ or only $z$ and see
that the differential in $CFK_{\mathcal{R}}((T_{2,3})_{2,1})$ is given by

$\partial a=0,\quad\partial b=Ua+V^{2}e,\quad\partial c=Vd,\quad\partial
e=Ud,\quad\partial d=0,\partial f=U^{2}c+Vg,\quad\partial g=0.$

The Alexander grading changes by one decreases when traveling along the
$\beta$ curve each time the $\beta$ curve crosses the short arc connecting the
$z$ and $w$ basepoints, increasing if $z$ is on the left and decreasing if $z$
is on the right, so we have

$A(a)=2,\quad A(b)=A(c)=1,\quad A(d)=0,\quad A(e)=A(f)=-1,\quad A(g)=-2.$

Relative Maslov gradings can also be computed from the diagram, with the
absolute grading fixed by the normalization $M(a)=0$.

In the case of (1,1) patterns, the first author proved a weaker version of
Theorem 1.1 in [Che23, Theorem 1.2], where the knot Floer chain complexes are
only defined over $\mathbb{F}[U]\cong\mathbb{F}[U,V]/V$. Recall that the
complex over $\mathbb{F}[U]$ does not count any disks covering the $z$
basepoint, while the complex over $\mathcal{R}$ allows disks to cover either
basepoint as long as they do not cover both. [Che23, Theorem 1.2] can be used
to recover the $\tau$-invariant formula for cable knots, Mazur satellites, and
Whitehead doubles [Hed07, Hom14a, Lev16]. Theorem 1.1 generalizes this earlier
result by showing that the same process recovers the complex over
$\mathcal{R}$, which carries strictly more information. In particular, this
version of the knot Floer complex allows one to compute the
$\epsilon$-invariant introduced by Hom [Hom14a] and infinitely many
concordance homomorphisms $\phi_{j}$ ($j\in\mathbb{Z}^{+}$) defined by Dai-
Hom-Stoffregen-Truong [DHST21]. For example, we use Theorem 1.1 to recover and
generalize the $\tau$ and $\epsilon$ formulas for cables from [Hom14a, Theorem
2] in Section 6.5 and to recover the $\tau$ and $\epsilon$ formulas for Mazur
patterns from [Lev16, Theorem 1.4] in Section 6.6.

The computation described above is even easier for a certain family of (1,1)
patterns. In [HW23, Theorem 1], the second author and Watson showed that the
immersed multicurve of a cable knot can be obtained from that of the companion
knot via a planar transform after lifting the immersed multicurves to an
appropriate covering space of the marked torus. In fact, we show in Theorem
1.3 that the same procedure works for a broader family of (1,1) patterns
called 1-bridge braids (see Definition 6.2). In addition to cables this class
of patterns contains all Berge-Gabai knots. These patterns are specified by
three integers and are denoted $B(p,q,b)$. We let $K_{p,q,b}$ denote the
satellite of the companion knot $K$ with pattern $B(p,q,b)$. In Section 6.4 we
define a diffeomorphism $f_{p,q,b}$ of $\mathbb{R}^{2}$ taking the integer
lattice $\mathbb{Z}^{2}$ to itself and show that this transformation computes
the immersed curve for $K_{p,q,b}$ from that of $K$.

######  Theorem 1.3.

Let $\gamma_{K}$ and $\gamma_{K_{p,q,b}}$ be the immersed multicurve
associated with $K$ and $K_{p,q,b}$ respectively. Let $\tilde{\gamma}_{K}$ and
$\tilde{\gamma}_{K_{p,q,b}}$ be the lifts of $\gamma_{K}$ and
$\gamma_{K_{p,q,b}}$ to $\tilde{T}_{\bullet}$ respectively. Then
$\tilde{\gamma}_{K_{p,q,b}}$ is homotopic to $f_{p,q,b}(\tilde{\gamma}_{K})$.

We demonstrate how this result is obtained from Theorem 1.1, in the example of
$(2,1)$ cabling the trefoil, in the bottom row of Figure 1. Note that the
$(2,1)$ cable pattern is the 1-bridge braid $B(2,1,0)$. On the left is the
diagram $\mathcal{H}_{w,z}(\alpha_{K})$, which by Theorem 1.1 computes the
complex $CFK_{\mathcal{R}}((T_{2,3})_{2,1})$, lifted to an appropriate
covering space (specifically $(\mathbb{R}/p\mathbb{Z})\times\mathbb{R}$, where
$p$ is the winding number). There is a homotopy that pulls the the $\beta$
curve coming from $\mathcal{H}_{w,z}$ straight to a vertical line, sliding the
basepoints and the $\alpha$ curve along the way, and rescales to obtain a
different covering space of the marked torus (namely
$(\mathbb{R}/\mathbb{Z})\times\mathbb{R}$). This homotopy does not change the
Floer complex associated with the diagram, so the new diagram still computes
$CFK_{\mathcal{R}}((T_{2,3})_{2,1})$, and since the $\beta$ curve is vertical
and passes through each pair of basepoints twice, the $\alpha$ curve in this
diagram is precisely the immersed curve representing
$CFK_{\mathcal{R}}((T_{2,3})_{2,1})$. The homotopy that pulls the $\beta$
curve to the vertical line is precisely the planar transformation $f_{2,1,0}$.
We note that in the special case of cables (for which $b=0$), the
transformation $f_{p,q,b}$ agrees with the planar transformation $f_{p,q}$
described in [HW23, Theorem 1].

We will use Theorem 1.3 to derive formulas for $\epsilon$ and $\tau$ for
1-bridge braid satellites in Theorem 6.6 and Theorem 6.8, generalizing similar
formulas for cables. We also determine the precise criteria for a 1-bridge
braid satellite to be an L-space knot in Theorem 6.5; this unifies and
generalizes similar results known for cables and Berge-Gabai knots [Hom11,
HLV14].

###  1.2. Immersed Heegaard Floer theory

The underlying machinery for proving Theorem 1.1 is the bordered Heegaard
Floer theory introduced by Lipshitz-Ozsváth-Thurston [LOT18, LOT15]. The new
input is an immersed Heegaard Floer theory that we develop in this paper in
which we allow Heegaard diagrams with an immersed multicurve in place of one
$\alpha$ curve. We closely follow the construction of bordered invariants in
[LOT18], highlighting the points at which more care is needed in this broader
setting.

Bordered Heegaard Floer theory is a toolkit to compute Heegaard Floer
invariants of manifolds that arise from gluing in terms of a set of relative
invariants for manifolds with boundaries. In the simplest setting, assume
$Y_{1}$ and $Y_{2}$ are two oriented 3-manifolds with parametrized boundary
such that $\partial Y_{1}$ is identified with an oriented parametrized surface
$\mathcal{F}$ and $\partial Y_{2}$ is identified with $-\mathcal{F}$ the
orientation reversal of $\mathcal{F}$, and let
$Y=Y_{1}\cup_{\mathcal{F}}Y_{2}$. Up to a suitable notion of homotopy
equivalence, the bordered Heegaard Floer theory associates to $Y_{1}$ a graded
$A^{\infty}$-module $\widehat{CFA}(Y_{1})$ (called type $A$ module) and
associates to $Y_{2}$ a graded differential module $\widehat{CFD}(Y_{2})$
(called type D module). Moreover, there is a box-tensor product operation
$\widehat{CFA}(Y_{1})\boxtimes\widehat{CFD}(Y_{2})$ which produces a chain
complex that is graded homotopy equivalent to the hat-version Heegaard Floer
chain complex $\widehat{CF}(Y)$ of the glued-up manifold. The second author,
Rasmussen, and Watson introduced an immersed-curve technique for working with
these invariants for manifolds with torus boundary [HRW23]. When $\mathcal{F}$
mentioned above is a parametrized torus $T^{2}$, $\widehat{CFA}(Y_{1})$ and
$\widehat{CFD}(Y_{2})$ are equivalent to immersed multicurves $\gamma_{1}$ and
$\gamma_{2}$ (decorated with local systems) in the parametrized torus $T^{2}$
away from a marked point $z$. Moreover, the Lagrangian Floer chain complex
$\widehat{CF}(T^{2}\backslash\\{z\\},\gamma_{1},\gamma_{2})$ is homotopy
equivalent to $\widehat{CFA}(Y_{1})\boxtimes\widehat{CFD}(Y_{2})$, which is in
turn homotopic equivalent to $\widehat{CF}(Y)$.

The bordered Heegaard Floer theory also contains a package to deal with the
situation of gluing a 3-manifold $M_{1}$ with two parametrized boundary
components $-\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ to a 3-manifold $M_{2}$
with $\partial M_{2}=-\mathcal{F}_{2}$. It associates to $M_{1}$ a type $DA$
bimodule $\widehat{CFDA}(M_{1})$ up to a suitable equivalence, and there is a
box-tensor product $\widehat{CFDA}(M_{1})\boxtimes\widehat{CFD}(M_{2})$
resulting in a type D module that is homotopy equivalent to
$\widehat{CFD}(M_{1}\cup_{\mathcal{F}_{2}}M_{2})$. In this paper, we introduce
an immersed-Heegaard-diagram approach to recapture this bimodule pairing when
the manifold boundaries are tori.

Recall that we can encode a manifold $M_{1}$ whose boundary are two
parametrized tori by some arced bordered Heegaard diagram
$\mathcal{H}_{M_{1}}$ (from which the type $DA$ bimodule is defined). Let
$\alpha_{M_{2}}$ be the immersed multicurve for an oriented 3-manifold $M_{2}$
with a single torus boundary component. In Section 4.2, we give a pairing
construction that merges such an arced bordered Heegaard diagram
$\mathcal{H}_{M_{1}}$ and an immersed multicurve $\alpha_{M_{2}}$ to obtain an
immersed bordered Heegaard diagram $\mathcal{H}_{M_{1}}(\alpha_{M_{2}})$; see
Figure 2 for a schematic example of pairing an arced bordered diagram and an
immersed curve. Extending the original way of defining type D modules from
(non-immersed) bordered Heegaard diagrams, we define type D modules for a
class of immersed bordered Heegaard diagrams that contain such pairing
diagrams, and we prove the following theorem in Section 4.6.

######  Theorem 1.4.

Let $\mathcal{H}^{a}$ be a left provincially admissible arced bordered
Heegaard diagram, and let $\alpha_{im}$ be a $z$-adjacent immersed multicurve.
Then

$\widehat{CFD}(\mathcal{H}^{a}({\alpha_{im}}))\cong\widehat{CFDA}(\mathcal{H}^{a})\boxtimes\widehat{CFD}(\alpha_{im}).$

######  Remark 1.5.

Similar to the $z$-passable condition in Theorem 1.1, the $z$-adjacency is a
diagrammatic condition that is used to handle boundary degeneration in
immersed Heegaard Floer theory; it is defined in Section 4.3 and can be easily
achieved via finger moves.

Among manifolds with torus boundary, of particular interests to us are knot
complements. By the results in [LOT18, Section 11.4-11.5], the knot Floer
chain complex $\mathcal{CFK}_{\mathcal{R}}(J)$ of any knot $J\subset S^{3}$ is
equivalent to the type D module $\widehat{CFD}(S^{3}\backslash\nu(J))$ of the
knot complement. (Consequently, $\mathcal{CFK}_{\mathcal{R}}(J)$ is equivalent
to an immersed-multicurve in a marked torus.)

More concretely, the current state of bordered Floer theory recovers certain
versions of knot Floer chain complex of a knot from the type D module of its
complement as follows. Note that a knot $J$ may be obtained from the knot
complement $S^{3}\backslash\nu(J)$ by gluing in the solid torus containing the
identity pattern knot, which is the core of the solid torus. Let
$\mathcal{H}_{id}$ denote the standard doubly-pointed Heegaard diagram for the
identity pattern knot; see Figure 2. In [LOT18], an $A^{\infty}$-module
$CFA^{-}(\mathcal{H}_{id})$ is associated to $\mathcal{H}_{id}$, and it is
shown that

$gCFK^{-}(J)\cong
CFA^{-}(\mathcal{H}_{id})\boxtimes\widehat{CFD}(S^{3}\backslash\nu(J)),$

where $gCFK^{-}(-)$ denotes the version of knot Floer chain complex over
$\mathbb{F}[U]$ [LOT18, Theorem 11.9]. To recover knot Floer complexes over
the larger ground ring $\mathcal{R}=\mathbb{F}[U,V]/UV$, we use a stronger
pairing theorem which occurred implicitly in [HRW22] (and even more implicitly
in [LOT18]): There are suitable extensions $\widetilde{CFA}(\mathcal{H}_{id})$
and $\widetilde{CFD}(S^{3}\backslash\nu(J))$ of ${CFA^{-}}(\mathcal{H}_{id})$
and $\widehat{CFD}(S^{3}\backslash\nu(J))$ respectively such that

$CFK_{\mathcal{R}}(J)\cong\widetilde{CFA}(\mathcal{H}_{id})\boxtimes\widetilde{CFD}(S^{3}\backslash\nu(J)).$

We provide an immersed-Heegaard-diagram approach to recapture the above
pairing theorem as well. In Section 2, we define the so-called weakly extended
type D structures $\widetilde{CFD}(-)$ of a certain class of immersed bordered
Heegaard diagrams that contains pairing diagrams
$\mathcal{H}_{M_{1}}(\alpha_{M_{2}})$ mentioned earlier. In Section 3, we
define knot Floer chain complexes of a class of immersed doubly-pointed
Heegaard diagrams that includes any diagram
$\mathcal{H}_{id}\cup\mathcal{H}_{im}$ obtained by gluing $\mathcal{H}_{id}$
and an immersed bordered Heegaard diagram $\mathcal{H}_{im}$. Moreover, we
prove the following theorem in Section 4.7.

######  Theorem 1.6.

Let $\mathcal{H}_{im}$ be an unobstructed, bi-admissible immersed bordered
Heegaard diagram, and let $\mathcal{H}_{id}$ be the standard bordered Heegaard
diagram for the identity pattern. Then

$CFK_{\mathcal{R}}(\mathcal{H}_{id}\cup\mathcal{H}_{im})\cong\widetilde{CFA}(\mathcal{H}_{id})\boxtimes\widetilde{CFD}(\mathcal{H}_{im}).$

Figure 2. The pairing diagram $\mathcal{H}_{w,z}(\alpha_{K})$ can
alternatively be obtained in the following three steps: First, pair an arced
bordered Heegaard diagram $\mathcal{H}_{X(P)}$ obtained from
$\mathcal{H}_{w,z}$ with the immersed curve $\alpha_{K}$; secondly, construct
a closed doubly-pointed immersed Heegaard diagram
$\mathcal{H}_{id}\cup\mathcal{H}_{X(P)}$; third, apply Heegaard moves to
$\mathcal{H}_{id}\cup\mathcal{H}_{X(P)}$ to get
$\mathcal{H}_{w,z}(\alpha_{K})$.

###  1.3. Strategy to prove the main theorem

The above theorems are used to compute the knot Floer chain complex of
satellite knots as follows.

First, the knot complement of a satellite knot $P(K)$ decomposes into the
union of two 3-manifolds along a torus: the exterior $X(P)=(S^{1}\times
D^{2})\backslash\nu(P)$ of the pattern knot and the complement
$S^{3}\backslash\nu(K)$ of the companion knot. Therefore,

$\widehat{CFD}(S^{3}\backslash\nu(P(K)))\cong\widehat{CFDA}(X(P))\boxtimes\widehat{CFD}(S^{3}\backslash\nu(K)),$

and hence we can apply Theorem 1.4 to compute
$\widehat{CFD}(S^{3}\backslash\nu(P(K)))$. More concretely, given a doubly-
pointed bordered diagram $\mathcal{H}_{w,z}$ for the pattern knot $P$, one can
apply a standard stabilization-and-drilling procedure to obtain an arced
bordered Heegaard diagram $\mathcal{H}_{X(P)}$ for $X(P)$, which is then
paired with the immersed multicurve $\alpha_{K}$ for $K$ to obtain an immersed
bordered Heegaard diagram $\mathcal{H}_{X(P)}(\alpha_{K})$. The type D module
$\widehat{CFD}(\mathcal{H}_{X(P)}(\alpha_{K}))$ is then homotopy equivalent to
a type D module of $S^{3}\backslash\nu(P(K))$ by Theorem 1.4.

Second, one can define a weakly extended type D module
$\widetilde{CFD}(\mathcal{H}_{X(P)}(\alpha_{K}))$ of the pairing diagram
$\mathcal{H}_{X(P)}(\alpha_{K})$. As mentioned above, the underlying (hat-
version) type D module $\widehat{CFD}(\mathcal{H}_{X(P)}(\alpha_{K}))$ defined
using the same diagram is homotopy equivalent to a type D module of
$S^{3}\backslash\nu(P(K))$. Since extensions of type D modules are unique up
to homotopy, $\widetilde{CFD}(\mathcal{H}_{X(P)}(\alpha_{K}))$ is homotopy
equivalent to a weakly extended type D module of $S^{3}\backslash\nu(P(K))$.
Now Theorem 1.6 implies that the knot Floer chain complex
$CFK_{\mathcal{R}}(\mathcal{H}_{id}\cup\mathcal{H}_{X(P)}(\alpha_{K}))$ is
homotopy equivalent to the knot Floer chain complex of $P(K)$.

Finally, the immersed doubly-pointed Heegaard diagram
$\mathcal{H}_{w,z}(\alpha_{K})$ can be obtained from
$\mathcal{H}_{id}\cup\mathcal{H}_{X(P)}(\alpha_{K})$ via Heegaard moves. We
show knot Floer chain complexes defined from immersed doubly-pointed Heegaard
diagrams that differ by Heegaard moves are homotopy equivalent, and hence
$CFK_{\mathcal{R}}(\mathcal{H}_{w,z}(\alpha_{K}))$ is homotopic equivalent to
a knot Floer chain complex of $P(K)$.

See Figure 2 for an illustration of the operations on Heegaard diagrams
involved in the strategy of the proof.

###  1.4. Further discussions

####  1.4.1. Immersed Heegaard diagrams

The work presented in this paper opens a new avenue for studying Heegaard
Floer homology using immersed Heegaard diagrams. While the results in this
paper already demonstrate this strategy can be useful useful for studying
satellite operations, many questions remain that are worthy of further study.
For example, a natural question is whether $CFD$ can be defined for a more
general class of immersed Heegaard diagrams in which more than one $\alpha$
and/or $\beta$ curve may be immersed, rather than just for a single $\alpha$
curve as in the present setting. We expect this is possible but the the
technical difficulties will be greater.

As a special case, one could consider doubly pointed genus one immersed
Heegaard diagrams in which both the $\alpha$ and $\beta$ curve are allowed to
be immersed. In this case there are no technical difficulties in defining a
Floer complex from such a diagram, as the construction is combinatorial. We
expect that this class of diagrams will be useful for studying satellite knots
with arbitrary patterns, so that it will not be necessary to restrict to
$(1,1)$ patterns to perform computations in a genus one surface. More
precisely, an arbitrary pattern should give rise to an immersed (1,1)-diagram
curve which can be used to recover the action of the satellite operation on
knot Floer complexes, and an analog of Theorem 1.1 should hold so that pairing
the immersed (1,1)-diagram for the satellite with the immersed curve for the
companion computes the knot Floer complex of a satellite. This will be
explored in future work.

A related question concerns immersed diagrams for bimodules in bordered Floer
theory. Stabilizing a (1,1) diagram gives a genus two arced bordered diagram
for the complement of the pattern knot, which gives rise to a bordered Floer
bimodule. In analogy to immersed (1,1) diagrams, we could consider arced
bordered diagram with an immersed $\beta$ curve. We could ask which bimodules
can be represented by such diagrams and if these diagrams are useful in
determining how bimodules act on immersed curves. Just as modules over the
torus algebra correspond to (decorated) immersed curves in the punctured torus
$T_{\bullet}$, it is expected that bimodules are related to immersed surfaces
in $T_{\bullet}\times T_{\bullet}$. It may be that arced bordered diagrams
with immersed curves are helpful in understanding this connection.

####  1.4.2. Pattern detection

In another direction, we can ask if the nice behavior demonstrated for one-
bridge braid patterns extends to any other patterns. Recall that given two
patterns $P_{1}$ and $P_{2}$, we define the composition $P_{1}\circ P_{2}$ to
be the pattern knot so that $(P_{1}\circ P_{2})(K)$ is $P_{1}(P_{2}(K))$ for
any companion knot $K$. Theorem 1.3 implies that one-bridge-braid patterns and
their compositions act as planar transforms on the (lifts of) immersed curves
of companion knots. We wonder if this property characterize these pattern
knots.

######  Question 1.7.

Are one-bridge braid patterns and their compositions the only pattern knots
that induce satellite operations that act as planar transforms on immersed
curves in the marked torus?

More generally, one can ask the following question.

######  Question 1.8.

Which pattern knots are detected by the bordered Floer bimodule?

Pattern knot detection is closely related to the pursuit of understanding
which links are detected by Heegaard Floer homology, as a pattern knot $P$ is
uniquely determined by the link $L_{P}$ in $S^{3}$ consisting of $P$ and the
meridian of the solid torus containing $P$. For example, detection of
$(n,1)$-cable patterns would follow from the corresponding link detection
result on $T_{2,2n}$ by Binns-Martin [BM20, Theorem 3.2]. Note that the
bimodule of a pattern knot complement is stronger than the knot/link Floer
homology group of $L_{P}$, so it is also natural to wonder if one can detect
patterns using bimodules that are not seen by the link detection results.

###  1.5. Organization

The rest of the paper can be divided into two parts.

The first part includes Section 2 to Section 4 and establishes the immersed
Heegaard Floer theory outlined in the introduction. Section 2 defines bordered
Heegaard Floer invariants of immersed bordered Heegaard diagrams. Section 3
defines knot Floer chain complexes of immersed doubly-pointed Heegaard
diagrams. Section 4 introduces the pairing constructions and proves the
corresponding pairing theorems, i.e., Theorem 1.4 and Theorem 1.6.

The second part concerns satellite knot and includes Section 5 and Section 6.
Section 5 proves Theorem 1.1 by applying the machinery established in the
previous sections. Section 6 applies Theorem 1.1 to study satellite knots with
$(1,1)$ patterns, in which we remove the $z$-passable assumption and analyze
satellites with one-bridge-braid patterns and the Mazur pattern in detail.

### Acknowledgment

The authors would like to thank Robert Lipshitz, Adam Levine, and Liam Watson
for helpful conversations while this work was in progress. The first author
was partially supported by the Max Planck Institute for Mathematics and the
Pacific Institute for the Mathematical Sciences during the preparation of this
work; the research and findings may not reflect those of the institutes. The
second author was partially supported by NSF grant DMS-2105501.

##  2\. Bordered Floer invariants of immersed Heegaard diagrams

This section aims to define (weakly extended) type D structures111Here, weakly
extended type D structures are same as type D structures with generalized
coefficient maps appearing in [LOT18, Chapter 11.6], and we call them weak
since they are a quotient of the extended type D structures defined in
[HRW23]. of a certain class of immersed bordered Heegaard diagrams. Even
though the Lagrangians are possibly immersed, we can still define such
structures by counting holomorphic curves with smooth boundaries. The main
technical complication compared to the embedded-Heegaard-diagram case are the
possible appearance of boundary degeneration, which interferes with the usual
proof that the differential squares to a desired element. Our method to deal
with this issue is to employ a key advantage of Heegaard Floer homology: The
boundary degenerations can be controlled by imposing certain diagrammatic
conditions on the Heegaard diagrams (in Section 2.1), and in Section 4 we show
these conditions are achievable via perturbing the $\alpha$ curves on the
Heegaard diagrams that we are interested in.

This section can be viewed as modifying [LOT18, Chapter 5 and Chapter 11.6]
and [LOT08, Errata] to our setting. Indeed, the local results on holomorphic
curves such as transversality and gluing results carry over without changes.
The main differences are that (1) the embedded index formula is different (see
Section 2.6) and that (2) we need a parity count of boundary degenerations
with corners at the self-intersection points of the immersed Lagrangian (see
Section 2.7–2.8). We also add in more details on counting holomorphic curves
with a single interior puncture as sketched in [LOT08, Errata]. The counting
of such curves also appeared in [HRW22], and the more general case where
multiple interior punctures are allowed is studied in-depth in the recent work
by Lipshitz-Ozsváth-Thurston on defining a $HF^{-}$ bordered invariant
[LOT23]; our analysis of the degeneration of one-punctured curves at east
infinity is extracted from [HRW22].

###  2.1. Immersed bordered Heegaard diagrams

######  Definition 2.1.

A local system over a manifold $M$ consists of a vector bundle over $M$ and a
parallel transport of the vector bundle. A trivial local system is a local
system where the vector bundle is of rank $1$.

######  Definition 2.2.

An _immersed bordered Heegaard diagram_ is a quadruple
$\mathcal{H}=(\bar{\Sigma},\bm{\bar{\alpha}},\bm{\beta},z)$, where

  * (1)

$\bar{\Sigma}$ is a compact, oriented surface of genus $g$ with one boundary
component;

  * (2)

$\bm{\bar{\alpha}}=\bm{\bar{\alpha}}^{a}\cup\bm{\alpha}^{c}$. Here
$\bm{\bar{\alpha}}^{a}=\\{\bar{\alpha_{1}}^{a},\bar{\alpha_{2}}^{a}\\}$ is a
set of two properly embedded arcs in $\bar{\Sigma}$, and
$\bm{\alpha}^{c}=\\{\alpha_{1},\ldots,\alpha_{g-2},\alpha_{g-1}\\}$, where
$\alpha_{1},\ldots,\alpha_{g-2}$ are embedded circles in the interior of
$\bar{\Sigma}$ and $\alpha_{g-1}=\\{\alpha^{0}_{im},\ldots,\alpha^{n}_{im}\\}$
is an immersed multicurve with a local system in the interior of
$\bar{\Sigma}$. We require that $\alpha_{im}^{0}$ has a trivial local system
and that
$\\{\bar{\alpha_{1}^{a}},\bar{\alpha_{2}^{a}},\alpha_{1},\ldots,\alpha_{g-2},\alpha_{im}^{0}\\}$
are pairwise disjoint and homologically independent in
$H_{1}(\bar{\Sigma},\partial{\bar{\Sigma}};\mathbb{Z})$. We also require the
curves $\alpha_{im}^{1},\ldots,\alpha_{im}^{n}$ are homologically trivial in
$H_{1}(\bar{\Sigma},\mathbb{Z})$222To maintain this property, we will not
consider handleslides of homologically-trivial curves over other embedded
$\alpha$-curves.;

  * (3)

$\bm{\beta}=\\{\beta_{1},\ldots,\beta_{g}\\}$ consists of $g$ pairwise
disjoint, homologically independent circles embedded in the interior of
$\bar{\Sigma}$;

  * (4)

A base point
$z\in\partial\bar{\Sigma}\backslash\partial\bar{\bm{\alpha}}^{a}$.

######  Remark 2.3.

We also denote $\alpha_{g-1}$ by $\alpha_{im}$ and call $\alpha_{im}^{0}$ the
_distinguished component_ of $\alpha_{im}$. Note if $\alpha_{im}$ is embedded
and consists of only the distinguished component, then the immersed bordered
Heegaard diagram is just an ordinary bordered Heegaard diagram representing a
bordered 3-manifold with torus boundary.

######  Remark 2.4.

One can define immersed bordered Heegaard diagrams that have more than one
immersed multicurves. We content ourselves with the one-immersed-multicurve
setting, for such diagrams occur naturally in applications and we would also
like to avoid the more tedious notations incurred by allowing more immersed
multicurves. One can also work with immersed bordered diagrams that
generalizes regular bordered diagrams for 3-manifolds with higher-genus or
multi-component boundaries. Again, we avoid such cases out of conciseness.

We need to impose additional conditions on immersed bordered diagrams to
define (weakly extended) type D structures. We give some terminology before
stating the conditions. Let $\\{\mathcal{D}_{i}\\}$ be the closures of the
regions in $\bar{\Sigma}\backslash(\bm{\bar{\alpha}}\cup\bm{\beta})$. A
_domain_ $B$ is a formal linear combination of the $\mathcal{D}_{i}$’s, i.e.,
an object of the form $\sum_{i}n_{i}\mathcal{D}_{i}$ for $n_{i}\in\mathbb{Z}$,
and the coefficient $n_{i}$ is called the multiplicity of $B$ at
$\mathcal{D}_{i}$. Given a point
$p\in\bar{\Sigma}\backslash(\bm{\bar{\alpha}}\cup\bm{\beta})$, $n_{p}(B)$
denotes the multiplicity of $B$ at the region containing $p$. A domain is
called positive if $n_{i}\geq 0$ for all $i$. Note that a domain $B$ specifies
an element $[B]$ in
$H_{2}(\bar{\Sigma},\partial\bar{\Sigma}\cup\bm{\beta}\cup\bm{\bar{\alpha}};\mathbb{Z})$.
Let $l:\amalg
S^{1}\rightarrow\partial\bar{\Sigma}\cup\bm{\beta}\cup\bm{\bar{\alpha}}\subset\bar{\Sigma}$
be an oriented multicurve; note that this multicurve is not necessarily
immersed, it can have corners at intersections of $\bar{\alpha}$ with $\beta$,
intersections of $\bar{\alpha}$ with $\partial\bar{\Sigma}$, or self-
intersections of $\alpha_{im}$. A domain $B$ is said to be bounded by $l$ if
$\partial[B]=[l]$ in
$H_{1}(\partial\bar{\Sigma}\cup\bm{\beta}\cup\bm{\bar{\alpha}};\mathbb{Z})$. A
domain $B$ is called a _periodic domain_ if it is bounded by a (possibly
empty) loop in $\partial\bar{\Sigma}\cup\bar{\bm{\alpha}}^{a}$ together with
some copies of the $\beta$-circles and the $\alpha$-circles, where we allow at
most one component of $\alpha_{im}$ to appear in $\partial B$.

We will need some language to keep track of when the boundaries of domains
include corners at self-intersection points of $\alpha_{i}$. Note that,
ignoring the local systems, we can identify
$\alpha_{im}=\\{\alpha^{0}_{im},\ldots,\alpha^{n}_{im}\\}$ as the image of an
immersion $f_{im}:\amalg^{n+1}S^{1}\rightarrow\bar{\Sigma}$.

######  Definition 2.5.

A closed curve $l:S^{1}\rightarrow\alpha_{im}\subset\bar{\Sigma}$ is said to
be _stay-on-track_ or _zero-cornered_ if it lifts to a map
$\tilde{l}:S^{1}\rightarrow S^{1}\subset\amalg S^{1}$ such that
$f_{im}\circ\tilde{l}=l$. Note that this is nearly the same as saying the
curve $l$ is immersed; the difference is that stay-on-track paths can stop and
change directions along $\alpha_{im}$. A curve $l:S^{1}\rightarrow\alpha_{im}$
is said to be _$n$ -cornered_ if there exists $n$-points
$\xi_{1},\ldots\xi_{n}$ in $S^{1}$ dividing $S^{1}$ into arcs
$a_{1},\ldots,a_{n}$ (ordered cyclically, with indices understood mod $n$)
such that $l|_{a_{i}}$ lifts through $f_{im}$ for each $i$, but $l|_{a_{i}\cup
a_{i+1}}$ does not. Note that $l$ maps each $\xi_{i}$ to some self-
intersection point $q_{i}$ of $\alpha_{im}$ and makes a sharp turn at $q_{i}$;
we refer to this as a corner of the curve $l$. We define an arc to be either
stay-on-track or $n$-cornered similarly.

Figure 3. Examples of a $3$-cornered curve and a stabilized teardrop on a
genus-2 immersed bordered Heegaard diagram, where the $\beta$ curves are
omitted. The $3$-cornered curve is the boundary of the shaded triangular
region. The highlighted region is a stabilized teardrop, where the dashed
circle is the separating curve.

See Figure 3 for an example of a $3$-cornered curve. Next, we define a class
of domains on immersed bordered Heegaard diagrams.

######  Definition 2.6.

A domain $B$ on a genus $g$ immersed bordered Heegaard diagram is called a
_stabilized teardrop_ if it satisfies the following conditions:

  * (1)

$B$ is a positive domain bounded by $\partial\bar{\Sigma}$ (with induced
boundary orientation) and a one-cornered subloop of $\alpha_{im}$. (In
particular, $B$ is a formal linear combination of regions of
$\bar{\Sigma}\backslash\bm{\bar{\alpha}}$.)

  * (2)

There exists a separating curve $C$ of $\bar{\Sigma}$ which does not intersect
$\bm{\bar{\alpha}}$, and the local multiplicity of $B$ on the region
containing $C$ is $1$.

  * (3)

Surgery on $(\bar{\Sigma},\bm{\bar{\alpha}})$ along $C$ produces two oriented
surfaces with $\alpha$-curves:
$(E_{1},\bar{\bm{\alpha}}^{a},\alpha_{1},\ldots,\alpha_{g-1})$, where $E_{1}$
is a genus-$(g-1)$ with one boundary component, and $(E_{2},\alpha_{im})$,
where $E_{2}$ a genus-one surface. The domain $B$ gives rise to two domains
$B_{1}$ and $B_{2}$ on $E_{1}$ and $E_{2}$ respectively, such that
$[B_{1}]=[E_{1}]$ and $B_{2}$ is an immersed teardrop in $E_{2}$ bounded by a
one-cornered subloop of $\alpha_{im}$. (Here, we also allow teardrops with
concave corner.)

A pictorial example of a stabilized teardrop is shown in Figure 3. For
convenience, we introduce the following terminology.

######  Definition 2.7.

A domain $B$ is said to be an _$n$ -cornered $\alpha$-bounded domain_ if it is
bounded by (possibly trivial) loops contained in
$\partial\bar{\Sigma}\cup\bar{\bm{\alpha}}^{a}\cup\alpha_{1}^{c}\cup\ldots\cup\alpha_{g-2}^{c}$
and an $n$-cornered loop contained in some connected component of
$\alpha_{im}$.

The condition on bordered diagrams needed to deal with boundary degenerations
is the following.

######  Definition 2.8.

Given an immersed bordered Heegaard diagram
$\mathcal{H}=(\bar{\Sigma},\bm{\bar{\alpha}},\bm{\beta},z)$,
$\bm{\bar{\alpha}}$ is called _unobstructed_ if

  * (1)

there are no positive zero- or one-cornered $\alpha$-bounded domains $B$ with
$n_{z}(B)=0$,

  * (2)

the only positive zero-cornered $\alpha$-bounded domain $B$ with $n_{z}(B)=1$
is $[\Sigma]$,

  * (3)

any positive one-cornered $\alpha$-bounded domain $B$ with $n_{z}(B)=1$ is a
stabilized teardrop, and

  * (4)

any positive two-cornered $\alpha$-bounded domain $B$ with $n_{z}(B)=0$ is a
bigon.

Abusing the terminology, we also say an immersed Heegaard diagram is
unobstructed if its $\bm{\bar{\alpha}}$ is so.

By a Reeb chord in $(\partial\bar{\Sigma},\partial\bar{\bm{\alpha}^{a}})$, we
mean an oriented chord on $\partial\bar{\Sigma}$ whose endpoints are on
$\partial\bar{\bm{\alpha}^{a}}$ and whose orientation is induced from that of
$\partial\bar{\Sigma}$. When defining type D structures, it will be convenient
to use Reeb chords in $(-\partial\bar{\Sigma},\partial\bar{\bm{\alpha}^{a}})$.
Let $\rho_{0},\rho_{1},\ldots,\rho_{3}$ denote the Reeb chords corresponding
to the four arcs in
$-\partial\bar{\Sigma}\backslash\partial\bar{\bm{\alpha}^{a}}$, where
$\rho_{0}$ contains the base point $z$ and the sub-index increases according
to the orientation of $-\partial\bar{\Sigma}$. We call these four Reeb chords
the elementary Reeb chords. Other Reeb chords in
$(-\partial\bar{\Sigma},\partial\bar{\bm{\alpha}^{a}})$ can be obtained by
concatenation of the four elementary Reeb chords. For example, $\rho_{12}$
denotes the concatenation of $\rho_{1}$ and $\rho_{2}$. We use the notation
$-\rho_{I}$ to indicate the orientation reversal of $\rho_{I}$, where $I$ is a
string of words in $\\{0,1,2,3\\}$ that permits concatenation; note that
$-\rho_{I}$ is a Reeb chord in
$(\partial\bar{\Sigma},\partial\bar{\bm{\alpha}^{a}})$.

We shall need two types of admissibility on the bordered Heegaard diagrams:
the first one is needed for defining the (extended) type D structures, and the
second one is needed for the paring operation studied in Section 4.7. Given a
domain $B$, denote by $n_{-\rho_{i}}(B)$ ($i=0,1,2,3$) the local multiplicity
of $B$ in the region containing the Reeb chord $-\rho_{i}$. (In particular,
$n_{-\rho_{0}}(B)=n_{z}(B)$). A periodic domain $B$ is called _provincial_ if
$n_{-\rho_{i}}(B)=0$ for all $i=0,1,2,3$.

######  Definition 2.9.

An immersed bordered Heegaard diagram is _provincially admissible_ if all non-
trivial provincial periodic domains have both positive and negative local
multiplicities.

######  Definition 2.10.

An immersed bordered Heegaard diagram is _bi-admissible_ if any non-trivial
periodic domain $B$ satisfying $n_{-\rho_{0}}(B)=n_{-\rho_{1}}(B)=0$ or
$n_{-\rho_{2}}(B)=n_{-\rho_{3}}(B)=0$ has both positive and negative local
multiplicities.

Note that bi-admissibility implies provincial admissibility.

###  2.2. Moduli spaces of stay-on-track holomorphic curves

In this subsection, we set up the moduli spaces that we use to define the
(weakly extended) type D structures. Roughly, the moduli spaces consist of
pseudo-holomorphic curves in $\Sigma\times[0,1]\times\mathbb{R}$. We define
them by modifying the corresponding definitions in Section 5.2 of [LOT18] with
two main differences: The first one is a new constraint on the boundaries with
respect to $\alpha_{im}$, and the second one is that we include pseudo-
holomorphic curves with a single interior puncture for the sake of defining a
weakly extended type D structure.

####  2.2.1. Definition of moduli spaces of holomorphic curves

######  Definition 2.11.

A _decorated source $S^{\triangleright}$ of type $0$-P_ is a smooth Riemann
surface $S$ with boundary such that

  * (1)

it has boundary punctures and has no interior punctures,

  * (2)

there is a labeling of each puncture by $+$, $-$, or $e$, and

  * (3)

there is a labeling of each $e$ puncture by a Reeb chord on the boundary of
the immersed bordered Heegaard diagram.

A _decorated source $S^{\triangleright}$ of type $1$-P_ is a smooth Riemann
surface $S$ with boundary such that

  * (1)

it has boundary punctures and a single interior puncture,

  * (2)

there is a labeling of each boundary puncture by $+$ or $-$, and

  * (3)

there is a labeling of the interior puncture by $e$.

By a decorated source, we mean it is either a decorated source of type $0$-P
or a decorated source of type $1$-P.

Let $\mathcal{H}=(\bar{\Sigma},\bm{\bar{\alpha}},\bm{\beta},z)$ be an immersed
bordered Heegaard diagram. Let $\Sigma$ denote the interior of $\bar{\Sigma}$.
Equip the target manifold $\Sigma\times[0,1]\times\mathbb{R}$ with an
admissible almost complex structure $J$ as in Definition 5.1 of [LOT18]. Let
$\pi_{\mathbb{D}}$, $\pi_{\Sigma}$, $s$, and $t$ denote the canonical
projection maps from $\Sigma\times[0,1]\times\mathbb{R}$ to
$[0,1]\times\mathbb{R}$, $\Sigma$, $[0,1]$, and $\mathbb{R}$ respectively. We
will count maps

$u:(S,\partial
S)\rightarrow(\Sigma\times[0,1]\times\mathbb{R},\bm{\beta}\times\\{0\\}\times\mathbb{R},\bm{\bar{\alpha}}\times\\{1\\}\times\mathbb{R})$

from decorated sources to the target manifold satisfying the following
conditions:

  * (M-1)

$u$ is $(j,J)$-holomorphic, where $j$ is a complex structure on the surface
$S$.

  * (M-2)

$u$ is proper.

  * (M-3)

$u$ extends to a proper map
$u_{\bar{e}}:S_{\bar{e}}\rightarrow\Sigma_{\bar{e}}\times[0,1]\times\mathbb{R}$,
where $S_{\bar{e}}$ and $\Sigma_{\bar{e}}$ are surfaces obtained by filling in
the corresponding east puncture(s).

  * (M-4)

the map $u_{\bar{e}}$ has finite energy in the sense in [BEH+03].

  * (M-5)

$\pi_{\mathbb{D}}\circ u_{\bar{e}}$ is a $g$-fold branched cover.

  * (M-6)

$t\circ u$ approaches $\infty$ at $+$ punctures.

  * (M-7)

$t\circ u$ approaches $-\infty$ at $-$ punctures.

  * (M-8)

$\pi_{\Sigma}\circ u$ approaches the labeled Reeb chord at a boundary $e$
punctures.

  * (M-9)

$\pi_{\Sigma}\circ u$ covers each of the regions next to
$\bar{e}\in\Sigma_{\bar{e}}$ at most once.

  * (M-10)

(Strong boundary monotonicity) For each $t\in\mathbb{R}$ , each of
$u^{-1}(\beta_{i}\times\\{0\\}\times\\{t\\})$ $(i=1,\ldots,g)$ and
$u^{-1}(\alpha^{c}_{i}\times\\{1\\}\times\\{t\\})$ $(i=1,\ldots,g-1)$ consists
of exactly one point, and $u^{-1}(\alpha_{i}^{a}\times\\{1\\}\times\\{t\\})$
$(i=1,2)$ consists of at most one point333In [LOT18], a weak boundary
monotonicity condition was introduced as well as the strong boundary
monotonicity. When restricting to torus-boundary bordered manifolds, however,
these two conditions are equivalent. So, we only state the strong boundary
monotonicity condition here..

  * (M-11)

(Stay-on-track boundary condition) Let $A$ be the boundary component of $S$
that is mapped to the $\alpha_{im}\times\\{1\\}\times\mathbb{R}$. Then
$\pi_{\Sigma}\circ u|_{A}$ is stay-on-track.

######  Remark 2.12.

Only (M-9) and (M-11) are different from the corresponding conditions in
[LOT18]. We impose (M-9) since we aim to define an extended type D structure,
which will not need holomorphic curves covering the boundary regions multiple
times.

Given an immersed (bordered) Heegaard diagram, generators and homology classes
of maps connecting generators are defined similarly as in the
embedded-$\alpha$-curve case.

######  Definition 2.13.

Let $\bm{x}$ and $\bm{y}$ be two generators and let
$B\in\tilde{\pi}_{2}(\bm{x},\bm{y})$ be a homology class connecting $\bm{x}$
to $\bm{y}$. $\widetilde{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright})$
is defined to be the moduli space of holomorphic curves with decorated source
$S^{\triangleright}$, satisfying (M-1)-(M-11), asymptotic to $\bm{x}$ at
$-\infty$ and $\bm{y}$ at $+\infty$, and inducing the homology class $B$.

Let $E(S^{\triangleright})$ be the set of east punctures of
$S^{\triangleright}$ lying on the boundary. Let
$\widetilde{ev}\colon\widetilde{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright})\rightarrow\mathbb{R}^{|E(S^{\triangleright})|}$
be the evaluation map given by the values of $t\circ u_{\bar{e}}$ at the east
punctures; the values are called heights of the east punctures.

######  Definition 2.14.

Let $P=\\{P_{i}\\}$ be a partition of $E$. Then

$\widetilde{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};P)\coloneqq\widetilde{ev}^{-1}(\Delta_{P}),$

where $\Delta_{P}:=\\{(x_{p})\in\mathbb{R}^{|E|}\mid x_{p}=x_{q}\text{ if
}p,q\in P_{i}\text{ for some i}\\}$.

######  Definition 2.15.

Let $\overrightarrow{P}=(P_{1},\ldots,P_{k})$ be an ordered partition $E$, and
let $P$ denote the corresponding underlying unordered partition. Define
$\widetilde{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};\overrightarrow{P})$
to be
$\\{u\in\widetilde{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};P)\mid
t\circ u(p)<t\circ u(q)\text{ for }p\in P_{i},\text{ }q\in
P_{i^{\prime}},\text{ and }i<i^{\prime}\\}.$

There is an $\mathbb{R}$-action on the above moduli spaces given by
translations along the $\mathbb{R}$-coordinate of
$\Sigma\times[0,1]\times\mathbb{R}$. The reduced moduli spaces are the
quotient of the relevant moduli spaces by the $\mathbb{R}$-action; they are
denoted by $\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright})$,
$\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright};{P})$, and
$\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright};\overrightarrow{P})$,
respectively. The evaluation maps $\widetilde{ev}$ induce maps $ev$ from the
reduced moduli spaces to $\mathbb{R}^{|E(S^{\triangleright})|}/\mathbb{R}$,
which record the relative heights between boundary east punctures.

##### Notation.

When we need to distinguish moduli spaces of 0-P holomorphic curves without
east punctures and moduli spaces of 1-P holomorphi curves, we will use the
notation $\widetilde{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};U)$ to
emphasize the source $S^{\triangleright}$ is of type 1-P.

####  2.2.2. Regularity and the expected dimension

######  Proposition 2.16.

For a generic admissible almost complex structure on
$\Sigma\times[0,1]\times\mathbb{R}$, the moduli space
$\widetilde{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};P)$ is
transversally cut out.

###### Proof.

See Proposition 5.6 of [LOT18]. We point out that the $\alpha$ curves being
immersed does not affect the usual proof. When analyzing the linearization of
the $\bar{\partial}$-operator in the standard proof of such a result, one
would be working with a pull-back bundle over $S$, on which one will not see
the immersed boundary condition anymore. ∎

######  Proposition 2.17.

Let $B\in\widetilde{\pi}_{2}(\bm{x},\bm{y})$. Let $g$ denote the genus of the
bordered Heegaard diagram. Let $\chi(\cdot)$ and $e(\cdot)$ denote the Euler
number and Euler measure, respectively.

  * (1)

Let $S_{0}^{\triangleright}$ be a decorated source of 0-P. Then the expected
dimension of the moduli space
$\widetilde{\mathcal{M}}^{B}(\bm{x},\bm{y};S_{0}^{\triangleright};P)$ is

$\text{ind}(B,S_{0},P)=g-\chi(S_{0})+2e(B)+|P|.$

  * (2)

Let $S_{1}^{\triangleright}$ be a decorated source of 1-P. Then the expected
dimension of the moduli space
$\widetilde{\mathcal{M}}^{B}(\bm{x},\bm{y};S_{1}^{\triangleright};U)$ is

$\text{ind}(B,S_{1})=g-\chi(S_{1})+2e(B)+1.$

###### Proof.

For (1) see Proposition 5.8 of [LOT18]. For the same reason mentioned in the
proof of the previous proposition, our $\alpha$-curves being immersed does not
affect the proof.

For (2), note that using the removable singularity theorem we can identify
holomorphic curves in
$\widetilde{\mathcal{M}}^{B}(\bm{x},\bm{y};S_{1}^{\triangleright};U)$ with
holomorphic curves in $\Sigma_{\bar{e}}\times[0,1]\times\mathbb{R}$ that
intersects $\\{e\\}\times[0,1]\times\mathbb{R}$ geometrically once. The
formula then follows from the index formula in the close-Heegaard-diagram case
given in Section 4.1 of [Lip06]. (Again, $\alpha_{im}$ being immersed does not
affect the formula).

∎

######  Remark 2.18.

If we regard the interior puncture of a 1-P holomorphic curve is asymptotic to
a single closed Reeb orbit, denoted by $U$, then (1) and (2) in the above
proposition may be unified using the formula in (1).

###  2.3. Compactification

The moduli spaces defined in Section 2.2 admit compactification similar to
that in [LOT18]. The overall idea is that a sequence of holomorphic curves in
$\Sigma\times[0,1]\times\mathbb{R}$ may converge to a holomorphic building in
$\Sigma\times[0,1]\times\mathbb{R}$ together with some holomorphic curves in
the east-$\infty$ attaching to it; such nodal holomorphic objects are called
_holomorphic combs_. In our setup, the degeneration in the east-$\infty$ is
the same as those when the $\alpha$-curves are embedded; we recollect the
relevant material (with straightforward modifications to accommodate for 1-P
holomorphic curves) in Subsection 2.3.1. However, the immersed $\alpha$ curves
do complicate the situation. For example, a limit holomorphic building in
$\Sigma\times[0,1]\times\mathbb{R}$ may have corners at self-intersection
points of $\alpha_{im}$. We will give a precise description of this phenomenon
in Subsection 2.3.2.

####  2.3.1. Holomorphic curves in the end at east-infinity

Let $Z$ denote the oriented boundary $\partial\bar{\Sigma}$ of the bordered
Heegaard surface. We define the moduli spaces of holomorphic curves in the
east end $\mathbb{R}\times Z\times[0,1]\times\mathbb{R}$. They host possible
degenerations of the limits of holomorphic curves at east-$\infty$. Since the
closed $\alpha$ curves do not approach the cylindrical end at east-$\infty$,
these moduli spaces are not affected by the closed $\alpha$ curves being
immersed and their definition is the same as the usual embedded case. We first
specify the sources of the holomorphic curves.

######  Definition 2.19.

A _bi-decorated source_ $T^{\diamond}$ is a smooth Riemann surface $T$ with
boundary such that

  * (1)

it has boundary punctures and at most one interior puncture,

  * (2)

the boundary punctures are labeled by $e$ or $w$,

  * (3)

the interior puncture, if exits, is labeled by $e$ and,

  * (4)

the boundary punctures are also labeled by Reeb chords.

Equip $\mathbb{R}\times Z\times[0,1]\times\mathbb{R}$ with a split almost
complex structure $J=j_{\mathbb{R}\times Z}\times j_{\mathbb{D}}$. The four
points $\textbf{a}=\partial{\bm{\alpha}^{a}}$ on $Z$ give rise to four
Lagrangians $\mathbb{R}\times\textbf{a}\times\\{1\\}\times\mathbb{R}$.

######  Definition 2.20.

Given a bi-decorated source $T^{\diamond}$, define
$\widetilde{\mathcal{N}}(T^{\diamond})$ to be the moduli spaces of maps
$v:(T,\partial T)\rightarrow(\mathbb{R}\times
Z\times[0,1]\times\mathbb{R},\mathbb{R}\times\textbf{a}\times\\{1\\}\times\mathbb{R})$
satisfying the following conditions:

  * (N-1)

$v$ is $(j,J)$-holomorphic with respect to some complex structure $j$ on $T$.

  * (N-2)

$v$ is proper.

  * (N-3)

Let $T_{\bar{e}}$ and $(\mathbb{R}\times Z)_{\bar{e}}$ denote the spaces
obtained from $T$ and $\mathbb{R}\times Z$ by filling in the east punctures.
Then $v$ extends to a proper map
$v_{\bar{e}}:T_{\bar{e}}\rightarrow(\mathbb{R}\times
Z)_{\bar{e}}\times[0,1]\times\mathbb{R}$ such that $\pi_{\Sigma}\circ
v_{\bar{e}}(e)=e$.444This condition excludes mapping the interior puncture end
to the west infinity end.

  * (N-4)

At each boundary $w$ puncture, $\pi_{\Sigma}\circ v$ approaches the
corresponding Reeb chords in $-\infty\times Z$ that labels $w$.

  * (N-5)

At each boundary $e$ puncture, $\pi_{\Sigma}\circ v$ approaches the
corresponding Reeb chords in $\infty\times Z$ that labels $e$.

We have the following proposition regarding the regularity of the moduli
spaces.

######  Proposition 2.21 (Proposition 5.16 of [LOT18]).

If all components of a bi-decorated source $T^{\diamond}$ are topological
disks (possibly with an interior puncture), then
$\widetilde{\mathcal{N}}(T^{\diamond})$ is transversally cut out for any split
almost complex structure on $\mathbb{R}\times Z\times[0,1]\times\mathbb{R}$.

The heights of $v\in\widetilde{\mathcal{N}}(T^{\diamond})$ at east or west
boundary punctures induce evaluation functions $\widetilde{ev}_{e}$ and
$\widetilde{ev}_{w}$. Given partitions $P_{e}$ and $P_{w}$ of the boundary
east and west punctures, one defines
$\widetilde{\mathcal{N}}(T^{\diamond};P_{e};P_{w})$ in an obvious way. One
defines the reduced moduli space $\mathcal{N}$ by taking the quotient of the
$\mathbb{R}\times\mathbb{R}$-action induced by translations in both
$\mathbb{R}$-directions in
$\mathbb{R}\times\mathcal{Z}\times[0,1]\times\mathbb{R}$. The evaluation maps
$\widetilde{ev}_{e}$ and $\widetilde{ev}_{w}$ also descend to $\mathcal{N}$,
taking values in $\mathbb{R}^{|E(T^{\diamond})|}/\mathbb{R}$ and
$\mathbb{R}^{|W(T^{\diamond})|}/\mathbb{R}$, respectively.

Given $u\in\mathcal{N}(T^{\diamond})$, the open mapping theorem implies that
the map $\pi_{\mathbb{D}}\circ u$ is constant on connected components of
$T^{\diamond}$ (taking values in $\\{1\\}\times\mathbb{R}$). So, the map $u$
is determined by its projection $\pi_{\Sigma}\circ u$ and $t$-coordinates on
connected components of $T$. Of primary interest to us are the following three
types of holomorphic curves.

  * (Join curve)

A _join component_ of a bi-decorated source is a topological disk with three
boundary punctures, and the punctures are labeled by $(e,\sigma)$,
$(w,\sigma_{1})$, and $(w,\sigma_{2})$ counter-clockwise, where the Reeb
chords satisfy the relation $\sigma=\sigma_{1}\uplus\sigma_{2}$ (here $\uplus$
denotes concatenation of Reeb chords). A _trivial component_ of a bi-decorated
source is a topological disk with two boundary punctures, one $w$ puncture and
one $e$ puncture, and both are labeled by the same Reeb chord. Holomorphic
maps from a join component or a trivial component to $\mathbb{R}\times
Z\times[0,1]\times\mathbb{R}$ exist and are unique up to translations. A _join
curve_ is a holomorphic curve with a bi-decorated source consisting of a
single join component and possibly some trivial components.

  * (Split curve)

A _split component_ of a bi-decorated source is a topological disk with three
boundary punctures, where the punctures are counter-clockwisely labeled by
$(e,\sigma_{1})$, $(e,\sigma_{2})$ and $(w,\sigma)$ with
$\sigma=\sigma_{1}\uplus\sigma_{2}$. Holomorphic maps from a split component
to $\mathbb{R}\times Z\times[0,1]\times\mathbb{R}$ exist and are unique up to
translations. A _split curve_ is a holomorphic curve with a bi-decorated
source consisting of one or more split components and possibly some trivial
components.

  * (Orbit curve)

An _orbit component_ of a bi-decorated source is a topological disk with a
single boundary $w$ puncture labeled by $\sigma\in\\{\
-\rho_{0123},-\rho_{1230},-\rho_{2301},-\rho_{3012}\\}$ and a single interior
$e$ puncture. Holomorphic maps from an orbit component to $\mathbb{R}\times
Z\times[0,1]\times\mathbb{R}$ exist and are unique up to translations. An
_orbit curve_ is holomorphic curve with a bi-decorated source consisting of a
single orbit component and possibly some trivial components.

####  2.3.2. Compactification by Holomorphic combs

We describe holomorphic combs in this subsection. We begin with a description
of nodal holomorphic curves.

######  Definition 2.22.

A _nodal decorated source_ $S^{\triangleright}$ is a decorated source together
with a set of unordered pairs of marked points
$D=\\{\\{\overline{d}_{1},\underline{d}_{1}\\},\\{\overline{d}_{2},\underline{d}_{2}\\},\ldots,\\{\overline{d}_{k},\underline{d}_{k}\\}\\}$.
The points in $D$ are called nodes.

######  Definition 2.23.

Let $\bm{x}$ and $\bm{y}$ be generators and let
$B\in\widetilde{\pi}_{2}(\bm{x},\bm{y})$. Let $S^{\triangleright}$ be a nodal
decorated source. Let $S_{i}$ be the components of
$S\backslash\\{\text{nodes}\\}$. Then a nodal holomorphic curve $u$ with
source $S^{\triangleright}$ in the homology class of $B$ is a continuous map

$u:(S,\partial
S)\rightarrow(\Sigma\times[0,1]\times\mathbb{R},\beta\times\\{0\\}\times\mathbb{R},\alpha\times\\{1\\}\times\mathbb{R})$

such that

  * (1)

the restriction of $u$ to each $S_{i}$ is a map satisfying condition
(M-1)-(M-11) except for (M-5),

  * (2)

$\lim_{p\rightarrow\overline{d}_{i}}u(p)=\lim_{p\rightarrow\underline{d}_{i}}u(p)$
for every pair of nodes,

  * (3)

$u$ is asymptotic to $\bm{x}$ at $-\infty$ and $\bm{y}$ at $\infty$, and

  * (4)

$u$ induces the homology class specified by $B$.

The nodes in a nodal source $S^{\triangleright}$ induce punctures on the
connected components $S_{i}$ of $S\backslash\\{\text{nodes}\\}$; they can be
interior punctures as well as boundary punctures. Note that $u|_{S_{i}}$
extends across these punctures continuously. We further divide the boundary
punctures induced by the nodes into two types.

######  Definition 2.24.

Let $u$ be a nodal holomorphic curve. Let $d$ be a boundary puncture induced
by a node on a component $S_{i}$ of the nodal Riemann surface. Let $l_{1}$ and
$l_{2}$ denote the components of $\partial S_{i}$ adjacent to $d$. If the path
$\pi_{\Sigma}\circ u|_{l_{1}\cup\\{d\\}\cup l_{2}}$ is stay-on-track in the
sense of (M-11), then we say $d$ is a _type I puncture_ ; otherwise, we say
$d$ is a _type II puncture_.

There are only type I punctures when the attaching curves are embedded. Type
II punctures naturally appear in our setup since we have an immersed $\alpha$
curve.

One can still define the evaluation map from the space of nodal holomorphic
maps to $\mathbb{R}^{|E(S^{\triangleright})|}$, where the value at a nodal
holomorphic curve is the heights of the east punctures.

######  Definition 2.25.

A _holomorphic story_ is an ordered $(k+1)$-tuple $(u,v_{1},\ldots,v_{k})$ for
some $k\geq 0$ such that

  * (1)

$u$ is a (possibly nodal) holomorphic curve in
$\Sigma\times[0,1]\times\mathbb{R}$,

  * (2)

each $v_{i}$ ($i=1,2,\ldots,k$) is a holomorphic curve in
$\mathbb{R}\times\mathcal{Z}\times[0,1]\times\mathbb{R}$,

  * (3)

the boundary east punctures of $u$ match up with the boundary west punctures
of $v_{1}$ (i.e., the two sets of punctures are identified by a one-to-one map
such that both the Reeb-chord labels and the relative heights are the same
under this one-to-one correspondence), and

  * (4)

the boundary east punctures of $v_{i}$ match up with the boundary west
punctures of $v_{i+1}$ for $i=1,2,\ldots,k-1$.

######  Definition 2.26.

Let $N\geq 1$ be an integer, and let $\bm{x}$ and $\bm{y}$ be two generators.
A _holomorphic comb_ of height $N$ connecting $\bm{x}$ to $\bm{y}$ is a
sequence of holomorphic stories $(u_{i},v_{i,1},\ldots,v_{i,k_{i}})$,
$i=1,2,\ldots,N$, such that $u_{i}$ is a (possibly nodal) stable curve in
$\mathcal{M}^{B_{i}}(\bm{x}_{i},{\bm{x}_{i+1}};S_{i}^{\triangleright})$ for
some generators ${\bm{x}_{1}},\ldots,{\bm{x}_{N+1}}$ such that
$\bm{x}_{1}=\bm{x}$ and $\bm{x}_{N+1}=\bm{y}$.

Given a holomorphic comb, the underlying (nodal) decorated sources and bi-
decorated sources can be glued up and deformed in an obvious way to give a
smooth decorated source; it is called the _preglued source_ of the holomorphic
comb.

######  Definition 2.27.

Given generators $\bm{x}$ and $\bm{y}$, and a homology class
$B\in\tilde{\pi}_{2}(\bm{x},\bm{y})$.
$\overline{\overline{\mathcal{M}}}^{B}(\textbf{x},\textbf{y};S^{\triangleright})$
is defined to be the space of all holomorphic combs with preglued source
$S^{\triangleright}$, in the homology class of $B$, and connecting $\bm{x}$ to
$\bm{y}$. $\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright})$ is
defined to be the closure of
$\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright})$ in
$\overline{\overline{\mathcal{M}}}^{B}(\bm{x},\bm{y};S^{\triangleright})$.
$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};P)$ and
$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};\overrightarrow{P})$
are defined to be the closure of
$\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright};P)$ and
$\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright};\overrightarrow{P}$ in
$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright})$ respectively.

The compactness result is stated below.

######  Proposition 2.28.

The moduli space
$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright})$ is compact. The
same statement holds for
$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};P)$ and
$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};\overrightarrow{P})$.

We omit the proof of the above proposition and remark that the proof of
[LOT18, Proposition 5.24] adapts to our setup easily.

###  2.4. Gluing results

As the regularity results, gluing results in pseudo-holomorphic curve theory
are proved by analyzing the $\bar{\partial}$-operator over certain section
space of some pull-back bundles over the underlying source surfaces. In
particular, having immersed $\alpha$ curves does not affect the proof of such
results. We hence recall the following results that we shall need without
giving the proof.

######  Proposition 2.29 (Proposition 5.30 of [LOT18]).

Let $(u_{1},u_{2})$ be a two-story holomorphic building with
$u_{1}\in\mathcal{M}^{B_{1}}(\bm{x},\bm{y};S_{1}^{\triangleright};P_{1})$ and
$u_{2}\in\mathcal{M}^{B_{2}}(\bm{y},\bm{z};S_{2}^{\triangleright};P_{2})$.
Assume the moduli spaces are transversally cut out. Then for sufficiently
small neighborhood $U_{i}$ of $u_{i}$ ($i=1,2$), there is neighborhood of
$(u_{1},u_{2})$ in
$\overline{\mathcal{M}}^{B_{1}+B_{2}}(\bm{x},\bm{z};S_{1}^{\triangleright}\natural
S_{2}^{\triangleright};P_{1}\cup P_{2})$ homeomorphic to $U_{1}\times
U_{2}\times[0,1)$.

######  Definition 2.30.

A holomorphic comb is said to be _simple_ if it only has a single story and it
is of the form $(u,v_{1})$, where $u$ is a non-nodal holomorphic curve.

######  Proposition 2.31 (Proposition 5.31 of [LOT18]).

Let $(u,v)$ be a simple holomorphic comb with
$u\in\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright})$ and
$v\in\mathcal{N}(T^{\diamond};P_{e})$. Let $m$ denote the number of east
punctures of $S^{\triangleright}$. Assume the moduli spaces are transversally
cut out, and the evaluation maps
$ev:\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright})\rightarrow\mathbb{R}^{m}/\mathbb{R}$
and
$ev_{w}:\mathcal{N}(T^{\diamond};P_{e})\rightarrow\mathbb{R}^{m}/\mathbb{R}$
are transverse at $(u,v)$. Then for sufficiently small neighborhood $U_{u}$ of
$u$ and $U_{v}$ of $v$, there is a neighborhood of $(u,v)$ in
$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright}\natural
T^{\diamond};P_{e})$ homeomorphic to $U_{u}\times_{ev}U_{v}\times[0,1)$.

###  2.5. Degeneration of moduli spaces

This subsection provides constraints on the degeneration in 1-dimensional
moduli spaces using the index formulas and strong boundary monotonicity. The
results here are simpler than the corresponding results in Section 5.6 of
[LOT18] since we restrict to the torus-boundary case. However, as mentioned
earlier, nodal curves with corners at self-intersection points of
$\alpha_{im}$ may occur in the compactification; we defer the further analysis
of these degenerations to later subsections. Readers who wish to skip the
details in this subsection are referred to Definition 2.34 and Proposition
2.40 for a quick summary.

The index formulas in Proposition 2.17 lead to the following constraints on
the ends of moduli spaces of 0-P curves.

######  Proposition 2.32 (cf. Proposition 5.43 of [LOT18]).

Let $B\in\tilde{\pi}_{2}(\bm{x},\bm{y})$, let $S^{\triangleright}$ be a
decorated source of 0-P, and let $P$ be a discrete partition of the east
punctures of $S^{\triangleright}$. Suppose that
$Ind(B,S^{\triangleright},P)=2$. Then for a generic almost complex structure
$J$, every holomorphic comb in
$\partial\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};P)=\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};P)-\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright};P)$
has one of the following forms:

  * (1)

a two-story holomorphic building $(u_{1},u_{2})$;

  * (2)

a simple holomorphic comb $(u;v)$ where $v$ is a join curve;

  * (3)

a simple holomorphic comb $(u;v)$ where $v$ is a split curve with a single
split component;

  * (4)

a nodal holomorphic comb, obtained by degenerating some arcs with ends on
$\partial S$.

###### Proof.

This proposition is proved the same way as Proposition 5.43 of [LOT18]: It is
a consequence of compactness, transversality, index formula, and gluing
results. Note that our statement is simpler as we restrict to a discrete
partition $P$: the shuffle-curve end and the multi-component split-curve end
that appear in [LOT18] do not occur in our setting. ∎

For moduli spaces of 1-P curves, we have the following proposition.

######  Proposition 2.33.

Let $B\in\tilde{\pi}_{2}(\bm{x},\bm{y})$ and let $S^{\triangleright}$ be a
decorated surface of type 1-P. Suppose $Ind(B,S^{\triangleright})=2$. Then for
a generic almost complex structure $J$, every holomorphic comb in
$\partial\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};U)$ has
one of the following forms:

  * (1)

a two-story holomorphic building $(u_{1},u_{2})$;

  * (2)

a simple holomorphic comb $(u;v)$ where $v$ is an orbit curve;

  * (3)

a nodal holomorphic comb, obtained by degenerating some arcs with boundary on
$\partial S$.

###### Proof.

Suppose a given holomorphic comb in
$\partial\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};U)$ is
not nodal, then it possibly has degeneration at the east infinity and level
splittings. The form of such holomorphic combs is analyzed in the Proof of
Proposition 42 in [HRW22]; the results are precisely item (1) and (2) in the
above statement. ∎

We will only be interested in those moduli spaces prescribed by an ordered
discrete partition $\overrightarrow{P}$; this is a void requirement for 1-P
holomorphic curves. This condition is also automatic for 0-P; it follows
easily from boundary monotonicity and holomorphicity (see, e.g., Lemma 5.51 of
[LOT18]).

The strong boundary monotonicity imposes further constraints on the
degeneration of one-dimensional moduli spaces. We first describe the
constraints on nodal holomorphic curves. Recall there are three types of nodal
holomorphic curves.

######  Definition 2.34.

A nodal holomorphic comb $u$ is called a _boundary degeneration_ if it has an
irreducible component $S_{0}$ that contains no $\pm$-punctures and
$\pi_{\Sigma}\circ u|_{S_{0}}$ is non-constant.

######  Definition 2.35.

A _boundary double point_ is a holomorphic comb with a boundary node $p$ such
that the projection to $[0,1]\times\mathbb{R}$ is not constant near either
preimage point $p_{1}$ or $p_{2}$ of $p$ in the normalization of the nodal
curve.

######  Definition 2.36.

A holomorphic comb $u$ is called _haunted_ if there is a component $S_{0}$ of
the source such that $u|_{S_{0}}$ is constant.

######  Proposition 2.37.

Let $J$ be a generic almost complex structure. For one-dimensional moduli
spaces
$\mathcal{M}^{B}(\bm{x},\bm{y};S_{0}^{\triangleright};\overrightarrow{P})$ of
0-P curves and for one-dimensional moduli spaces
$\mathcal{M}^{B}(\bm{x},\bm{y};S_{1}^{\triangleright};U)$ of 1-P curves,
boundary double points and haunted curves do not appear in
$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S_{0}^{\triangleright};\overrightarrow{P})$
and $\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S_{1}^{\triangleright};U)$.

###### Proof.

This is Lemma 5.56 and Lemma 5.57 of [LOT18]. ∎

In summary, the only nodal holomorphic combs that could possibly appear are
boundary degenerations. We defer the analysis of such degenerations to later
subsections. Instead, we conclude this subsection with some further
constraints in
$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S_{0}^{\triangleright};\overrightarrow{P})$
obtained by combining the strong boundary monotonicity and the torus-boundary
condition.

######  Proposition 2.38.

For one-dimensional moduli spaces
$\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright};\overrightarrow{P})$ of 0-P
curves, join curve ends do not appear in
$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};\overrightarrow{P})$.

###### Proof.

Suppose this is not true. Write
$\overrightarrow{P}=(\sigma_{1},\ldots,\sigma_{k})$ for some $k\geq 1$. The
appearance of join curve end means there is a holomorphic comb $(u;v)$ such
that
$u\in\mathcal{M}^{B}(\bm{x},\bm{y};{S^{\triangleright}}^{\prime};(\sigma_{1},\ldots,\\{\sigma_{i}^{\prime},\sigma_{i}^{\prime\prime}\\},\ldots,\sigma_{k}))$
where $\sigma_{i}=\sigma_{i}^{\prime}\uplus\sigma_{i}^{\prime\prime}$. The
strong boundary monotonicity condition implies $\bm{x}$ has one and only one
component that lies on exactly one of the $\alpha$ arcs. Hence all the east
punctures of $u$ are of different heights due to holomorphicity. This
contradicts that the east punctures marked by $\sigma_{i}^{\prime}$ and
$\sigma_{i}^{\prime\prime}$ are of the same height. ∎

######  Proposition 2.39.

Let
$\partial\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};\overrightarrow{P})=\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};\overrightarrow{P})-\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright};\overrightarrow{P})$.
Then
$\partial\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};\overrightarrow{P})\cap\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright};P)=\emptyset$.

###### Proof.

If not, then collision of levels appears, i.e., there is a sequence of
$u_{i}\in\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright};\overrightarrow{P})$
converging to a holomorphic curve
$u\in\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright};P)$ such that at least
two of the east punctures are of the same height. However, we already observed
such $u$ does not exist in the proof of Proposition 2.38. In other words,
$\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright};P)$ is equal to
$\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright};\overrightarrow{P})$ for a
particular order determined by $S^{\triangleright}$ when we restrict to the
torus-boundary case. ∎

We summarize the results above into the following proposition for convenience.

######  Proposition 2.40.

Let $B\in\tilde{\pi}_{2}(\bm{x},\bm{y})$ and let $J$ be a generic almost
complex structure. For a one-dimensional moduli space
$\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright};\overrightarrow{P})$ of 0-P
curves, where $\overrightarrow{P}$ is a discrete ordered partition of the east
punctures of $S^{\triangleright}$, every holomorphic comb in
$\partial\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};\overrightarrow{P})$
has one of the following forms:

  * (1)

a two-story holomorphic building $(u_{1};u_{2})$;

  * (2)

a simple holomorphic comb $(u;v)$ where $v$ is a split curve with a single
split component;

  * (3)

a boundary degeneration.

For a one-dimensional moduli space
$\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright};U)$ of 1-P curves, every
holomorphic comb in
$\partial\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};S^{\triangleright};U)$ has
one of the following forms:

  * (1)

a two-story holomorphic building $(u_{1},u_{2})$;

  * (2)

a simple holomorphic comb $(u;v)$ where $v$ is an orbit curve;

  * (3)

a boundary degeneration.

######  Remark 2.41.

The restriction to manifolds with torus boundary greatly simplifies the study
of moduli spaces. In higher-genus cases, join ends, collision of levels, and
split curve with many splitting components may appear. In particular, the
latter prevents one from proving the compactified moduli spaces are manifolds
with boundaries (as the gluing results fail to apply). In [LOT18], notions
like smeared neighborhood, cropped moduli spaces, and formal ends were
employed to circumvent this difficulty. The results in this subsection allow
us to avoid introducing any of these terms.

###  2.6. Embedded holomorphic curves

We only use embedded holomorphic curves when defining (weakly extended) type D
structures. When $\alpha$-curves are embedded, a holomorphic curve is embedded
if and only if the Euler characteristic of its underlying source surface is
equal to the one given by an _embedded Euler characteristic formula_. This is
still true in the current setup, but the embedded Euler characteristic formula
needs to be generalized to take care of new phenomena caused by immersed
$\alpha$-multicurves.

The embedded Euler characteristic formula involves signs of self-intersections
of oriented immersed arcs in our setup. Here we fix the sign convention. Let
$f:(0,1)\rightarrow\Sigma$ be an immersed arc with transverse double self-
intersection at $p=f(a)=f(b)$ with $0<a<b<1$. Then the sign of the
intersection point at $p$ is positive if the orientation at $T_{p}\Sigma$
coincides with the one specified by the ordered pair
$(f^{\prime}(a),f^{\prime}(b))$; otherwise, the sign of the intersection point
is negative. Let $f\cdot f$ denote the signed count of self-intersection
points of the arc $f$. Let $B\in\pi_{2}(\bm{x},\bm{y};\overrightarrow{P})$ be
a domain. Then up to homotopy within $\alpha_{im}$, the boundary of $B$ at
$\alpha_{im}$ gives rise to a curve $\partial_{\alpha_{im}}B$. We will
restrict to those domains for which $\partial_{\alpha_{im}}B$ has a single
component. We define $s(\partial_{\alpha_{im}}B)$ to be
$(\partial_{\alpha_{im}}B)\cdot(\partial_{\alpha_{im}}B)$, the signed count of
transverse double points in $\partial_{\alpha_{im}}B$.

For convenience, we also define the _length_ of each elementary Reeb chord to
be one and the _length_ $|\sigma|$ of a general Reeb chord $\sigma$ to be the
number of elementary Reeb chords it consists of. Note if
$\overrightarrow{\rho}$ is a sequence of Reeb chords appearing as the east
boundary of a holomorphic curve, then by Condition (M-9) we know for any
$\sigma\in\overrightarrow{\rho}$, $|\sigma|\leq 4$.

With these terminologies set, our proposition is the following.

######  Proposition 2.42.

A holomorphic curve
$u\in\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright},\overrightarrow{P})$ is
embedded if and only if
$\chi(S)=g+e(B)-n_{\bm{x}}(B)-n_{\bm{y}}(B)-\iota([\overrightarrow{P}])+s(\partial_{\alpha_{im}}B)$.

Here, $S^{\triangleright}$ can either be a 0-P source or a 1-P source. If
$S^{\triangleright}$ is of 0-P, then $[\overrightarrow{P}]$ stands for the
sequence of Reeb chords obtained from the labels of the east punctures, and
the term $\iota([\overrightarrow{P}])$ is defined in Formula (5.65) of [LOT18]
(in the non-extended case) and Section 4.1 of [LOT21] (for Reeb chords of
length $4$). In particular, if $\overrightarrow{P}$ contains a Reeb chord of
length $4$, then it consists of a single Reeb chord by (M-9), and hence the
only new case we need to know is $\iota((\sigma))=-1$ when $|\sigma|=4$. If
$S^{\triangleright}$ is of 1-P, then by Condition (M-9)
$\overrightarrow{P}=\emptyset$ and the term $\iota([\overrightarrow{P}])$
vanishes.

###### Proof of Proposition 2.42.

The proof is adapted from the corresponding proof in the embedded alpha curve
case, [LOT18, Proposition 5.69]. To keep it concise, we state the main steps
and skip the details that can be found in [LOT18], but we give details on the
modifications. The proof is divided into four steps; only the third step is
signicantly different from the embedded case.

Let $u\in\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright};\overrightarrow{P})$
be an embedded curve.

Step 1. Apply the Riemann-Hurwitz formula to express $\chi(S)$ in terms of
$e(B)$ and $br(u)$, where $br(u)$ stands for the total number of branch points
of the projection $\pi_{\Sigma}\circ u$, counted with multiplicities:

(2.1)
$\chi(S)=e(S)+\frac{g}{2}+\sum_{i}\frac{|P_{i}|}{2}=e(B)-br(u)+\frac{g}{2}+\sum_{i}\frac{|P_{i}|}{2}.$

Step 2. Let $\tau_{\epsilon}(u)$ be the translation of $u$ by $\epsilon$ in
the $\mathbb{R}$-direction. Since we are in the torus-boundary case, we may
always assume $\overrightarrow{P}$ is discrete. There is only one case where
there is a branch point escaped to east infinity: This occurs when
$\overrightarrow{P}=\\{\sigma\\}$ for some $|\sigma|=4$. Therefore,

(2.2) $br(u)=\tau_{\epsilon}(u)\cdot
u-\frac{|\\{\sigma|{\sigma}\in\overrightarrow{P},\ |\sigma|=4\\}|}{2}$

when $\epsilon$ is sufficiently small. This is because when $u$ is embedded
and $\epsilon$ is small, excluding those intersection points which escape to
east infinity when $\epsilon\rightarrow 0$, the remaining intersection points
correspond to points at which $u$ is tangent to $\frac{\partial}{\partial t}$,
which are exactly the branch points of $\pi_{\Sigma}\circ u$.

Step 3. Compute $\tau_{\epsilon}(u)\cdot u$. For all sufficiently large $R$,

(2.3) $\tau_{R}(u)\cdot u=n_{x}(B)+n_{y}(B)-\frac{g}{2}.$

To see this, note that the intersection of $u$ and $\tau_{R}(u)$ looks like
the intersection of $u$ with the trivial disk from the generator $\bm{x}$ to
itself when $t$ is small and it looks like the intersection of $u$ with the
trivial disk corresponding to the generator $\bm{y}$ when $t$ is large (see
Proposition 4.2 of [Lip06] for the computation).

To recover $\tau_{\epsilon}(u)\cdot u$, we need to understand how
$\tau_{t}(u)\cdot u$ changes as $t$ varies. In the closed-manifold and
embedded-alpha-curve case, $\tau_{t}(u)\cdot u$ does not change. In the
bordered-manifold case, there is a change in $\tau_{\epsilon}(u)\cdot
u-\tau_{R}(u)\cdot u$ caused by the relative position change of $\partial u$
and $\partial\tau_{t}(u)$ at east infinity. This change is captured by the
term $\iota([\overrightarrow{P}])$. More precisely, when the $\alpha$-curves
are embedded,

$\tau_{\epsilon}(u)\cdot u-\tau_{R}(u)\cdot
u=\iota([\overrightarrow{P}])+\frac{|\overrightarrow{P}|}{2}+\frac{|\\{\sigma|{\sigma}\in\overrightarrow{P},\
|\sigma|=4\\}|}{2}.$

When the $\alpha$-curves are immersed, there is another source of change in
$\tau_{t}(u)\cdot u$ corresponding to self-intersection points of
$\partial_{\alpha_{im}}u$. We explain such change in the examples below. As
one shall see, such phenomenon is local (i.e., depends only on the behavior of
$u$ near the pre-images of self-intersection points of
$\partial_{\alpha_{im}}u$). Therefore, the examples also speak for the general
situation.

Figure 4. Examples of embedded disks in $\Sigma\times[0,1]\times\mathbb{R}$.
The $\alpha$-curve is drawn in red, and the $\beta$-curves are in green. We
also depicted a few $s$-grid lines; points on a single $s$-grid line have
constant $s$-coordinate.

We spell out the example. Consider an embedded disk $u$ in
$\Sigma\times[0,1]\times\mathbb{R}$ as shown in Figure 4 (a). In the figure,
$\pi_{\Sigma}\circ u$ is shown by its domain, and we also introduced immersed
$s$-grid lines to help visualize the $[0,1]\times\mathbb{R}$-coordinates: the
points on a single immersed $s$-grid line are all mapped to the same value in
$[0,1]$, and as we move from $x$ to $y$, the $t$-coordinate on an $s$-grid
line increases from $-\infty$ to $\infty$. In the figure, we highlighted a few
$s$-grid lines and self-intersection points on them. Despite having these
self-intersection points on the $s$-grid lines, $u$ is still embedded as each
such self-intersection point corresponds to two points in
$\Sigma\times[0,1]\times\mathbb{R}$ with different $t$-coordinate. However,
each self-intersection point gives rise to an intersection point in
$\tau_{t}u\cdot u$ for an appropriate $t$. In our example, the projection to
$\Sigma$ of $\partial_{\alpha}u$ has two self-intersection points $p$ and $q$.
There are $t_{i}\in\mathbb{R}$ ($i=1,2,3,4$), $t_{1}<t_{2}<t_{3}<t_{4}$ so
that $p$ has $t$-coordinate $t_{2}$ and $t_{3}$, and $q$ has coordinate
$t_{1}$ and $t_{4}$. We first examine what is incurred by the negative
intersection point $p$. Note $\tau_{t}u\cdot u$ does not change for
$t<t_{3}-t_{2}$, and at $t=t_{3}-t_{2}$, $\tau_{t}u\cap u$ picks up a boundary
intersection point $(p,1,t_{3})$. Inspecting the example, we can see for
$t\in[t_{3}-t_{2}-\epsilon,t_{3}-t_{2}+\epsilon]$ for a small $\epsilon$, a
boundary intersection point appears and then enters to become an interior
intersection point. So $\tau_{t_{3}-t_{2}+\epsilon}u\cdot
u-\tau_{t_{3}-t_{2}-\epsilon}u\cdot u=1$. Similarly, for the positive self-
intersection point $q$, for $t\in[t_{4}-t_{1}-\epsilon,t_{4}-t_{1}]$, we see
an interior intersection point of $\tau_{t}u\cdot u$ hits the boundary and
then disappears.

In general, an intersection point $p$ of $\partial_{\alpha_{im}}u$ contributes
a boundary intersection of $u$ with $\tau_{t}u$ for some value of $t=t_{p}$,
and intersection points of arcs for $s$-values just less than one contribute
intersections of $u$ with $\tau_{t}u$ for nearby values of $t$. These
intersections occur at shifts $t>t_{p}$ if $p$ is a positive intersection
point and at $t<t_{p}$ if $p$ is a negative intersection point, so positive
intersection points always give rise to times $t$ at which an intersection
point appears on the boundary and then moves into the interior, while negative
intersection points give rise to times $t$ at which an interior intersection
moves to the boundary and disappears. Thus the net change in $\tau_{t}u\cdot
u$ caused by negative and positive boundary intersection points is given by
$s(\partial_{\alpha_{im}}u)$. In the example above from Figure 4(a),
$s(\partial_{\alpha_{im}}u)=0$. An example with $s(\partial_{\alpha_{im}}u)=1$
is shown in Figure 4 (b).

Overall, taking into account the changes at the east infinity and at
$\alpha_{im}\times\\{1\\}\times\mathbb{R}$, we can derive the following
equation from Equation (2.3):

(2.4) $\tau_{\epsilon}(u)\cdot
u=n_{x}(B)+n_{y}(B)-\frac{g}{2}+\iota([\overrightarrow{P}])+\frac{|\overrightarrow{P}|}{2}+\frac{|\\{\sigma|{\sigma}\in\overrightarrow{P},\
|\sigma|=4\\}|}{2}+s(\partial_{\alpha_{im}}B)$

Step 4. Synthesize the steps. Synthesizing Equation (2.1), Equation (2.2), and
Equation (2.4) gives the formula, proving the “only if” direction. To see the
“if” direction, note if the holomorphic curve is not embedded, then in Step 2
we know $br(u)=\tau_{\epsilon}u\cdot u-2\text{sigu}(u)$, where
$\text{sigu}(u)>0$ is the order of singularity $u$. So, in this case we have
$\chi(S)$ is strictly greater than
$g+e(B)-n_{x}(B)-n_{y}(B)-\iota([\overrightarrow{P}])+s(\partial_{\alpha_{im}}B).$
∎

From now on, we use $\overrightarrow{\rho}$ to denote either a sequence of
Reeb chords of length less than or equal to $4$ or the length one sequence
containing a single closed Reeb orbit $U$ which is used for marking the
interior puncture of a 1-P curve. Given such a $\overrightarrow{\rho}$, denote
by $\overrightarrow{{\rho}_{\star}}$ the sub-sequence of non-closed Reeb
chords, i.e., if $\overrightarrow{\rho}=(U)$ then
$\overrightarrow{{\rho}_{\star}}=\emptyset$ and otherwise
$\overrightarrow{{\rho}_{\star}}=\overrightarrow{\rho}$. A domain
$B\in\tilde{\pi}_{2}(\bm{x},\bm{y})$ is said to be _compatible_ with a
sequence of Reeb chords $\overrightarrow{\rho}$ if the _homology class_
induced by the east boundary $\partial^{\partial}B$ of $B$ agrees with that
induced by $\overrightarrow{\rho}$, and
$(\bm{x},\overrightarrow{{\rho}_{\star}})$ is strongly boundary monotone (in
the sense of Definition 5.52 of [LOT18]). In view of Proposition 2.42, we make
the following definitions.

######  Definition 2.43.

Given $B\in\tilde{\pi}_{2}(\bm{x},\bm{y})$ and a sequence of Reeb chords
$\overrightarrow{\rho}$ such that $(B,\overrightarrow{\rho})$ is compatible.
The _embedded Euler characteristic_ is defined to be

$\chi_{emb}(B,\overrightarrow{\rho}):=g+e(B)-n_{x}(B)-n_{y}(B)-\iota(\overrightarrow{{\rho}_{\star}})+s(\partial_{\alpha_{im}}B).$

The _embedded index_ is defined to be

$\text{ind}(B,\overrightarrow{\rho}):=e(B)+n_{x}(B)+n_{y}(B)+|\overrightarrow{\rho}|+\iota(\overrightarrow{{\rho}_{\star}})-s(\partial_{\alpha_{im}}B).$

The _embedded moduli space_ is defined to be

$\mathcal{M}^{B}(\bm{x},\bm{y};\overrightarrow{\rho}):=\bigcup_{\chi(S)=\chi_{emb}(B,\overrightarrow{\rho}),\
[\overrightarrow{P}]=\overrightarrow{{\rho}_{\star}}}\mathcal{M}^{B}(\bm{x},\bm{y};S^{\triangleright};\overrightarrow{P}).$

Clearly, the embedded moduli space
$\mathcal{M}^{B}(\bm{x},\bm{y};\overrightarrow{\rho})$ has expected dimension
$\text{ind}(B,\rho)-1$ by Proposition 2.42 and Proposition 2.17.

######  Proposition 2.44.

The embedded index formula is additive, i.e., for compatible pairs
$(B_{i},\overrightarrow{\rho_{i}})$ where $B_{1}\in\pi_{2}(\bm{x},\bm{w})$ and
$B_{2}\in\pi_{2}(\bm{w},\bm{y})$ we have
$\text{ind}(B_{1}+B_{2},(\overrightarrow{\rho_{1}},\overrightarrow{\rho_{2}}))=\text{ind}(B_{1},\overrightarrow{\rho_{1}})+\text{ind}(B_{2},\overrightarrow{\rho_{2}}).$

###### Proof.

The proof is a modification of the proof of Proposition 5.75 in [LOT18], which
is adapted from the proof in the closed-manifold case in [Sar11]. We will only
mention the modifications instead of going into the details. Given oriented
arcs $a$ and $b$, the _jittered intersection number_ $a\cdot b$ is defined to
be $\frac{1}{4}(a_{NE}+a_{NW}+a_{SE}+a_{SW})\cdot b$, where
$a_{NE},\ldots,a_{SW}$ are slight translations of the arc $a$ in directions
suggested by the subscript (and we assume these translations intersect $b$
transversely). The proof of Proposition 5.75 of [LOT18] uses that if $a$ and
$a^{\prime}$ are two arcs contained in the $\alpha$-curve, then $a\cdot
a^{\prime}=0$ (Lemma 5.73 of [LOT18] (2)). This is no longer true in our
setting. Overall555For careful readers: Specifically, Lemma 5.73 (5) of
[LOT18] needs to be changed to $a\cdot a^{\prime}+a\cdot b^{\prime}+b\cdot
a^{\prime}=0$ and Lemma 5.74 of [LOT18] needs to be changed to $a\cdot
a^{\prime}+a\cdot b^{\prime}+b\cdot
a^{\prime}=L(\partial^{\partial}B,\partial^{\partial}B^{\prime})$ when $a\cdot
a^{\prime}$ can no longer be assumed to be zero., running the proof in [LOT18]
in our setting now gives

(2.5) $\displaystyle
e(B_{1}+B_{2})+n_{\bm{x}}(B_{1}+B_{2})+n_{\bm{y}}(B_{1}+B_{2})+|(\overrightarrow{\rho_{1}},\overrightarrow{\rho_{2}})|+\iota((\overrightarrow{\rho_{1}},\overrightarrow{\rho_{2}})_{b})$
$\displaystyle=$ $\displaystyle
e(B_{1})+n_{\bm{x}}(B_{1})+n_{\bm{y}}(B_{1})+|\overrightarrow{\rho_{1}}|+\iota((\overrightarrow{\rho_{1}})_{b})+e(B_{2})+n_{\bm{x}}(B_{2})$
$\displaystyle+n_{\bm{y}}(B_{2})+|\overrightarrow{\rho_{2}}|+\iota((\overrightarrow{\rho_{2}})_{b})+\partial_{\alpha_{im}}B_{1}\cdot\partial_{\alpha_{im}}B_{2}.$

(If the $\alpha$-curves are embedded, then the term
$\partial_{\alpha_{im}}(B_{1})\cdot\partial_{\alpha_{im}}B_{2}$ vanishes,
recovering the additivity of the index in that case.) And it is easy to see

(2.6)
$s(\partial_{\alpha_{im}}(B_{1}+B_{2}))=s(\partial_{\alpha_{im}}B_{1})+s(\partial_{\alpha_{im}}B_{2})+\partial_{\alpha_{im}}B_{1}\cdot\partial_{\alpha_{im}}B_{2}.$

Now the additivity of the index follows readily from the definition of the
embedded index, Equation (2.5), and Equation (2.6). ∎

When proving the structure equations (e.g., $\partial^{2}=0$ for type D
structures), one needs to relate the coefficient of the structural equation
and the ends of 1-dimensional moduli spaces. A key result that allows us to do
this is the following proposition, which is the counterpart of Lemma 5.76 of
[LOT18].

######  Proposition 2.45.

Given $(B,\overrightarrow{\rho})$ such that
$\text{ind}(B,\overrightarrow{\rho})=2$. Then the two-story buildings that
occur in the degeneration of
$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};\overrightarrow{\rho})$ are
embedded.

###### Proof.

Given a two-story building $(u_{1},u_{2})$, by transversality we know
$\text{ind}(u_{i})=1$, $i=1,2$. Let $(B_{i},\overrightarrow{\rho_{i}})$ be the
corresponding pair of domain and Reeb chords of $u_{i}$. In view of
Proposition 2.42,
$\text{ind}(u_{i})\leq\text{ind}(B_{i},\overrightarrow{\rho_{i}})$, and the
equality is achieved if and only if $u_{i}$ is embedded. Now by Proposition
2.44, if some $u_{i}$ is not embedded, we will have
$\text{ind}(B,\overrightarrow{\rho})=\text{ind}(B_{1},\overrightarrow{\rho_{1}})+\text{ind}(B_{2},\overrightarrow{\rho_{2}})>2,$
which contradicts our assumption. ∎

###  2.7. Ends of moduli spaces of 0-P curves

We analyze boundary degenerations of moduli spaces of 0-P curves in this
subsection, which was left untouched in Section 2.5. We will separate the
discussion into two cases. The first case is when $n_{z}(B)=0$, needed for
defining the hat-version type D structure. The second case is when
$n_{z}(B)=1$, needed for defining extended type D structures. Readers who wish
to skip the details are referred to Proposition 2.47, Proposition 2.48, and
Proposition 2.49 for the statement of the results.

####  2.7.1. Ends of $\mathcal{M}^{B}$ when $n_{z}(B)=0$

######  Proposition 2.46.

If $n_{z}(B)=0$, then
$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};\overrightarrow{\rho})$ does not
have boundary degeneration.

###### Proof.

Suppose boundary degeneration appears. Let $v$ denote the (union of the)
component(s) of the nodal holomorphic curve with no $\pm$-punctures. The union
of components that have $\pm$-punctures will be called the main component. We
say that the boundary degeneration has corners if at least one of the nodes
connecting $v$ and the main component is of type II. Otherwise, a boundary
degeneration is said to have no corners. (See Definition 2.24 for type II
nodes.)

Let $B_{v}$ denote the domain corresponding to $v$. If the boundary
degeneration has no corners, then $B_{v}$ is a positive zero-cornered
$\alpha$-bounded domain. Such $B_{v}$ does not exist as $\mathcal{H}$ is
unobstructed. This observation left us with the possibility of boundary
degeneration with corners. If such degeneration appears, then each type II
node connecting the main component and the degenerated components is obtained
from pinching a properly embedded arc on the original source surface $S$,
whose endpoints are on the component of $\partial S$ that is mapped to
$\alpha_{im}\times[0,1]\times\mathbb{R}$. Therefore, $B_{v}$ is a (union of)
positive one-cornered $\alpha$-bounded domain with $n_{z}=0$. Such domains do
not exist either since $\mathcal{H}$ is unobstructed. Therefore, there is no
boundary degeneration. ∎

######  Proposition 2.47.

For a generic almost complex structure $J$. Let
$\mathcal{M}^{B}(\bm{x},\bm{y};\overrightarrow{\rho})$ be a one-dimensional
moduli space with $n_{z}(B)=0$ and $a(-\overrightarrow{\rho})\neq 0$. Then
$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};\overrightarrow{\rho})$ is a compact
1-manifold with boundary such that all the boundary points correspond to two-
story embedded holomorphic buildings.

###### Proof.

It follows from Proposition 2.40 and Proposition 2.46 that the only elements
that can appear in
$\partial\overline{\mathcal{M}}_{0}^{B}(\bm{x},\bm{y};\overrightarrow{P})$ are
split curves or two-story buildings. However, if split curves appear in the
torus-boundary case and $n_{z}(B)=0$, a simple examination of the torus
algebra implies we will always have $a(-\overrightarrow{\rho})=0$. Therefore,
the only degenerations that can appear are two-story buildings. The statement
that $\overline{\mathcal{M}}_{0}^{B}(\bm{x},\bm{y};\overrightarrow{P})$ is a
1-manifold with boundary then follows from the gluing results. ∎

####  2.7.2. Ends of $\mathcal{M}^{B}$ when $n_{z}(B)=1$

We separate the discussion into two sub-cases. First, we consider the sub-case
where $n_{-\rho_{i}}(B)=0$ for at least one of
$\rho_{i}\in\\{\rho_{1},\rho_{2},\rho_{3}\\}$, in which we have the following
proposition.

######  Proposition 2.48.

If $n_{z}(B)=1$ and $n_{-\rho_{i}}(B)=0$ for at least one
$\rho_{i}\in\\{\rho_{1},\rho_{2},\rho_{3}\\}$, then
$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};\overrightarrow{\rho})$ does not
have boundary degeneration. In particular, if $a(-\overrightarrow{\rho})\neq
0$, then $\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};\overrightarrow{\rho})$ is
a compact 1-manifold with boundary such that all the boundary points
correspond to two-story embedded holomorphic buildings.

###### Proof.

The proof for the first part of this is similar to that of Proposition 2.46,
which reduces to excluding the existence of positive zero- and one-cornered
$\alpha$-bounded domains $B_{v}$. The existence of such a $B_{v}$ with
$n_{z}(B)=0$ is excluded by unobstructedness; see Condition (1) of Definition
2.8. The existence of such $B_{v}$ with $n_{z}(B)=1$ is excluded by Condition
(2) and (3) of Definition 2.8: If it exists, then the multiplicity of $B_{v}$
at all the Reeb chords are equal to one, which contradicts our assumption that
$n_{-\rho_{i}}(B)=0$ for some $\rho_{i}$. The proof for the second part is
identical to that of Proposition 2.47. ∎

Next, we consider the case where the multiplicity of $B$ at all of the four
regions next to the east puncture of $\Sigma$ is equal to one. Let $q$ be a
self-intersection point of $\alpha_{im}$. We use $T(q)$ to denote the set of
stabilized teardrops with an acute corner at $q$. We will use
$\mathcal{M}^{B}(\bm{x},\bm{y};q)$ to denote the moduli space of embedded
holomorphic curves whose $\alpha$-boundary projection to $\Sigma$ is allowed
to take a single sharp turn at $q$; see also Definition 2.53 below.

######  Proposition 2.49.

Given a compatible pair $(B,\overrightarrow{\mathbb{\rho}})$ with
$B\in\pi_{2}(\bm{x},\bm{y})$,
$\overrightarrow{\rho}\in\\{(-\rho_{0},-\rho_{1},-\rho_{2},-\rho_{3}),(-\rho_{0},-\rho_{123}),(-\rho_{012},-\rho_{3})\\}$,
and $\text{Ind}(B,\overrightarrow{\mathbb{\rho}})=2$, the compactified moduli
space $\mathcal{M}^{B}(\bm{x},\bm{y};\overrightarrow{\mathbb{\rho}})$ is a
compact 1-manifold with boundary. The ends are of the following four types:

  * (E-1)

Two-story ends;

  * (E-2)

split curve ends;

  * (E-3)

ends corresponding to boundary degeneration with corners;

  * (E-4)

ends corresponding to boundary degeneration without corners.

Moreover,

  * (a)

If split curve ends occur, then $\overrightarrow{\mathbb{\rho}}$ is either
$(-\rho_{0},-\rho_{123})$ or $(-\rho_{012},-\rho_{3})$, and the number of such
ends is $\\#\mathcal{M}^{B}(\bm{x},\bm{y};-\rho_{1230})$ if
$\overrightarrow{\mathbb{\rho}}=(-\rho_{0},-\rho_{123})$; otherwise the number
of ends is equal to $\\#\mathcal{M}^{B}(\bm{x},\bm{y};-\rho_{3012})$;

  * (b)

If ends corresponding to boundary degeneration with corners occur and
$\overrightarrow{\rho}=(-\rho_{0},-\rho_{1},-\rho_{2},-\rho_{3})$, then the
number of such ends is mod 2 congruent to

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

For the other two choices of $\overrightarrow{\rho}$, the numbers of ends
corresponding to boundary degeneration with corners are both even;

  * (c)

If (E-4) occurs, then $[B]=[\Sigma]$, $\bm{x}=\bm{y}$, and the number of such
ends is odd.

######  Remark 2.50.

Proposition 2.49 considers $\overrightarrow{\mathbb{\rho}}$ with
$a(-\overrightarrow{\mathbb{\rho}})=\rho_{0123}$. Corresponding propositions
hold for
$a(-\overrightarrow{\mathbb{\rho}})\in\\{\rho_{1230},\rho_{2301},\rho_{3012}\\}$
by cyclically permuting the subscripts in the above statement.

The remainder of the subsection is devoted to proving Proposition 2.49.

####  2.7.3. Reformulation of the moduli spaces

To apply gluing results needed for studying the ends corresponding to boundary
degeneration, we need to know the moduli spaces of the degenerate components
are transversely cut out. However, this transversality result is not clear as
a key lemma needed for the proof, Lemma 3.3 of [Lip06], is not available for
degenerate curves. In [LOT18], this difficulty is overcome by using the
formulation of moduli spaces in terms of holomorphic disks in the symmetric
product $Sym^{g}(\Sigma)$. Such moduli spaces are identified with the moduli
spaces of holomorphic curves in $\Sigma\times[0,1]\times\mathbb{R}$ through a
tautological correspondence (provided one uses appropriate almost complex
structures). One can prove the desired transversality results of degenerate
discs in the symmetric product in a standard way. We shall employ the same
strategy here. The difference is that the Lagrangians holding the boundary of
holomorphic disks are no longer embedded. To cater for this change, the
definition of moduli spaces we give below corresponds to the one used in Floer
theory for immersed Lagrangians with clean self-intersections [AB18, Fuk17].

Specifically, equip $\Sigma$ with a Kahler structure $(j,\eta)$, where $j$
denotes a complex structure and $\eta$ is a compatible symplectic form. Let
$\Sigma_{\bar{e}}$ denote the closed Riemann surface obtained from $\Sigma$ by
filling in the east puncture $e$. Then $Sym^{g}(\Sigma)$ can be viewed as the
complement of $\\{e\\}\times Sym^{g-1}(\Sigma_{\bar{e}})$ in
$Sym^{g}(\Sigma_{\bar{e}})$. It is a symplectic manifold with a cylindrical
end modeled on the unit normal bundle of $\\{e\\}\times
Sym^{g-1}(\Sigma_{\bar{e}})$. In particular, there is a Reeb-like vector field
$\overrightarrow{R}$ tangent to the $S^{1}$-fibers of the unit normal bundle.

The products $\mathbb{T}_{\beta}=\beta_{1}\times\cdots\times\beta_{g}$ and
$\mathbb{T}_{\alpha,i}=\alpha^{a}_{i}\times\alpha^{c}_{1}\times\cdots\times\alpha^{c}_{g-1}$,
$i=1,2$, are Lagrangian submanifolds of $Sym^{g}(\Sigma)$. Note
$\mathbb{T}_{\alpha,i}$ is immersed with self-intersections
$\alpha^{a}_{i}\times\alpha_{1}^{c}\times\ldots\times\alpha^{c}_{g-2}\times
q$, where $q$ is some self-intersection point of
$\alpha_{im}=\alpha_{g-1}^{c}$. We identify $\mathbb{T}_{\alpha,i}$ with the
image of a map $\iota_{i}:(0,1)\times\mathbb{T}^{g-2}\times(\amalg
S^{1})\rightarrow Sym^{g}(\Sigma)$.

The immersed Lagrangian $\mathbb{T}_{\alpha,i}$ ($i=1,2$) intersects the ideal
boundary of $Sym^{2}(\Sigma)$ at
$\partial\overline{\alpha^{a}_{i}}\times\alpha_{1}^{c}\times\cdots\times\alpha_{g-1}^{c}$.
Each Reeb chord $\rho$, which connects two (possibly the same) alpha arcs, now
corresponds to a one-dimensional family of $\overrightarrow{R}$-chords
$\rho\times\bm{x}$ that connects two (possibly the same)
$\mathbb{T}_{\alpha,i}$, parametrized by
$\bm{x}\in\alpha_{1}^{c}\times\cdots\times\alpha_{g-1}^{c}$.

To define pseudo-holomorphic maps, we shall work with an appropriate class of
almost complex structures called _nearly-symmetric almost complex structures_
that restrict to $Sym^{g}(j)$ on the cylindrical end. (The concrete
definitions do not matter for our purpose, and we refer the interested readers
to Definition 3.1 of [OS04b] and Definition 13.1 of [Lip06]).

In this subsection, we only give definitions to the moduli spaces relevant to
the case of 0-P curves; the 1-P counterparts are postponed to the next
subsection.

######  Definition 2.51.

Let $J_{s}$, $s\in[0,1]$, be a path of nearly-symmetric almost complex
structures. Let
$\bm{x},\bm{y}\in\amalg\mathbb{T}_{\alpha,i}\cap\mathbb{T}_{\beta}$, let
$\overrightarrow{\rho}=(\sigma_{1},\ldots,\sigma_{n})$ be a sequence of Reeb
chords, and let $B\in\tilde{\pi}_{2}(\bm{x},\bm{y})$. We define
$\widetilde{\mathcal{M}}_{Sym,J_{s}}^{B}(\bm{x},\bm{y};\overrightarrow{\rho})$
as the set of maps

$u:([0,1]\times\mathbb{R}\backslash\\{(1,t_{1}),\ldots,(1,t_{n})\\})\rightarrow
Sym^{g}(\Sigma)$

such that

  * (1)

$t_{1}<\ldots<t_{n}$ and are allowed to vary;

  * (2)

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

  * (3)

$u(\\{1\\}\times(\mathbb{R}\backslash\\{t_{1},\ldots,t_{n}\\}))\subset\mathbb{T}_{\alpha,1}\cup\mathbb{T}_{\alpha,2}$.
Moreover, the restriction of $u$ to any connected components of
$\\{1\\}\times(\mathbb{R}\backslash\\{t_{1},\ldots,t_{n}\\})$ lifts through
$\iota_{i}:(0,1)\times\mathbb{T}^{g-2}\times(\amalg S^{1})\rightarrow
Sym^{g}(\Sigma)$ for an appropriate $i\in\\{1,2\\}$;

  * (4)

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

  * (5)

$\lim_{(s,t)\rightarrow(1,t_{i})}u(s+it)$ is an $\overrightarrow{R}$-chord
$\sigma_{i}\times\bm{a}$ for some
$\bm{a}\in\alpha_{1}^{c}\times\cdots\times\alpha_{g-1}^{c}$;

  * (6)

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

  * (7)

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

######  Remark 2.52.

The only difference between our setting and the setting of embedded
Lagrangians is the lifting property stated in (3). This condition ensures that
$\partial u$ does not have corners at self-intersection points of
$\alpha_{im}$.

The tautological correspondence between the two moduli spaces defined using
two different ambient symplectic manifolds holds in our setting as well.
Roughly, holomorphic disks $u$ in the symmetric product are in one-to-one
correspondence with pairs $(v,\pi)$, where $v$ is a stay-on-track 0-P
holomorphic curves $S\rightarrow\Sigma$, and
$\pi:S_{\bar{e}}\rightarrow[0,1]\times\mathbb{R}$ is a $g$-fold branched cover
where the filled-in punctures are mapped to $(1,t_{i})$, $i=1,\ldots,n$. The
tautological correspondence was first proved in Section 13 of [Lip06]. The
proof for our case follows the same line and is omitted. From now on, we will
simply denote
$\widetilde{\mathcal{M}}_{Sym,J_{s}}^{B}(\bm{x},\bm{y};\overrightarrow{\rho})$
by $\widetilde{\mathcal{M}}_{J_{s}}^{B}(\bm{x},\bm{y};\overrightarrow{\rho})$.
The reduced moduli space
${\mathcal{M}}_{J_{s}}^{B}(\bm{x},\bm{y};\overrightarrow{\rho})$ is the
quotient of
$\widetilde{\mathcal{M}}_{J_{s}}^{B}(\bm{x},\bm{y};\overrightarrow{\rho})$ by
the $\mathbb{R}$-action given by vertical translation.

The moduli spaces $\mathcal{M}^{B}(\bm{x},\bm{y};q)$ where $q$ is some self-
intersection point of $\alpha_{im}$ can be similarly defined in the symmetric-
product setup (and the tautological correspondence to curves in
$\Sigma\times[0,1]\times\mathbb{R}$ holds).

######  Definition 2.53.

$\mathcal{M}_{J_{s}}^{B}(\bm{x},\bm{y};q)$ is the space of $J_{s}$-holomorphic
maps

$u:([0,1]\times\mathbb{R}\backslash\\{(1,0)\\})\rightarrow Sym^{g}(\Sigma)$

satisfying conditions (2)(3)(4)(6)(7) of Definition 2.51 and
$\lim_{(s,t)\rightarrow(1,0)}u(s+it)=(q,\bm{a})$ for some
$\bm{a}\in\alpha^{a}_{i}\times\alpha_{1}^{c}\times\cdots\times\alpha_{g-2}^{c}$
for an appropriate $i\in\\{1,2\\}$.

Note there is a natural evaluation map
$ev_{J_{s}}:\mathcal{M}^{B}_{J_{s}}(\bm{x},\bm{y};q)\rightarrow\alpha^{a}_{i}\times\alpha_{1}^{c}\times\cdots\times\alpha_{g-2}^{c}$
for an appropriate $i\in\\{1,2\\}$, given by $u\mapsto\bm{a}$ if
$\lim_{(s,t)\rightarrow(1,0)}u(s+it)=(q,\bm{a})$.

We call a holomorphic disc _degenerate_ if its boundaries are in
$\mathbb{T}_{\alpha,i}$ ($i=1,2$). A degenerate holomorphic disc may be viewed
as a map from the upper-half plane $\mathbb{H}$ with boundary punctures to the
symmetric product. We further divide such discs into degenerate discs with or
without corners based on the behavior of asymptotics at the point at infinity,
corresponding to Type I and Type II nodes in Definition 2.24. We spell out the
definitions for completeness.

######  Definition 2.54 (Degenerate disks without corners).

Let $J$ be a nearly-symmetric almost complex structure. Let
$\bm{x}\in\mathbb{T}_{\alpha}$ and
$\overrightarrow{\rho}=(\sigma_{1},\ldots,\sigma_{n})$.
$\mathcal{N}_{J}(\bm{x};\overrightarrow{\rho})$ is the set of maps
$v:\mathbb{H}\backslash\\{t_{1},\ldots,t_{n}\\}\rightarrow Sym^{g}(\Sigma)$
such that

  * (1)

$0=t_{1}<\ldots<t_{n}$ and are allowed to vary;

  * (2)

$v(\mathbb{R}\backslash\\{t_{1},\ldots,t_{n}\\}))\subset\mathbb{T}_{\alpha,1}\cup\mathbb{T}_{\alpha,2}$.
Moreover, the restriction of $v$ to any connected components of
$\mathbb{R}\backslash\\{t_{1},\ldots,t_{n}\\}$ lifts through
$\iota_{i}:(0,1)\times\mathbb{T}^{g-2}\times(\amalg S^{1})\rightarrow
Sym^{g}(\Sigma)$ for an appropriate $i\in\\{1,2\\}$;

  * (3)

$\lim_{z\rightarrow\infty}v(z)=\bm{x}$, and the path obtained from
$v|_{(-\infty,t_{1})\cup(t_{n},\infty)}$ by continuous extension at $\infty$
lifts through $\iota_{i}$ for an appropriate $i\in\\{1,2\\}$;

  * (4)

$\lim_{z\rightarrow t_{i}}v(z)$ is an $\overrightarrow{R}$-chords
$\sigma_{i}\times a$ for some
$a\in\alpha_{1}^{c}\times\cdots\times\alpha_{g-1}^{c}$;

  * (5)

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

######  Definition 2.55 (Degenerate disks with corners).

Let $J$ be a nearly-symmetric almost complex structure. Let $q$ be a self-
intersection point of $\alpha_{im}$. Let
$\overrightarrow{\rho}=(\sigma_{1},\ldots,\sigma_{n})$.
$\mathcal{N}_{J}(q;\overrightarrow{\rho})$ is the set of maps
$v:\mathbb{H}\backslash\\{t_{1},\ldots,t_{n}\\}\rightarrow Sym^{g}(\Sigma)$
such that

  * (1)

$0=t_{1}<\ldots<t_{n}$ and are allowed to vary;

  * (2)

$v(\mathbb{R}\backslash\\{t_{1},\ldots,t_{n}\\}))\subset\mathbb{T}_{\alpha,1}\cup\mathbb{T}_{\alpha,2}$.
Moreover, the restriction of $v$ to any connected components of
$\mathbb{R}\backslash\\{t_{1},\ldots,t_{n}\\}$ lifts through
$\iota_{i}:(0,1)\times\mathbb{T}^{g-2}\times(\amalg S^{1})\rightarrow
Sym^{g}(\Sigma)$ for an appropriate $i\in\\{1,2\\}$;

  * (3)

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

  * (4)

$\lim_{z\rightarrow t_{i}}v(z)$ is an $\overrightarrow{R}$-chords
$\sigma_{i}\times a$ for some
$a\in\alpha_{1}^{c}\times\cdots\times\alpha_{g-1}^{c}$;

  * (5)

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

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

####  2.7.4. Boundary degeneration with corners

######  Definition 2.56.

A _simple boundary degeneration_ is a boundary degeneration of the form $u\vee
v$, where $u$ is a (non-nodal) holomorphic curve, and $v$ is a degenerate
disk.

######  Proposition 2.57.

If a boundary degeneration with corners appears in a one-dimensional moduli
space $\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};\overrightarrow{\rho})$ where
$(B,\overrightarrow{\rho})$ is as in Proposition 2.49, the boundary
degeneration must be a simple boundary degeneration. Moreover, the domain for
the degenerate disk must be a stabilized teardrop with an acute corner.

###### Proof.

First, we consider the case where we assume degeneration at east infinity,
multi-story splitting, and boundary degeneration without corners do not occur
simultaneously with the boundary degeneration with corners. Also, note that
sphere bubbles do not occur as $\Sigma$ is punctured at the east infinity.
Hence we may assume the boundary degeneration with corners is a holomorphic
map $u_{\infty}:\mathbb{B}\rightarrow Sym^{g}(\Sigma)$, where $\mathbb{B}$ is
a disc bubble tree: $\mathbb{B}$ has one main component containing the
$\pm$-puncture and some other components attached to the main component or
each other such that the graph obtained by turning the components of
$\mathbb{B}\backslash\\{\text{nodes}\\}$ into vertexes and nodes into edges is
a tree. (See Figure 5 (a).) The vertex corresponding to the main component
will be called the root.

Figure 5. The bubble tree $\mathbb{B}$; the $\sigma_{i}^{\prime}s$ are labels
of the east punctures. Hypothetically, a bubble tree with many branches might
appear, as shown in (a). In our case, we first prove the bubble tree must be
of the form with a single branch as shown in (b), and then further show it
must be of the simple form shown in (c).

Our first claim is that $\mathbb{B}$ must only have one leaf. (See Figure 5
(b).) To see this, note a leaf corresponds to a degenerate disk whose domain
is positive, and by homological consideration the domain is bounded by a one-
cornered subloop of $\alpha_{im}$ and possibly $\partial\Sigma$. Note that at
most one leaf would have a domain with boundary containing $\partial\Sigma$,
as we assumed $a(-\overrightarrow{\rho})=\rho_{0123}$; call this the
distinguished leaf. Therefore, all the other leaves, if they exist, would have
positive one-cornered $\alpha$-bounded domains with $n_{z}=0$ (as $n_{z}(B)=1$
and the distinguished leaf already has multiplicity one at $z$); such domains
do not exist since $\mathcal{H}$ is unobstructed. Therefore, only the
distinguished leaf exists, and its domain is a stabilized teardrop since
$\mathcal{H}$ is unobstructed.

Now denote the map restricting to the main component by $u$. Let $n$ denote
the number of components of $\mathbb{B}$ except the root. Denote the
degenerate disc corresponding to the leaf by $v_{n}$, and denote those
connecting the root and the leaf by $v_{i}$, $i=1,\ldots,n-1$. We want to
prove $n=1$ and that the stabilized teardrop for the leaf has an acute corner.
They follow from an index consideration as follows.

Note that the domains $D_{i}$ corresponding to $v_{i}$, $i=1,\ldots,n-1$ are
bigons: Such domains are two-cornered $\alpha$-bounded domains with $n_{z}=0$
and we know these domains are bigons by the assumption that $\mathcal{H}$ is
unobstructed. Let $\mathcal{N}^{D_{i}}$ denote the reduced moduli space of
holomorphic curves with domain $D_{i}$. Direct computation shows the virtual
dimension $\text{vdim}(\mathcal{N}^{D_{i}})$ of the reduced moduli space
satisfies $\text{vdim}(\mathcal{N}^{D_{i}})\geq g-1$, and the equality
attained when both corners of $D_{i}$ are acute. Here the term $g-1$ comes
from varying the constant value of the holomorphic map in
$\alpha_{i}^{a}\times\alpha_{1}^{c}\times\cdots\times\alpha^{c}_{g-2}$ for
some $i\in\\{1,2\\}$.

Now we move to consider $D_{n}$. We already know it is a stabilized teardrop.
Depending on whether the corner of $D_{n}$ is acute or obtuse, the virtual
dimension of the corresponding moduli space $\mathcal{N}^{D_{n}}$ is $g-1$ or
$g$.

Let $D$ be the domain of $u$, and let $q$ denote the corner corresponding to
the node. Then

$\displaystyle\text{vdim}({\mathcal{M}}^{B}(\bm{x},\bm{y};\overrightarrow{\rho}))=$
$\displaystyle\text{vdim}({\mathcal{M}}^{D}(\bm{x},\bm{y};q))+\sum_{i=1}^{n-1}\text{vdim}(\mathcal{N}^{D_{i}})+\text{vdim}(\mathcal{N}^{D_{n}})$
$\displaystyle-(g-1)n+n.$

Here, the term $-(g-1)n$ comes from the evaluation map, and the term $+n$
appears since we glued $n$ times. Note
$\text{vdim}({\mathcal{M}}^{B}(\bm{x},\bm{y};\overrightarrow{\rho}))=1$, and
hence we have $\text{vdim}({\mathcal{M}}^{D}(\bm{x},\bm{y};q))\leq 1-n$ since
$\text{vdim}(\mathcal{N}^{D_{i}})\geq g-1$ for $1\leq i\leq n$. Therefore, as
long as we fix a generic path of nearly-symmetric almost complex structure
$\mathcal{J}_{s}$ so that $\mathcal{M}^{D}(\bm{x},\bm{y};q)$ is transversally
cut out, it being non-empty implies $n=1$ (see Figure 5 (c)). This also forces
$\text{vdim}(\mathcal{N}^{D_{n}})=g-1$, which implies the corner of $D_{n}$ is
acute.

The above analysis shows the index of a degenerate disk with a corner is
greater than or equal to $g-1$, and hence an index consideration rules out the
possibility of several types of degeneration appearing simultaneously. ∎

######  Proposition 2.58.

Let $B\in T(q)$ be a stabilized teardrop with an acute corner at $q$, and let
$\overrightarrow{\rho}=(-\rho_{0},-\rho_{1},-\rho_{2},-\rho_{3})$. For a
generic nearly symmetric almost complex $J$, the moduli space of degenerate
disks $\mathcal{N}_{J}^{B}(q;\overrightarrow{\rho})$ is a $(g-1)$-manifold,
and a generic fiber of the evaluation map
$ev_{J}:\mathcal{N}_{J}^{B}(q;\overrightarrow{\rho})\rightarrow\alpha^{a}_{1}\times\alpha^{c}_{1}\times\cdots\alpha^{c}_{g-2}$
consists of an odd number of points.

###### Proof.

The argument for seeing the moduli space is smoothly cut out for a generic
almost complex structure is standard, and in this case, it closely follows
that of Proposition 3.14 of [OS04b].

We now study the parity of a generic fiber of the evaluation map. We first
prove the relevant statements for the case where $g(\Sigma)=2$. Note that if
we fix a point $p\in\alpha^{a}_{1}$, by standard arguments we may choose a
generic almost complex structure $J$ so that the fiber $ev_{J}^{-1}(p)$ over
$p$ is smoothly cut out as a $0$-manifold. Standard consideration of
degeneration shows $ev_{J}^{-1}(p)$ is compact: limits of such maps cannot
have further degenerate disks for index reasons, nor can a sequence of maps
converge into a sphere bubbling as the domain is not $\Sigma$. We now claim
that $|ev_{J}^{-1}(p)|$ is odd. A lemma is needed for this.

######  Lemma 2.59.

Assume $g(\Sigma)=2$. For a generic perturbation of the $\alpha$-curves,
$ev_{Sym^{2}(j)}^{-1}(p)$ is smoothly cut out, and $|ev_{Sym^{2}(j)}^{-1}(p)|$
is odd.

###### Proof of Lemma 2.59.

First note $ev_{Sym^{2}(j)}^{-1}(p)$ is smoothly cut out provided we perturb
the $\alpha$-curves if necessary, as $B$ being a stabilized teardrop
guarantees any holomorphic disks representing $[B]$ is somewhere boundary
injective; see Proposition 3.9 of [OS04b], or see Theorem I and Theorem 3.2 of
[Oh96] for more details on the relation between boundary injectivity and
regularity.

For the second part of the statement, note by the tautological correspondence
it suffices to find all the pairs $(\hat{u},\pi)$ of holomorphic maps
$\hat{u}:F\rightarrow\Sigma$ and two-fold branched covers
$\pi:F_{\bar{e}}\rightarrow D_{\bar{e}}$, where $F_{\bar{e}}$ stands for the
surface obtained from $F$ by filling in the east punctures and $D_{\bar{e}}$
is the holomorphic disk with one boundary puncture.

Figure 6. (a) The domain $B$. (b) The annulus $F$ with punctures on the
boundaries.

Examining the region $B$ shows that $\hat{u}$ is the obvious map from a unique
holomorphic annulus $F$ with boundary punctures. (One may think $F$ is
obtained by cutting $B$ open along the alpha arcs when $B$ is embedded; see
Figure 6). Without loss of generality, we may regard $F$ is obtained from the
annulus $\\{z\in\mathbb{C}|\frac{1}{r}\leq|z|\leq r\\}$ for some positive
number $r$ by adding boundary punctures: we assume the outer boundary has only
one boundary puncture and is asymptotic to $q$ under $\hat{u}$, and the inner
boundary has five punctures, corresponding to $p$, $-\rho_{0}$, $-\rho_{1}$,
$-\rho_{2}$, and $-\rho_{3}$ (see Figure 6 (b)), whose relative positions
depend on the complex structure induced from $j$ on $\Sigma$. There is only
one involution $\iota$ on $F_{\bar{e}}$ interchanging the inner and outer
boundary and swapping the boundary punctures labeled by $p$ and $q$; see
[OS04b, Lemma 9.3]. This involution induces $\pi:F_{\bar{e}}\rightarrow
F_{\bar{e}}/\iota$, and $D$ is obtained from $F_{\bar{e}}/\iota$ by removing
the boundary points corresponding to the filled-in east punctures. In summary,
$ev_{Sym^{2}(j)}^{-1}(p)$ consists of a unique map, and hence it has odd
cardinality. ∎

Let $J$ be a generic almost complex structure such that both
$\mathcal{N}_{J}^{B}(q;\overrightarrow{\rho})$ and $ev_{J}^{-1}(p)$ are
smoothly cut out. Then a generic path $J_{s}$ of almost complex structures
such that $J_{0}=J$ and $J_{1}=Sym^{2}(j)$ induces a cobordism between
$ev_{J}^{-1}(p)$ and $ev_{Sym^{2}(j)}^{-1}(p)$. This shows $|ev_{J}^{-1}(p)|$
is odd. For a generic point $p^{\prime}\in\alpha^{a}_{1}$, let $l$ be the sub-
arc in $\alpha^{a}_{1}$ connecting $p$ and $p^{\prime}$, then $ev^{-1}_{J}(l)$
is a cobordism from $ev_{J}^{-1}(p)$ to $ev_{J}^{-1}(p^{\prime})$, implying
$|ev_{J}^{-1}(p^{\prime})|$ is odd as well. This finishes the proof of the
theorem for $g(\Sigma)=2$.

The relevant statements for $g(\Sigma)>2$ can be proved inductively from the
base case in which $g(\Sigma)=2$ using the neck-stretching argument in Section
10 of [OS04b], which relates moduli spaces built from Heegaard diagrams differ
by a stabilization. ∎

######  Proposition 2.60.

Let $B\in T(q)$ be a stabilized teardrop with an acute corner at $q$ and let
$\overrightarrow{\rho}=(-\rho_{0},-\rho_{123})$ or $(-\rho_{012},-\rho_{3})$.
For a generic nearly symmetric almost complex $J$, the moduli space of
degenerate disk $\mathcal{N}_{J}^{B}(q;\overrightarrow{\rho})$ is a
$(g-1)$-manifold, and a generic fiber of the evaluation map
$ev_{J}:\mathcal{N}^{B}(q;\overrightarrow{\rho})\rightarrow\alpha^{a}_{1}\times\alpha^{c}_{1}\times\cdots\alpha^{c}_{g-2}$
consists of an even number of points.

###### Proof.

The regularities and dimensions of the moduli spaces are proved the same way
as in Proposition 2.58.

We study the parity of a generic fiber $ev^{-1}_{J}(p)$ of
$\mathcal{N}_{J}^{B}(q;\overrightarrow{\rho})$. Again, as in the previous
proposition, we only need to study the case in which $g(\Sigma)=2$. We first
prove the following lemma.

######  Lemma 2.61.

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 $ev^{-1}_{Sym^{2}(j)}(p)$ is empty.

###### Proof of Lemma 2.61.

The proof is similar to Proposition 3.16 of [OS04b]. We provide a sketch. Let
$j_{t}$ denote the complex structure on $\Sigma$, corresponding to when the
connected sum tube is isometric to $S^{1}\times[-t,t]$. If the statement is
not true, then there exists a sequence of $u_{t}\in
ev^{-1}_{Sym^{2}(j_{t})}(p)$ which converges to a holomorphic disk
$u_{\infty}$ in $Sym^{2}(E_{1}\vee E_{2})$ by Gromov compactness. In
particular, the main component of $u_{\infty}$ is a holomorphic disk in
$E_{1}\times E_{2}$. Projecting this disk to $E_{1}$, we would have a
holomorphic disk in $E_{1}$ with the east punctures prescribed by
$\overrightarrow{\rho}=(-\rho_{0},-\rho_{123})$ or $(-\rho_{012},-\rho_{3})$,
which cannot exist by direct examination. ∎

With the above lemma at hand, a cobordism argument as in Proposition 2.58 can
be applied to conclude that when the stretching parameter $t$ is sufficiently
large, for a generic nearly $j_{t}$-symmetric almost complex structure
$J_{t}$, the parity of a generic fiber of $ev_{J_{t}}$ is even. Then one can
show the statement is independent of the stretching parameter $t$ as in
Theorem 3.15 of [OS04b]: When $t_{1}$ and $t_{2}$ are sufficiently close, and
$J_{t_{i}}$ is a $j_{t_{i}}$-nearly symmetric almost complex structure
sufficiently close to $Sym^{2}(j_{t_{i}})$, $i=1,2$, then the moduli spaces
can be identified. ∎

####  2.7.5. Boundary degeneration without corners

######  Proposition 2.62.

Under the assumption of Proposition 2.49, if a boundary degeneration without
corners occurs, then:

  * (1)

The degenerate disk has domain $[B]=[\Sigma]$.

  * (2)

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

  * (3)

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

  * (4)

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

###### Proof.

We first prove (1),(2), and (3). Suppose the limit curve $u_{\infty}$ has a
nodal source being a disc bubble tree $\mathbb{B}$. A schematic picture of
$\mathbb{B}$ to keep in mind would be an analogue of Figure 5 (a). The same
analysis as in the proof of Proposition 2.57 shows there must be only one leaf
in the bubble tree $\mathbb{B}$, which is the one that contains the boundary
puncture. (All the others are excluded as there are no corresponding positive
domains.) The degenerate disk corresponding to the leaf cannot have a corner,
for otherwise, we are back to the case considered in the previous subsection
by index consideration. Denote the degenerate disk corresponding to the leaf
by $v$. Then, the domain $B_{v}$ of $v$ is a zero-cornered positive
$\alpha$-bounded domain with $n_{z}=1$. Hence $B_{v}=\Sigma$ as $\mathcal{H}$
is unobstructed. Note that a degenerate disk with domain $[\Sigma]$ has Maslov
index $2$, which implies the nodal curve must be of the form $u\vee v$ with
$u$ being a constant curve and $v$ being the degenerate disk. We finished
proving (1), (2), and (3).

(4) follows from a standard gluing argument and Proposition 11.35 of [LOT18],
which states for a generic almost complex structure, the moduli space of
degenerate discs at $\bm{x}$ is transversally cut out, and it consists of an
odd number of points if $\overrightarrow{\rho}=(-\rho_{0},\ldots,-\rho_{3})$
and an even number of points for other $\overrightarrow{\rho}$. ∎

####  2.7.6. Proof of Proposition 2.49

In this subsection, we synthesize the previous results to prove Proposition
2.49.

###### Proof of Proposition 2.49.

By Proposition 2.40, we know degenerations appearing in
$\overline{\mathcal{M}}^{B}(\bm{x},\bm{y};\overrightarrow{\rho})$ are two-
story splittings, simple holomorphic combs with a single split component, or
boundary degenerations.

For a simple holomorphic comb $(u,v)$ with a single split component to appear,
there must be two consecutive Reeb chords in $\overrightarrow{\rho}$ such that
the endpoint of the first one is equal to the start point of the second one,
this excludes the possibility of
$\overrightarrow{\rho}=(-\rho_{0},\ldots,-\rho_{3})$. Moreover, when such a
degeneration appears, as the split curve’s domain is a single disk, the moduli
space of the split curve $\mathcal{N}(v)$ is a transversally cut-out
0-dimensional manifold (by Proposition 2.21), and it consists of a single
point by the Riemann mapping theorem. Therefore, the gluing result,
Proposition 2.31, shows there is a neighborhood of such a holomorphic comb in