1 Introduction
A great deal of interesting and beautiful mathematics has been devoted to understanding the fundamental dichotomy in threedimensional contact geometry: the subdivision of contact structures into tight and overtwisted. Overtwisted structures are determined by homotopical data and thus may be addressed by tools from algebraic topology. In contrast, tight contact structures do not satisfy an hprinciple, and many existence and classification questions for tight contact structures are still open. Tight structures are nevertheless extremely natural objects of study, as they include the contact structures arising as the boundary of a complex or symplectic manifold.
Many of the recent advances in classifying tight contact structures were made possible by the advent of Heegaard Floer homology in the early 2000s and the subsequent development of Floertheoretic contact invariants. Using open books, Ozsváth and Szabó defined an invariant of closed contact threemanifolds [Reference Ozsváth and SzabóOS05]. Given a closed, contact manifold $(M,\xi )$ , this invariant is a class $c(\xi )$ in the Heegaard Floer homology $\widehat {\mathit {HF}}(M)$ . In [Reference Honda, Kazez and MatićHKM09b], Honda, Kazez and Matić gave an alternative description of $c(\xi )$ , again using open books. This ‘contact class’ was used to show that knot Floer homology detects both genus [Reference Ozsváth and SzabóOS04] and fiberedness [Reference GhigginiGhi08, Reference NiNi07]. It gives information about overtwistedness: If $\xi $ is overtwisted, then $c(\xi ) = 0$ , whereas if $\xi $ is Stein fillable, then $c(\xi )\neq 0$ [Reference Ozsváth and SzabóOS05]. The contact class was also used to distinguish notions of fillability: Ghiggini used it to construct examples of strongly symplectically fillable contact threemanifolds which do not have Stein fillings [Reference GhigginiGhi05].
Contact manifolds with convex boundary can be partitioned by the equivalence class of the dividing set on their boundary. Given a contact manifold M with convex boundary and dividing set $\Gamma $ , Honda, Kazez and Matić used partial open books to define an invariant $c(M, \Gamma , \xi )$ of the equivalence class in the sutured Floer homology $\mathit {SFH}(M, \Gamma )$ [Reference Honda, Kazez and MatićHKM09a]. They also defined a gluing map for sutured Floer homology that respects the contact invariants [Reference Honda, Kazez and MatićHKM08]. This map requires the Heegaard diagrams to satisfy a number of technical conditions, collectively referred to as ‘contactcompatibility.’ Establishing contactcompatibility for specific examples is difficult, so in practice, computations with the gluing map are rarely possible. As a result, most applications of this gluing map have relied only on its formal properties.
Gluing techniques – and Heegaard Floer theory more broadly – benefited soon after from the introduction of bordered Floer homology, a new theory for manifolds with boundary defined by Lipshitz, Ozsváth and Thurston in [Reference Lipshitz, Ozsvath and ThurstonLOT18]. Although bordered Floer theory has produced a wide variety of new results, (e.g., [Reference Hanselman, Rasmussen, Rasmussen and WatsonHRRW20, Reference HanselmanHan17, Reference LevineLev12, Reference LevineLev16, Reference HomHom13, Reference PetkovaPet13, Reference Alishahi and LipshitzAL19]), applications to contact topology remain mostly uncharted territory.
To a threemanifold with parametrized boundary, bordered Floer homology associates an $\mathcal A_{\infty }$ module (or type A structure) or, equivalently, a type D structure over a differential graded algebra associated to the parametrization. When manifolds are glued along compatible parameterized boundaries, the derived tensor product of their bordered invariants recovers the Heegaard Floer homology of the closed threemanifold, up to homotopy equivalence. Zarev introduced a generalization, bordered sutured Floer homology, which is an invariant of threemanifolds whose boundary is ‘part sutured, part parametrized’ [Reference ZarevZar09]. Bordered sutured Floer homology similarly associates to a bordered sutured manifold a type A structure and a type D structure such that taking derived tensor products recovers the sutured Floer homology of the manifold formed by gluing along the parameterized parts of the boundary.
1.1 Results
In this paper, we define a contact invariant in the bordered sutured Heegaard Floer homology of a contact threemanifold with boundary. We consider contact manifolds whose convex boundary is equipped with a certain type of singular foliation. As shown in [Reference Licata and VértesiLV20], to any such contact manifold with foliated boundary, one may associate a topological decomposition known as a foliated open book. Intuitively, foliated open books are constructed by cutting ordinary open books along separating convex surfaces. The pages and binding of the resulting pair of foliated open books are simply the restrictions of the pages and binding of the original open book to the corresponding piece. The intersection of the cutting surface with the pages determines an ordered signed singular foliation, here called the boundary foliation. This induced ‘open book foliation’ is closely related to the characteristic foliation of a supported contact structure and has been extensively studied in the work of Ito–Kawamuro, for example, [Reference Ito and KawamuroIK14a, Reference Ito and KawamuroIK14b]. Under mild technical hypotheses, the topological data of the resulting foliated open book uniquely determine the restriction of the original contact structure to each piece, up to isotopy.
We associate to a foliated contact threemanifold $(M,\xi , \mathcal {F})$ a bordered sutured manifold $(M, \mathsf {\Gamma }, \mathcal {Z})$ . We then show that the data of a sorted foliated open book for $(M,\xi , \mathcal {F})$ give rise to an admissible bordered sutured Heegaard diagram for the manifold $(M, \mathsf {\Gamma }, \mathcal {Z})$ , and so via dualizing, an admissible bordered sutured Heegaard diagram for $(M,\mathsf {\Gamma },\overline {\mathcal Z})$ , which we will call $\mathcal H$ . Moreover, we define preferred generators $\mathbf {x}_D$ and $\mathbf {x}_A$ in $\widehat {\mathit {BSD}}(\mathcal H)$ and $\widehat {\mathit {BSA}}(\mathcal H)$ , respectively. The structures $\widehat {\mathit {BSD}}(\mathcal H)$ and $\widehat {\mathit {BSA}}(\mathcal H)$ give invariants of $(M,\mathsf {\Gamma },\overline {\mathcal Z})$ up to homotopy equivalence. (Some further algebraic properties of these elements may be found in Proposition 3.5.) We further define an equivalence between elements of two homotopy equivalent type A structures or between elements of two homotopy equivalent type D structures in Section 2.2.2. Our main result is that these preferred generators are invariants of the contact structure up to this equivalence.
Theorem 1. Let $(M,\xi , \mathcal {F})$ be a foliated contact threemanifold with associated bordered sutured manifold $(M, \mathsf {\Gamma }, \mathcal {Z})$ . Given admissible bordered sutured Heegaard diagrams $\mathcal H$ and $\mathcal H'$ for $(M,\mathsf {\Gamma },\overline {\mathcal Z})$ with preferred generators $\mathbf {x}_A\in \widehat {\mathit {BSA}}(\mathcal H)$ and $\mathbf {x}^{\prime }_A\in \widehat {\mathit {BSA}}(\mathcal H')$ , there is a homotopy equivalence between $\widehat {\mathit {BSA}}(\mathcal H)$ and $\widehat {\mathit {BSA}}(\mathcal H')$ induced by Heegaard moves, under which $\mathbf {x}_A$ and $\mathbf {x}^{\prime }_A$ are equivalent; an analogous statement holds for the preferred generators $\mathbf {x}_D\in \widehat {\mathit {BSD}}(\mathcal H)$ and $\mathbf {x}^{\prime }_D\in \widehat {\mathit {BSD}}(\mathcal H')$ . We refer to these type A and type D equivalence classes as $c_A(M,\xi , \mathcal {F})$ and $c_D(M,\xi , \mathcal {F})$ , respectively.
More precisely, we show that varying the choices made in our construction induces homotopy equivalences between the bordered sutured Floer homologies associated to the resulting Heegaard diagrams; these maps carry the preferred generator of one module to the preferred generator of the other module.
An ordered signed singular foliation on the convex boundary of a contact manifold determines a dividing set, but such foliations also induce a finer partition on the set of contact manifolds with convex boundary. These additional data improve the compatibility with cutandpaste operations, as the ordered signed singular foliation carries data about both the contact structure and the foliated open book near the boundary. Given a pair of foliated contact threemanifolds $(M^L, \xi ^L, \mathcal {F}^L)$ and $(M^R, \xi ^R, \mathcal {F}^R)$ whose foliated boundaries agree in an appropriate sense, there is a canonical perturbation of the contact structure near the boundary so that the pieces glue to a closed contact threemanifold $(M, \xi )$ . In fact, the foliated open books supporting these manifolds also glue to an open book for the resulting closed contact manifold; see Section 2.3.6. Because our bordered sutured contact invariant is sensitive not only to the dividing set of a convex boundary but also to the singular foliation, it behaves nicely with respect to these cutandpaste operations.
We prove that the contact invariants of the two foliated contact threemanifolds pair to recover the contact invariant of $(M, \xi )$ .
Theorem 2. The tensor product $c_D(M^L,\xi ^L,\mathcal {F}^L)\boxtimes c_A(M^R,\xi ^R,\mathcal {F}^R)$ recovers the contact invariant $c(M,\xi )$ .
A more precise version of this statement on the level of generators is given in Theorem 5.1.
One may also choose to forget the singular foliation and retain only the data of the dividing set on the convex boundary; this is captured by a natural map from a foliated open book to a partial open book. On the level of Heegaard diagrams, this corresponds to converting a bordered sutured Heegaard diagram to a sutured Heegaard diagram; the procedure to do so was described by Zarev in [Reference ZarevZar10] and induces an isomorphism
where $\iota _+$ is an idempotent naturally determined by the foliation data. We show that under Zarev’s isomorphism the bordered sutured invariant associated to a foliated open book maps to the contact invariant in the sutured Floer homology associated to the corresponding partial open book.
Theorem 3. Under the above isomorphism, $c_A(M,\xi , \mathcal {F})\cdot \iota _+$ is identified with the contact invariant $\mathrm {EH}(M,\Gamma (\mathcal {F}), \xi )$ from [Reference Honda, Kazez and MatićHKM09a].
In particular, $EH(M,\Gamma (\mathcal {F}), \xi )$ vanishes in $\mathit {SFH}(M,\Gamma (\mathcal {F}))$ if and only if $c_A(M,\xi , \mathcal {F})\cdot \iota _+$ is zero in $H_\ast \left (\widehat {\mathit {BSA}}(M,\mathsf {\Gamma },\overline {\mathcal Z})\right )\cdot \iota _+$ . Since $c_A(M,\xi ,\mathcal {F})=c_A(M,\xi , \mathcal {F})\cdot \iota _+$ and the differential of $\widehat {\mathit {BSA}}(M,\mathsf {\Gamma },\overline {\mathcal Z})$ respects the splitting by the idempotents, the class $c_A(M,\xi , \mathcal {F})\cdot \iota _+$ being zero in $H_\ast \left (\widehat {\mathit {BSA}}(M,\mathsf {\Gamma },\overline {\mathcal Z})\right )\cdot \iota _+$ is equivalent to the class $c_A(M,\xi , \mathcal {F})$ being zero in $H_\ast \left (\widehat {\mathit {BSA}}(M,\mathsf {\Gamma },\overline {\mathcal Z})\right )$ , which together with [Reference Ghiggini, Honda and Van HornMorrisGHVHM07, Theorem 1] and [Reference Honda, Kazez and MatićHKM09a, Corollary 4.3 and Theorem 4.9] implies the following two corollaries.
Corollary 4. If $(M,\xi ,\mathcal {F})$ is overtwisted or has positive $2\pi $ torsion, then the class $c_A(M,\xi , \mathcal {F})$ is zero in $H_\ast \left (\widehat {\mathit {BSA}}(M,\mathsf {\Gamma },\overline {\mathcal Z})\right )$ .
Corollary 5. If $c_A(M,\xi , \mathcal {F})$ is zero in $H_\ast \left (\widehat {\mathit {BSA}}(M,\mathsf {\Gamma },\overline {\mathcal Z})\right )$ , then $(M,\xi ,\mathcal {F})$ does not embed into any closed contact manifold $(N,\xi ')$ with nonvanishing contact invariant.
1.2 Further directions
The results in this paper establish a framework for studying foliated open books via Heegaard Floer homology in concert with other combinatorial representations of contact manifolds. This is an essential first step towards developing cutandpaste technology for the Heegaard Floer contact invariant, and we briefly note further avenues for developing this theory.
The defining data of a contact manifold with foliated boundary includes a choice of a distinguished leaf in the foliation, which plays an essential role in constructing the associated Heegaard diagram. It is natural to ask how the invariant depends on this choice; accordingly, this dependence is the subject of planned future work. We anticipate that the foliation on $\partial M$ may be reparameterized by the addition of a suitable foliated open book for $\partial M\times I$ . This is a special case of the more general process of gluing a boundaryparallel layer onto $\partial M$ . We hope to understand the maps induced by such gluings in general and to compare our gluing operation to the sutured Floer homology gluing map from [Reference Honda, Kazez and MatićHKM09a].
Theorems 1 and 2 are phrased terms of equivalences of elements, rather than elements. This subtlety arises because as of this writing there is not a naturality result for the bordered variants of Heegaard Floer homology; hence, the type D bordered sutured invariant associated to $(M,\mathsf {\Gamma },\mathcal Z)$ is the type D homotopy equivalence class of $\widehat {\mathit {BSD}}(M,\mathsf {\Gamma },\mathcal Z)$ , as defined in Section 2.2.2. An analogue of Juhász, Thurston and Zemke’s proof of the naturality of the Heegaard Floer homology of closed threemanifolds [Reference Juhász, Thurston and ZemkeJTZ21] for bordered sutured Floer homology would immediately upgrade our invariants.
Organization
Section 2 reviews some necessary background in contact geometry and Heegaard Floer homology. Section 2.1 discusses assumed background in contact geometry while Section 2.2 contains a rapid review of Heegaard Floer homology and bordered sutured Heegaard Floer homology; in particular, Section 2.2.2 introduces a notion of equivalence between elements in bordered sutured Heegaard Floer homology under homotopy equivalences of type A and type D structures. Finally, Section 2.3 summarizes the relevant material from [Reference Licata and VértesiLV20] concerning foliated open books. In Section 3, we associate a bordered sutured manifold $(M, \mathsf {\Gamma }, \mathcal {Z})$ to a foliated contact threemanifold. We then show how the data of a sorted foliated open book give rise to an admissible bordered Heegaard diagram $\mathcal {H}$ for $(M, \mathsf {\Gamma }, \overline {\mathcal {Z}})$ and we identify preferred generators $\mathbf {x}_A$ and $\mathbf {x}_D$ in the associated bordered Floer homology modules $\widehat {\mathit {BSA}}(\mathcal {H})$ and $ \widehat {\mathit {BSD}}(\mathcal {H})$ . Section 4 proves Theorem 1, namely invariance of $\mathbf {x}_A$ and $\mathbf {x}_D$ up to the choices made in their definitions. Section 5 proves Theorem 2, showing that we recover the ordinary contact invariants after gluing. Finally, Section 6 discusses the relationship of our invariants to the invariant in sutured Floer homology, proving Theorem 3.
2 Preliminaries
This section provides the background required to read the rest of the paper. We provide references for various classical objects in contact geometry in Section 2.1 and Heegaard Floer theory in Section 2.2, along with more indepth summaries of partial open books and the contact invariants in various flavors of Floer homology. Because we are concerned with the relationships between various theories, we pay particular attention to the conventions for bordered, sutured, and bordered sutured versions. Finally, Section 2.3 gives an efficient introduction to foliated open books.
2.1 Assumed background in contact geometry
Throughout this article, we assume familiarity with many standard definitions in threedimensional contact geometry, including contact structures; characteristic foliations; convex surfaces and dividing sets; open book decompositions for closed threemanifolds, as in, for example, [Reference GeigesGei08] and other standard references. Although we will introduce foliated open books carefully in Section 2.3 below, we briefly first recall the definition of a partial open book supporting a contact structure on a manifold with boundary.
Definition 2.1. A partial open book is a triple $(S, P, h)$ , where

1. S is a compact, oriented, connected surface with boundary;

2. $P=\cup P_i$ is a subsurface of S such that the surface S is obtained from $\overline {S\setminus P}$ by successively attaching onehandles $P_i$ ; and

3. $h \colon \thinspace P\rightarrow S$ is an embedding which is the identity along $\partial P\cap \partial S$ .
To a partial open book, we can associate a sutured manifold $(M, \Gamma )$ , as follows. (See [Reference Honda, Kazez and MatićHKM09a] or [Reference Etgü and OzbagciEO11] for more details.) Let $H=S\times [1,0]$ with the identification $(x,t)\sim (x,t')$ for $x\in \partial S$ and $t, t'\in [1,0]$ . Similarly, let $N=P\times [0,1]$ with the identification $(x,t)\sim (x,t')$ for $x\in \partial P\cap \partial S$ and $t, t'\in [0,1]$ . Then $M=H\cup N$ where we identify $P\times \{0\}\subset \partial H$ with $P\times \{0\}\subset \partial N$ and $h(P)\times \{1\}\subset \partial H$ with $P\times \{1\}\subset \partial N$ . The suture $\Gamma $ on $\partial M$ can be given as the union of oriented closed curves obtained by gluing the following arcs, modulo identifications:
Definition 2.2. A contact structure $\xi $ is compatible with the partial open book $(S,P,h)$ if for the corresponding sutured manifold $(M=H\cup N, \Gamma )$ , the following hold:

1. $\xi $ is tight on H and N;

2. $\partial H$ is a convex surface in $(M, \xi )$ with dividing set $\partial S\times \{0\}$ ;

3. $\partial N$ is a convex surface in $(M, \xi )$ with dividing set $\partial P\times \{1/2\}$ .
2.2 Heegaard Floer theories
We assume familiarity with the various Heegaard Floer theories and provide only a brief review to establish notation and review details related to the contact invariants defined in [Reference Ozsváth and SzabóOS05, Reference Honda, Kazez and MatićHKM09a, Reference Honda, Kazez and MatićHKM09b].
2.2.1 Earlier Heegaard Floer theoretic contact invariants
Using open books, Ozsváth and Szabó defined a Heegaard Floer invariant of a closed contact threemanifold [Reference Ozsváth and SzabóOS05]. For a contact manifold $(M,\xi )$ , this invariant is a class $c(\xi )$ in the Heegaard Floer homology $\widehat {\mathit {HF}}(M)$ . In [Reference Honda, Kazez and MatićHKM09b], Honda, Kazez and Matić gave an alternative description of $c(\xi )$ . Their construction again uses open books and goes roughly as follows. An open book $(S,h)$ for $(M,\xi )$ induces a Heegaard splitting of M into and with Heegaard surface $\Sigma =(S\times \{1/2\})\cup _B(S\times \{0\})$ . Let $\{a_i\}$ be a collection of properly embedded arcs on S that cut S into a disk. For all i, let $b_i$ be a small perturbation of $a_i$ that moves the endpoints in the positive direction along $\partial S$ so that $b_i$ intersects $a_i$ in one point. Fix a basepoint z on $\partial S$ away from the thin strips cobounded by the $\{a_i\}$ and $\{b_i\}$ . It is clear that $\Sigma = \partial U_1 = \partial U_2$ and that the $\alpha _i := \partial (a_i\times \left [0,1/2\right ])$ bound compressing disks for $U_1$ and the $\beta _i := \partial (b_i\times \left [1/2,1\right ])$ bound compressing disks for $U_2$ . Thus, $(\Sigma , \boldsymbol {\alpha }, \boldsymbol {\beta }, z)$ is a Heegaard diagram for M. One can see the curves as
Then $c(\xi )\in \widehat {\mathit {HF}}(M)$ is defined as the homotopy equivalence class of the unique generator of $\widehat {\mathit {CF}}(\Sigma , {\boldsymbol {\beta }}, {\boldsymbol {\alpha }}, z)$ fully supported on $S\times \{1/2\}$ .
Using partial open books, Honda, Kazez and Matić then extended the above construction to define an invariant $EH(M, \Gamma , \xi )$ of contact threemanifolds with convex boundary. In this construction, one begins with a partial open book $(S,P, h)$ for $(M, \Gamma , \xi )$ , where S is built up from $\overline {S\setminus P}$ by the addition of onehandles $P_i$ , as in Definition 2.1. The roles of $U_1$ and $U_2$ are now played by $\big (P\times [1/2,0]\big )\cup \big (S\times [0,1/2]\big )/\sim $ and $\big (S\times [1/2,1]\big )\cup \big (P\times [1,1/2]\big )/\sim $ , respectively, and the curves $\{a_i\}$ are the cocores of the onehandles $P_i$ . The rest of the construction is as above, except that the final generator supported on $P\times \{1/2\}$ defines an element in the sutured Floer homology $\mathit {SFH}(M, \Gamma )$ . Under certain technical conditions, the authors also defined a gluing map for sutured Floer homology that respects the contact invariants [Reference Honda, Kazez and MatićHKM08].
Before briefly reviewing bordered sutured Floer homology, we discuss a straightforward generalization of the contact invariant to multipointed Heegaard diagrams. We begin by constructing a Heegaard diagram analogous to the one above, except that we allow additional arcs $\{a_i\}$ that cut S into n disks. We place a basepoint in each of the disks thus obtained. This results in a multipointed Heegaard diagram $(\Sigma , {\boldsymbol {\alpha }}, {\boldsymbol {\beta }}, \mathbf {z})$ for M. The unique generator $\mathbf {x}$ on $S\times \{1/2\}$ defines an element in ${\mathrm {\widetilde {HF}}}(\Sigma , {\boldsymbol {\beta }}, {\boldsymbol {\alpha }}, \mathbf {z}) \cong \widehat {\mathit {HF}}(M)\otimes H_*(T^{n1})$ . Here, ${\mathrm {\widetilde {HF}}}$ is the homology with respect to the differential that avoids all basepoints, and $H_*(T^{n1}) \cong H_*(S^1)^{\otimes (n1)}$ is the ordinary singular homology of $T^{n1}$ .
An adaptation of the argument of part (5) of [Reference Baldwin, VelaVick and VértesiBVVV13, Theorem 3.1] shows the following.
Proposition 2.3. Let $(S,h)$ and $(S',h')$ be two open book decompositions compatible with $(M,\xi )$ . Let $\{a_i\}$ and $\{a_i'\}$ be sets of cutting arcs that cut up S and $S'$ into n and $n'$ disks, respectively, with $n<n'$ . The graded isomorphism between the Heegaard Floer homologies induced by Heegaard moves, including index $0$ and $3$ stabilizations,
maps the homology class to $[\mathbf {x}'] \otimes \theta ^{\otimes {(n'n)}}$ to $[\mathbf {x}]$ , where $\theta $ corresponds to the lowerdegree generator of $H_*(S^1)$ .
This means that up to homotopy equivalence, the multipointed contact invariant is simply $c(\xi )\otimes \theta ^{\otimes {(n1)}}$ , where n is the number of basepoints.
We will also make use of the fact that the multipointed Heegaard Floer homology for a closed threemanifold M can be interpreted as the sutured Floer homology of M with balls removed, as follows. Let $(\Sigma , {\boldsymbol {\alpha }}, {\boldsymbol {\beta }}, \mathbf {z})$ be a multipointed Heegaard diagram for M with n basepoints, and for each basepoint $z\in \mathbf {z}$ , let $D^2_z$ be a disk neighborhood of z. Then $(\Sigma \setminus \cup _{\mathbf {z}} D^2_z, {\boldsymbol {\alpha }}, {\boldsymbol {\beta }})$ is a sutured Heegaard diagram for $M(n) = (M\setminus \cup _{z\in \mathbf {z}}B^3_{z}, \cup _{z\in \mathbf {z}}\partial D^2_z)$ . As the two Heegaard diagrams are identical outside the basepointed/sutured regions, the chain complex ${\mathrm {\widetilde {CF}}}(\Sigma , {\boldsymbol {\alpha }}, {\boldsymbol {\beta }}, \mathbf {z})$ is isomorphic to the chain complex ${\mathrm {SFC}}(\Sigma \setminus \cup _{\mathbf {z}} D^2_z, {\boldsymbol {\alpha }}, {\boldsymbol {\beta }})$ . Thus, we can compute the multipointed Heegaard Floer homology ${\mathrm {\widetilde {HF}}}(\Sigma , {\boldsymbol {\beta }}, {\boldsymbol {\alpha }}, \mathbf {z})$ as the sutured Floer homology $\mathit {SFH}(\Sigma \setminus \cup _{\mathbf {z}} D^2_z, {\boldsymbol {\alpha }}, {\boldsymbol {\beta }})$ . See also [Reference JuhászJuh06, Proposition 9.14].
2.2.2 Bordered sutured Floer homology
Lipshitz, Oszváth and Thurston refine Heegaard Floer homology to a bordered variant associated to a threemanifold with parametrized boundary [Reference Lipshitz, Ozsvath and ThurstonLOT18, Reference Lipshitz, Ozsváth and ThurstonLOT11], and Zarev [Reference ZarevZar09, Reference ZarevZar10] further refines the invariant to an invariant of sutured manifolds with partially parameterized boundary. We briefly discuss Zarev’s constructions [Reference ZarevZar10, Section 3]. Note that the following description is indicative, rather than complete; for a review of the algebraic definitions involved, see [Reference Lipshitz, Ozsvath and ThurstonLOT18, Section 2].
Recall that an arc diagram $\mathcal Z = (Z,a,m)$ consists of a finite collection of oriented line segments Z, commonly called arcs in the literature, a collection of points $a=(a_1,\cdots , a_{2k})$ on Z, a matching m of the points in a into pairs and a ‘type,’ $\alpha $ or $\beta $ . To every arc diagram $\mathcal Z$ one associates an $\mathcal A_{\infty }$ algebra $\mathcal A(\mathcal Z)$ generated by tuples of oriented arcs in $[0,1]\times Z$ such that each arc connects some $(0,a_i)$ to some $(1, a_j)$ with $a_j\geq a_i$ , up to an equivalence relation imposed by the matching. The ground ring of idempotents $\mathcal I(\mathcal Z)$ of this algebra consists of elements $\iota $ corresponding to tuples of horizontal strands $[0,1]\times \{a_k\}$ in $[0,1]\times Z$ . One further associates to $\mathcal Z$ a graph $G(\mathcal Z)$ and a surface $F(\mathcal Z)$ . The graph is constructed by beginning with Z and adding an edge between each pair of matched points, while $F(\mathcal Z)$ is constructed from $Z \times \left [0,1\right ]$ by attaching onehandles to neighborhoods of the matched points on $Z\times \{0\}$ .
A bordered sutured manifold $(M,\mathsf {\Gamma },\mathcal Z)$ consists of the following:

• a threemanifold M whose boundary decomposes as a bordered part F and a sutured part T;

• an arc diagram $\mathcal {Z}=(Z,a,m)$ and an identification of F with $F(\mathcal Z)$ ;

• a dividing set $\mathsf {\Gamma }$ which is a properly embedded, oriented onemanifold in T with boundary $\partial \mathsf {\Gamma } = \partial (Z\times \{\frac 1 2 \})$ so that $\mathsf {\Gamma }$ decomposes T into the union of $R_+(\mathsf {\Gamma })$ and $R_(\mathsf {\Gamma })$ with $\partial R_{\pm }(\mathsf {\Gamma })\setminus \partial F=\pm \mathsf {\Gamma }$ .
The components of the dividing set are also referred to as sutures.
A $\beta $ bordered Heegaard diagram $\mathcal {H}=(\Sigma , {\boldsymbol {\alpha }}, {\boldsymbol {\beta }}, \mathcal Z)$ consists of a compact surface $\Sigma $ with no closed components; a collection of pairwise disjoint, properly embedded circles ${\boldsymbol {\alpha }}$ ; a collection of pairwise disjoint, properly embedded circles ${\boldsymbol {\beta }}^{c}$ and of properly embedded arcs ${\boldsymbol {\beta }}^{a}$ , with ${\boldsymbol {\beta }} = {\boldsymbol {\beta }}^a \cup {\boldsymbol {\beta }}^c$ and an arc diagram $\mathcal Z$ of $\beta $ type, together with an embedding $G(\mathcal Z)\rightarrow \Sigma $ which maps Z to a subset of $\partial \Sigma $ and the edges of $G(\mathcal Z)$ connecting matched points to the arcs ${\boldsymbol {\beta }}^c$ . One also requires that the maps $\pi _0(\partial \Sigma \setminus Z)\to \pi _0(\Sigma \setminus {\boldsymbol {\alpha }})$ and $\pi _0(\partial \Sigma \setminus Z)\to \pi _0(\Sigma \setminus {\boldsymbol {\beta }})$ be surjective. (An $\alpha $ bordered diagram is defined similarly, mutatis mutandis.) One constructs a bordered sutured manifold $(M,\mathsf {\Gamma },\mathcal Z)$ from $\mathcal {H}$ as follows. The threemanifold M is obtained by attaching twohandles to $\Sigma \times [0,1]$ along ${\boldsymbol {\alpha }} \times \{1\}$ and ${\boldsymbol {\beta }}^c\times \{0\}$ circles; the dividing set $\mathsf {\Gamma }$ appears as $\mathsf {\Gamma } = (\partial \Sigma \setminus Z) \times \left \{\frac {1}{2}\right \}$ , and $F(\mathcal Z)$ is a neighborhood of $(Z \times [0,1])\cup ({\boldsymbol {\beta }}^a\times \{0\})$ . Another way to view $(M,\mathsf {\Gamma },\mathcal Z)$ as coming from $\mathcal {H}$ , which fits better with Morse theory, is to also attach ‘halves of twohandles’ along ${\boldsymbol {\beta }}^a \times \{0\}$ and see $F(\mathcal Z)$ as $Z\times [0,1]$ together with the intersection of the thickened cores of the partial twohandles with $\partial M$ .
Let $\textbf {x}$ and $\textbf {y}$ denote tuples of intersection points between the ${\boldsymbol {\alpha }}$ and ${\boldsymbol {\beta }}$ such that each $\alpha $ circle is used exactly once, each $\beta $ circle is used exactly once, and each $\beta $ arc is used no more than once; these will ultimately be the generators of the bordered sutured modules. We recall that the set of homology classes $\pi _2(\textbf {x}, \textbf {y})$ connecting $\textbf {x}$ to $\textbf {y}$ is defined as follows. We let $I_s =[0,1]$ and $I_t = [\infty , \infty ]$ be intervals and consider the relative homology group
The set $\pi _2(\textbf {x}, \textbf {y})$ denotes elements of this group which are sent to the fundamental class of $(\textbf {x}\times I_s)\cup (\textbf {y}\times I_s)$ by the map which applies the boundary homomorphism and then collapses the remainder of the boundary. Any homology class has a unique corresponding domain, that is, a linear combination of the components of $\Sigma \setminus ({\boldsymbol {\alpha }} \cup {\boldsymbol {\beta }})$ obtained by projection. A domain is provincial if its boundary has no intersection with Z.
The set $\pi _2(\textbf {x}, \textbf {x})$ is the set of periodic domains. We recall that $\mathcal {H}$ is said to be provincially admissible if every provincial periodic domain has both positive and negative coefficients and admissible if every periodic domain has both positive and negative coefficients.
To a provincially admissible $\beta $ bordered Heegaard diagram $\mathcal {H}=(\Sigma , {\boldsymbol {\alpha }}, {\boldsymbol {\beta }}, \mathcal Z)$ equipped with an admissible almost complex structure, Zarev associates a left type D module $\widehat {\mathit {BSD}}(\mathcal {H})$ over $\mathcal A(\mathcal Z)$ and right type A module $\widehat {\mathit {BSA}}(\mathcal {H})$ over $\mathcal A(\mathcal Z)$ . In both cases, the generators are tuples of intersection points as described above.Footnote ^{1} The type A chain homotopy equivalence class of $\widehat {\mathit {BSA}}(\mathcal {H})$ is an invariant of the associated bordered sutured manifold $(M, \mathsf {\Gamma }, \mathcal Z)$ , written $\widehat {\mathit {BSA}}(M, \mathsf {\Gamma }, \mathcal Z)$ . More precisely, given bordered sutured Heegaard diagrams $\mathcal {H}$ and $\mathcal {H}'$ for $(M, \mathsf {\Gamma }, \mathcal Z)$ , there is a sequence of Heegaard moves connecting them as in [Reference ZarevZar09, Proposition 4.5], which induce a type A homotopy equivalence f from $\widehat {\mathit {BSA}}(\mathcal {H})$ to $\widehat {\mathit {BSA}}(\mathcal {H}')$ . This type A homotopy equivalence is a collection of maps
indexed by $i\ge 1$ , satisfying certain conditions (see [Reference Lipshitz, Ozsvath and ThurstonLOT18, Section 2]). Consider $\mathbf {x}\in \widehat {\mathit {BSA}}(\mathcal {H})$ and $\mathbf {x}'\in \widehat {\mathit {BSA}}(\mathcal {H}')$ so that
for all $i\ge 0$ and $a_i\in \mathcal {A}(\mathcal Z)$ . We say $\mathbf {x}$ and $\mathbf {x}'$ are equivalent if there exists a homotopy equivalence f from $\widehat {\mathit {BSA}}(\mathcal {H})$ to $\widehat {\mathit {BSA}}(\mathcal {H}')$ so that $f_1(\mathbf {x})=\mathbf {x}'$ and $f_{i+1}(\mathbf {x},a_1,\dots ,a_i)=0$ for all $i>0$ and $a_i\in \mathcal {A}(\mathcal Z)$ . Similarly, there is a type D homotopy equivalence from $\widehat {\mathit {BSD}}(\mathcal {H})$ to $\widehat {\mathit {BSD}}(\mathcal {H}')$ . If $\mathbf {x}\in \widehat {\mathit {BSD}}(\mathcal {H})$ and $\mathbf {x}'\in \widehat {\mathit {BSD}}(\mathcal {H}')$ are both closed elements (meaning that $\delta ^1(\mathbf {x})=\delta ^1(\mathbf {x}')=0$ ), we say $\mathbf {x}$ and $\mathbf {x}'$ are equivalent if there is a homotopy equivalence from $\widehat {\mathit {BSD}}(\mathcal {H})$ to $\widehat {\mathit {BSD}}(\mathcal {H}')$ that maps $\mathbf {x}$ to $\mathbf {x}'$ .
Given two bordered sutured manifolds $(M_1, \mathsf {\Gamma }_1, \mathcal Z)$ and $(M_2, \mathsf {\Gamma }_2, \mathcal Z)$ , we may glue along $\mathcal Z$ to obtain a sutured manifold $(M, \Gamma )$ . Given $\beta $ bordered sutured Heegaard diagrams $\mathcal {H}_1 = (\Sigma _1, {\boldsymbol {\alpha }}_1, {\boldsymbol {\beta }}_1, \mathcal Z)$ for $(M_1, \mathsf {\Gamma }_1, \mathcal Z)$ and $\mathcal {H}_2 = (\Sigma _2, {\boldsymbol {\alpha }}_2, {\boldsymbol {\beta }}_2, \mathcal Z)$ for $(M_1, \mathsf {\Gamma }_1, \mathcal Z)$ , there is a glued Heegaard diagram $\mathcal {H} = (\Sigma , {\boldsymbol {\alpha }}, {\boldsymbol {\beta }})$ , where $\Sigma = \Sigma _1 \bigcup _{Z} \Sigma _2$ ; the set ${\boldsymbol {\alpha }}$ is the union of ${\boldsymbol {\alpha }}_1$ and ${\boldsymbol {\alpha }}_2$ and the set ${\boldsymbol {\beta }}$ is the union of ${\boldsymbol {\beta }}_1$ and ${\boldsymbol {\beta }}_2$ , together with the circles formed by gluing the arcs in ${\boldsymbol {\beta }}_1^a$ and ${\boldsymbol {\beta }}_2^a$ along their endpoints on Z.
When at least one of the two diagrams is admissible, there is a gluing map
which induces a chain homotopy equivalence of vector spaces. Chain homotopy equivalences of type A and type D modules induce chain homotopy equivalences of the box tensor product, so there is a welldefined equivalence class of $\mathbb F_2$ vector spaces
and an equivalence of chain homotopy equivalence classes of vector spaces
2.3 Abstract foliated open books
In this section, we provide an overview of the essential definitions and properties of foliated open books that we will rely on in the remainder of this article. Readers are encouraged to see [Reference Licata and VértesiLV20] for more detail.
Definition 2.4. [Reference Licata and VértesiLV20, Definition 3.14] An abstract foliated open book is a tuple $(\{S_i\}_{i=0}^{2k}, h)$ , where $S_i$ is a surface with boundary $\partial S_i=B\cup A_i$ Footnote ^{2} and corners at $E=B\cap A_i$ such that

1. for all i, $A_i$ is a union of intervals;

2. with the boundary orientation, each component I of $A_i$ is oriented from a corner labeled $e_+ = e_+(I)\in E_+$ to a corner labeled $e_ = e_(I)\in E_$ , and $E=E_+\cup E_$ ;

3. the surface $S_{i}$ is obtained from $S_{i1}$ by either

 (add): attaching a onehandle along two points $\{p_{i1}, q_{i1}\}\in A_{i1}$ , or

 (cut): cutting $S_{i1}$ along a properly embedded arc $\gamma _{{i}}$ with endpoints in $A_{{i1}}$ and then smoothing.Footnote ^{3}

Furthermore, $h \colon S_{2k}\rightarrow S_0$ is a diffeomorphism between cornered surfaces that preserves B pointwise.
We denote by $H_+$ (resp. $H_$ ) the set of indices i for which $S_{i}$ is obtained from $S_{i1}$ by cutting (resp. adding) so that we have a partition $[2k]=H_+\cup H_$ . (Here $[n]$ denotes the set $\{1,\dots , n\}$ .)
Note that the operations (add) and (cut) are inverses of each other. Let $\gamma $ denote the cocore of a handle attached along points p and q. Then cutting along $\gamma $ cancels the handle attachment and vice versa. We will use the following notation to describe this:
Example 2.5. Figure 2.1 shows a first example of a foliated open book. The complete set of labels is shown for the indicative page $S_0$ , while the attaching spheres and cutting arcs are shown on all pages to the right. The binding is shown in bold. The monodromy $h \colon S_4\rightarrow S_0$ is the identity, and we have the partition $H_=\{2,4\}$ , $H_+=\{1, 3\}$ .
2.3.1 Supported contact structures
The construction of a compact manifold is natural from the data of a foliated open book: Pairs of successive pages define cornered cobordisms. As described in more detail below, we concatenate these to form a manifold with boundary and collapse the resulting components of $B\times I$ to circles and intervals again labeled B. The final page glues to the initial page via h. The resulting manifold M associated to the foliated open book $(\{S_i\}_{i=0}^{2k}, h)$ retains partial information about the sequence of cuts and adds in the abstract data. This information is encoded in the form of boundary decorations on $\partial M$ , and we introduce the kind of foliation we will consider before explaining how it arises.
Definition 2.6. A signed singular foliation is an equivalence class of smooth vector fields (up to multiplication by smooth positive functions) that vanish at only finitely many isolated points, called singular points. The complement of the singular points has an open cover such that in each ball, the integral curves of the vector field are a product of oriented intervals, while elliptic and fourpronged hyperbolic singularities patch these charts together. Elliptic singularities may be classified as positive (sources) or negative (sinks), and the hyperbolic singularities have signs determined by input external to the defining vector field.
See Figure 2.2 for local models of the integral curves.
As is standard, leaves that enter (or exit) the hyperbolic singularities are called stable (or unstable) separatrices.
Each elliptic point e induces a cyclic order on the subset of hyperbolic points with separatrices terminating at e. If e is positive (respectively, negative), the order increases as the separatrices are encountered in a counterclockwise (clockwise) path around e.
Definition 2.7. A signed singular foliation is ordered if there is a cyclic order on the set of all hyperbolic points which is compatible with the cyclic orders associated to each of its elliptic points.
Beginning with an abstract foliated open book, we now build a manifold whose boundary is naturally equipped with an ordered signed singular foliation. Each pair of successive pages defines an elementary cobordism $M_i$ from $S_{i1}$ to $S_{i}$ with vertical boundary $(B\times I)\cup V_i$ , where $V_i$ is the union of a single saddle and of the collection of products $A_{i1}\times I$ for any components $A_{i1}$ that are left unchanged. More specifically, the saddle connects either the components of $A_{i1}$ containing the endpoints of $\gamma _{i}$ or the component(s) containing $p_{i1}$ and $q_{i1}$ with the obtained component(s) of $A_i$ that have the same endpoints. See Figure 2.3.
Concatenating these $M_i$ and gluing the final $S_{2k}$ to the initial $S_0$ via h yields a manifold with boundary $(B\times S^1) \cup V$ , where V is the (circular) union of the $V_i$ . The singular foliation on V has $2k$ singular points associated to the transitions between topological types of the pages, while the regular leaves may be identified with curves of the form $A_i\times \{t\}$ . If $S_{i1}\xrightarrow {{\textbf {add}}} S_{i}$ , then the corresponding hyperbolic point $h_i$ is negative; otherwise, it is positive. We denote the set of hyperbolic points with respect to this partition $H=H_+\cup H_$ . The signs match the partition of $[2k]$ introduced after Definition 2.4. Collapsing the components $B\times S^1$ to a single copy of B yields a manifold with foliated boundary $(M, \partial M, \mathcal {F})$ . (See Definition 2.9, below, for a precise definition of this term.) Each endpoint of B labeled $e_+$ becomes a positive elliptic point of the foliation, which we again denote by $e_+$ ; likewise, an endpoint $e_$ becomes a negative elliptic point $e_$ .
Example 2.8. We return to the foliated open book introduced in the first example. The associated smooth manifold is the solid torus shown in Figure 2.4. The images of $A_1$ and $A_2$ are shown in green and blue, respectively, on both the manifold on the left and in the associated boundary foliation on the right.
In order to define the compatibility between contact structures and foliated open books, we examine the boundary foliation $\mathcal {F}$ more closely. To any smooth manifold constructed as above, we may associate an $S^1$ valued Morse function $\pi \colon M \setminus B \rightarrow S^1$ whose level sets are the pages of the open book and which restricts to the boundary as an $S^1$ valued Morse function ${\widetilde {\pi }}:=\pi _{\partial M}\colon \partial M\setminus E\to S^1$ with the same critical points as $\pi $ . The function $\pi $ has only index 1 and 2 critical points, and all of these are index 1 critical points for ${\widetilde {\pi }}$ . The index with respect to $\pi $ is visible in the sign of these hyperbolic points: index 2 critical points of $\pi $ give positive hyperbolic points of the foliation, while index 1 critical points of $\pi $ give negative hyperbolic points of the foliation. Heuristically, this means that the interesting features of $\pi $ may all be seen from the foliation on the boundary. One may build a signed singular foliation simply by patching together the local models of Figure 2.2, but in the context of this paper we will only encounter ordered signed singular foliations; equivalently, these are signed singular foliations that can be induced by an $S^1$ valued Morse function. We therefore consider the map ${\widetilde {\pi }}$ to be an essential part of the data and we write $\mathcal {F}=({\widetilde {\pi }}, H=H_\cup H_+, E=E_\cup E_+)$ . We may also assume that $0$ is a regular value of ${\widetilde {\pi }}$ , and we further require that ${\widetilde {\pi }}=\pm \phi $ on an $(r, \phi )$ disk neighborhood of each elliptic point $e\in E$ , where the sign depends on the sign of the elliptic point. Here, $(r,\phi )$ is the standard polar coordinate system on $D^2$ . By construction, the foliation on a manifold constructed from an abstract foliated open book has only elliptic and fourpronged hyperbolic singularities, with signs inherited from the partitions of E and H.
Definition 2.9. A manifold with foliated boundary is a compact oriented threemanifold equipped with an ordered signed singular foliation $\mathcal {F}=({\widetilde {\pi }}, H=H_\cup H_+, E=E_\cup E_+)$ on its boundary. We denote this collection of data by $(M, \partial M, \mathcal {F})$ .
We recall that these data include the $S^1$ valued Morse function ${\widetilde {\pi }}$ , and we require that a diffeomorphism $f:(M_1, \partial M_1, \mathcal {F}_1) \rightarrow (M_2, \partial M_2, \mathcal {F}_2)$ between manifolds with foliated boundary must satisfy ${\widetilde {\pi }}_2\circ f= {\widetilde {\pi }}_1$ .
We have seen that an abstract foliated open book gives rise to a diffeomorphism class of manifolds with foliated boundary; we say a manifold with foliated boundary is compatible with $(\{S_i\}, h)$ if it is diffeomorphic to an output of this construction. A fixed manifold with foliated boundary will admit many different functions $\pi $ extending the Morse function ${\widetilde {\pi }}$ on the boundary, but compatibility identifies an equivalence class of such functions whose level sets are diffeomorphic to the pages $\{S_i\}$ .
Definition 2.10. Suppose that $\mathcal {F}=({\widetilde {\pi }}, H=H_\cup H_+, E=E_\cup E_+)$ is a signed singular foliation with no $S^1$ leaves. We define $R_+(\mathcal {F})$ to be a closed neighborhood of the union of the stable separatrices of hyperbolic points in $H_+$ , and let $R_(\mathcal {F})=\overline {M\setminus R_+(\mathcal {F})}$ . The dividing curve $\Gamma (\mathcal {F})$ of $\mathcal {F}$ is $\partial R_+(\mathcal {F})=\partial R_(\mathcal {F})$ .
Note that the serifed symbol $\Gamma $ used here is distinct from the sansserif symbol $\mathsf {\Gamma }$ introduced in Section 2.2.2 as part of the defining data of a bordered sutured manifold. The above construction of $\Gamma (\mathcal {F})$ follows the construction of Giroux [Reference GirouxGir00] for dividing curves of characteristic foliations of convex surfaces and in our case too $\Gamma (\mathcal {F})$ indeed divides $\mathcal {F}$ : $\Gamma (\mathcal {F})$ is positively transverse to the leaves of $\mathcal {F}$ and it separates $\partial M\setminus \Gamma (\mathcal {F})$ into two parts, one of which contains all the positive singular points, while the other part contains all the negative singular points.Footnote ^{4} These properties in fact specify $\Gamma (\mathcal {F})$ up to isotopy transverse to $\mathcal {F}$ , and we will often work with its equivalence class, only specifying the representative when needed. The choice of the representative then changes $R_+(\mathcal {F})$ and $R_(\mathcal {F})$ accordingly, while still preserving the condition that $\Gamma (\mathcal {F})=\partial R_+(\mathcal {F})=\partial R_(\mathcal {F})$ . Note that distinct foliations on $\partial M$ may induce isotopic dividing sets; see Figure 2.5.
Definition 2.11. Two ordered signed singular foliations $\mathcal {F}_1$ and $\mathcal {F}_2$ on a surface $\Sigma $ are strongly topologically conjugate if there is an isotopy taking $\mathcal {F}_1$ to $\mathcal {F}_2$ which respects the cyclic orders on the hyperbolic points.
The term ‘topological’ refers to the fact that the above isotopy may not be smooth. In fact, in the next definition, the isotopy cannot be chosen to be smooth.
Definition 2.12. [Reference Licata and VértesiLV20, Definition 3.8, 3.10] A contact structure $\xi _0$ on a manifold with foliated boundary $(M, \partial M, \mathcal {F})$ is strictly supported by the abstract foliated open book $(\{S_i\}, h)$ if there is a diffeomorphism $f\colon (M', \partial M', \mathcal {F'})\to (M, \partial M, \mathcal {F})$ , where $(M', \partial M', \mathcal {F'})$ is a manifold constructed from $(\{S_i\}, h)$ as above such that the pullback $f^{\ast }\xi $ is the kernel of some contact oneform $\alpha $ on $M'$ satisfying the following conditions:

1. $\alpha (TB)>0$ ;

2. for all t, $d\alpha _{\pi ^{1}(t)}$ is an area form; and

3. for each pair of consecutive hyperbolic points at times $t_1$ and $t_2$ , there is a time $t_*$ such that $t_1<t_*<t_2$ with the property that the regular leaf ${\widetilde {\pi }}^{1}(t_*)$ of $\mathcal {F}$ is Legendrian.
A contact structure $\xi _1$ is supported by $(\{S_i\}, h)$ if there exists a path of contact structures $\xi _t$ such that $\xi _0$ is strictly supported by $(\{S_i\}, h)$ and for all t, the characteristic foliation $\mathcal {F}_{\xi _t}$ is strongly topologically conjugate to the characteristic foliation $\mathcal {F}_{\xi _0}$ .
Definition 2.13. A foliated contact threemanifold $(M, \xi , \mathcal {F})$ is a manifold with foliated boundary together with a contact structure $\xi $ on M such that $\mathcal {F}$ is strongly topologically conjugate to $\mathcal {F}_{\xi }$ .
Theorem 2.14. [Reference Licata and VértesiLV20, Theorems 3.12, 7.1, 7.2] Any foliated open book with circlefree boundary foliation supports a unique isotopy class of contact structures, and any foliated contact threemanifold $(M,\xi ,\mathcal {F})$ admits a supporting foliated open book.
Remark 2.15. Because the signed singular foliation $\mathcal {F}$ data include a function $\widetilde {\pi }$ , a foliated contact threemanifold has a distinguished $t=0$ leaf, which we always assume to be regular.
Distinct foliations may induce isotopic dividing sets, as illustrated in the following example.
Example 2.16. Let $(B^3, \xi _1, \mathcal {F}_1)$ and $(B^3, \xi _2, \mathcal {F}_2)$ be two foliated contact threeballs with $\xi _1$ and $\xi _2$ tight, and with the foliations $\mathcal {F}_1$ and $\mathcal {F}_2$ on the boundaries as in Figure 2.5. The two threeballs are equivalent in the sense that there is a contact embedding of one into the other with the property that the image of the embedded boundary is convexly isotopic to the boundary of the target manifold. This notion is an equivalence relation on contact manifolds with boundary, yet the given balls are distinct as foliated contact manifolds.
2.3.2 Stabilization
Although each foliated open book supports a unique isotopy class of contact structures, there are many foliated open books which support the same class. Next, we define an operation which changes a foliated open book while preserving the contactomorphism class of the associated contact manifold.
Definition 2.17. Let $\gamma \subset S_0$ be a properly embedded arc with $\partial \gamma \subset B$ . The stabilization of $(\{S_i\}, h)$ along $\gamma $ is the foliated open book $(\{S^{\prime }_i\}, h')$ defined as follows:

• $S_i'=S_i\cup H$ , where H is a onehandle attached along $\partial \gamma $ ; and

• $h'=\tau \circ \overline {h}$ , where $\tau $ is a positive Dehn twist along the circle formed by $\gamma $ and the core of H. (Here, $\overline {h}$ denotes the extension of h to H by the identity.)
In order to define stabilization along some arc in a page besides $S_0$ , we first describe an operation on foliated open books which preserves the contactomorphism type of the associated manifold.
Definition 2.18. [Reference Licata and VértesiLV20, Definition 3.16] The $1$ shift of an abstract foliated open book $(\{S_i\}_{i=0}^{2k},h)$ is the foliated open book
where

 if $S_{0}\xrightarrow [\gamma _{1}]{{\textbf {cut}}} S_1$ , then $S_1'$ is defined by the relation $S_{2k}\xrightarrow [{h^{1}(\gamma _{1})}]{{\textbf {cut}}} S_1'$ and $h'$ is the restriction of h to $S_1'$ ;

 if $S_{0}\xrightarrow [{p_0, q_0}]{{\textbf {add}}} S_1$ , then $S_1'$ is defined by the relation $S_{2k}\xrightarrow [{ h^{1}(p_0), h^{1}(q_0) }]{{\textbf {add}}} S_1'$ and $h'$ is h extended by the identity on the added onehandle.
An rfold iteration of the shift operation is called an rshift and denoted by $(\{S_i[r]\}_{i=0}^{2k},h[r])$ . One can analogously define rshifts for $r<0$ .
Shifts correspond to postcomposing $\pi $ with a diffeomorphism of $S^1$ .
Definition 2.19. Let $\gamma \subset S_r$ be a properly embedded arc with $\partial \gamma \subset B$ . The stabilization of $(\{S_i\}, h)$ along $\gamma \subset S_r$ is the foliated open book $(\{S^{\prime }_i\}, h')$ defined as follows:

1. first perform an rshift of $(\{S_i\},h)$ to $(\{S_i[r]\},h[r])$ ;

2. stabilize as in Definition 2.17 along the image of $\gamma \subset S_0[r]$ ;

3. perform a $(r)$ shift to obtain $(\{S_i'\},h')$ , where $S_i'$ is still obtained from $S_i$ by a handle attachment along p and q.
It is easy to see that stabilizing does not change the underlying manifold with foliated boundary.
Theorem 2.20. [Reference Licata and VértesiLV20, Proposition 6.8, Theorem 6.9] Let $(\{S_i\}, h)$ be a foliated open book and $(\{S_i'\}, h')$ be a positive stabilization of $(\{S_i\}, h)$ . Then the corresponding contact threemanifolds are contactomorphic. Furthermore, if two foliated open books support contactomorphic foliated contact threemanifolds, then they admit a common positive stabilization.
2.3.3 Sorted foliated open books
Next, we introduce some additional bookkeeping to record the cutting arcs and cocore arcs on all possible pages of the foliated open book. This additional information corresponds to a choice of a gradientlike vector field for $\pi $ on the associated smooth manifold, as we will describe in Section 2.3.4.
If $S_{{i1}}\xrightarrow {{\textbf {add}}} S_{i}$ , then for all $j\geq {i}$ , decorate $S_{j}$ with a cocore of the added handle and label this cocore $\gamma _i^$ . If $S_{{i1}}\xrightarrow {{\textbf {cut}}} S_{i}$ along $\gamma \subset S_{{i1}}$ , then for all $j\leq {i1}$ , decorate $S_j$ with the cutting curve and label it $\gamma _i^+$ .
We call the $\gamma _i^\pm $ sorting arcs. Note that by an abuse of notation we use a single label $\gamma _i^\pm $ to refer to distinct copies of the sorting arcs on multiple $S_j$ . In general, sorting arcs may intersect (and consequently some sorting arcs may appear as a union of disconnected intervals on some pages). We will shortly restrict to the subclass of sorted foliated open books [Reference Licata and VértesiLV20, Section 5.3], which are characterized by the requirement that the sorting arcs are disjoint on all pages, together with a prescribed ordering of the intersections between sorting arcs and regular leaves. In the sorted case, we enhance the notation for describing a foliated open book to reflect these additional choices.
Definition 2.21. An abstract foliated open book $(\{S_i\}, h, \{\gamma _i^\pm \})$ is sorted if the following properties hold:

1. On each page $S_i$ and for each component I of $A_i$ with $\partial I=\{e_+, e_\}$ , there exist disjoint subintervals $I_+$ and $I_$ such that $I_+$ contains $e_+\cup \bigcup _j \big (\gamma ^+_j\cap I\big )$ and $I_{}$ contains $e_\cup \bigcup _j \big (\gamma ^_j\cap I\big )$ .

2. For $i {<} j< l$ , on each component of $A_i$ any endpoint of $\gamma _l^+$ is closer to $e_+$ than any endpoint of $\gamma _j^+ $ ;

3. For $j<l\leq i$ , on each component of $A_i$ any endpoint of $\gamma _l^$ lies closer to $e_+$ than any endpoint of $\gamma _j^$ ;

4. On each page, the sorting arcs are connected, properly embedded, and disjoint.
See Figure 2.7 in Example 2.24 for an illustration of the ordering conventions given above. Sorted foliated open books are useful because of their close relation to partial open books, which will be described in Section 2.3.5. Below, we show that a foliated open book need not be sorted.
Example 2.22. The foliated open book from Example 2.5 is not sorted. To see this, we start on the $S_0$ page and attempt to decorate each successive page with cutting arcs which realize the topological transitions between pages and whose endpoints also satisfy the ordering conditions of Definition 2.21.
Cutting along $\gamma _1^+$ on $S_0$ yields $S_1$ , while attaching a handle to $S_1$ gives rise to $S_2$ , which is decorated with the new cocore $\gamma _2^$ . However, any cutting arc associated to $h_3$ necessarily intersects $\gamma _2^$ on $S_2$ , as the endpoints of $\gamma _3^+$ must lie closer to the positive ends of $A_2$ than the endpoints of $\gamma _2^$ . Thus, the foliated open book is not sorted. See Figure 2.6.
We may always stabilize a foliated open book to obtain a sorted one.
Theorem 2.23. Any foliated contact threemanifold $(M,\xi ,\mathcal {F})$ admits a supporting sorted foliated open book. If two sorted foliated open books support the same foliated contact threemanifold, then we can stabilize each through sorted foliated open books to obtain a common sorted foliated open book. Moreover, all of these stabilizations can be assumed to be performed on the $S_0$ page as in Definition 2.17.
Proof. The first two statements are taken from [Reference Licata and VértesiLV20, Theorem 3.12, Proposition 8.4, Theorem 8.14]. The final claim that the necessary stabilizations may be assumed to occur on the $S_0$ page is implicit in [Reference Licata and VértesiLV20], as explained next; we include the brief argument here, although it relies on the connection between foliated and partial open books described in Section 2.3.5. Specifically, the proof of [Reference Licata and VértesiLV20, Proposition 8.6] explains why stabilizations on the $S_0$ page suffice to turn an arbitrary sorted foliated open book into a sufficiently stabilizedFootnote ^{5} foliated open book. The proof that two sufficiently stabilized open books admit a common stabilization relies on Giroux correspondence for the associated partial open books and, hence, leads again to stabilizations on $S_0$ .
Example 2.24. Figure 2.7 shows that the foliated open book from Example 2.5 may be stabilized to a sorted foliated open book. Starting from index $S_0$ , the first obstruction to being sorted occurred as an intersection between $\gamma _2^$ and $\gamma _3^+$ on $S_2$ . This intersection may be removed by stabilizing ‘at’ the $S_2$ page; that is, by shifting twice and then stabilizing at the new $S_0$ . Each new page is obtained from the corresponding old page by attaching a onehandle with one attaching interval on each binding component. The new final maps to the new initial page by a positive Dehn twist; after shifting back, this becomes a positive Dehn twist identifying the two copies of the $S_2'$ page, as shown. Thus, $\gamma _2^$ changes by a positive Dehn twist as it ascends to higherindex pages, while $\gamma _3^+$ changes by a negative Dehn twist as it descends to lowerindex pages.
When depicting a sequence of pages, there is an implicit identification by translation in the page unless otherwise noted. In this example, the one nontranslation occurs at $S_2'$ while the map from $S^{\prime }_4$ to $S_0'$ remains translation in the plane of the page, as seen on Figure 2.7. The result is a sorted foliated open book for the original contact manifold with foliated boundary.
2.3.4 Gradient flows on foliated open books
As in Section 2.3.1, we consider the smooth manifold with foliated boundary $(M, \partial M, \mathcal {F})$ constructed from a foliated open book. The initial abstract data determines an equivalence class of circlevalued Morse functions $\pi \colon \thinspace M\setminus B\rightarrow S^1$ [Reference Licata and VértesiLV20, Section 5.2]. The level sets of $\pi $ are the pages, and the restriction ${\widetilde {\pi }} \colon \thinspace \partial M\setminus B \rightarrow S^1$ is again a circlevalued Morse function with the same set of critical points. In this setting, one may consider the critical submanifolds of a gradientlike vector field for $\pi $ parallel to $\partial M$ , which thus restricts to a gradientlike vector field for ${\widetilde {\pi }}$ . The intersection of the critical submanifolds with the regular level sets of $\pi $ determines sorting arcs on the pages of the corresponding abstract foliated open book [Reference Licata and VértesiLV20, Section 5.3]. Thus, a foliated open book is sorted when a gradientlike vector field on $\partial M$ that satisfies the ordering conditions for sorting arcs extends to a gradientlike vector field on M whose critical submanifolds are disjoint.
Any surface with an open book foliation decomposes along regular leaves into square tiles that each contain precisely one hyperbolic singularity. Figure 2.8 shows the flowlines of a preferred $\nabla {\widetilde {\pi }}$ on such a tile; these flowlines satisfy the ordering conditions in Definition 2.21, although ensuring their extension to M yields a sorted foliated open book may require stabilization.
The Morse perspective is useful not only for building geometric intuition but also for relating the boundary decorations to those in the interior of the manifold. For example, a sorting arc $\gamma ^{+}_i$ is isotopic through the associated stable critical submanifold of $\nabla \pi $ to the stable submanifold of $\nabla {\widetilde {\pi }}$ on $\partial M$ which passes through the corresponding positive hyperbolic point $h_i$ . Likewise, a sorting arc $\gamma ^{}_i$ is isotopic through the associated unstable critical submanifold of $\nabla \pi $ to the unstable submanifold of $\nabla {\widetilde {\pi }}$ on $\partial M$ which passes through the corresponding negative hyperbolic point $h_i$ . Conditions 1, 2 and 3 in Definition 2.21 are chosen so that the graphs formed by the (un)stable separatrices of positive (negative) hyperbolic points with respect to $\mathcal {F}$ and with respect to $\nabla {\widetilde {\pi }}$ are isotopic. See [Reference Licata and VértesiLV20, Lemma 8.8]. In the next section, this will allow us to define certain subsurfaces as neighborhoods of either graph.
2.3.5 Sufficiently stabilized foliated open books and the associated partial open book
Sorted foliated open books are closely related to partial open books, and [Reference Licata and VértesiLV20] introduced the technical designation of a ‘sufficiently stabilized’ foliated open book. In this section, we recall this definition and show that a sufficiently stabilized foliated open book naturally has a companion partial open book. Throughout this section, we will let $(\{S_i\}_{i=0}^{2k},h,\{\gamma _i^\pm \})$ be a sorted foliated open book and $(M, \xi , \mathcal {F})$ be the supported foliated contact threemanifold. We further assume that either $k>0$ or $\partial M>1$ ; the only foliated open books this excludes are those formed from an honest open book by removing a neighborhood of a single point on the binding.
Set
where $N_{\lrcorner }$ denotes a ‘cornered’ neighborhood of $A_i \cup (\bigcup _{i < j}\gamma _j^+)\cup (\bigcup _{i\geq j}\gamma _j^)$ , with corners at E so that $R_i$ meets B only at E. Furthermore, define
as shown on the left in Figure 2.9. Since only the $R_i$ change with i, we can identify all the $P_i$ with each other by the flow of the gradientlike vector field. We denote the composition of these identifications from $P_0\subset S_0$ onto $P_{2k}\subset S_{2k}$ by $\iota $ . Let $\widetilde {S}=S_{0}$ , $\widetilde {P}=P_{0}$ and $\tilde {h}=h\vert _{P_{2k}}\circ \iota $ .
Definition 2.25. [Reference Licata and VértesiLV20, Proof of Proposition 8.6] The sorted foliated open book $(\{S_i\}_{i=0}^{2k}, h, \{\gamma _i\})$ is sufficiently stabilized if each component of $P_0$ intersects the boundary of $R_0$ in at least two intervals.
This condition ensures that $(\widetilde {S}, \widetilde {P}, \tilde {h})$ is a partial open book whose pages embed into the pages of $(\{S_i\}_{i=0}^{2k}, h, \{\gamma _i\})$ . As the term suggests, this condition may always be achieved by a sequence of positive stabilizations, and in fact, these may be chosen to occur on the $S_0$ page.
Theorem 2.26. [Reference Licata and VértesiLV20, Proposition 8.10] With the notation above, the partial open book $(\widetilde {S},\widetilde {P},\widetilde {h})$ is compatible with a contact manifold $(\widetilde {M},\widetilde {\xi })$ which is contactomorphic to $(M,\xi )$ . Furthermore, under this contactomorphism, the image of the dividing set of the characteristic foliation of $\xi $ on $\partial M$ divides $\widetilde {\mathcal {F}}$ .
Example 2.27. In this example, we return to the foliated open book of Example 2.24, which is not sufficiently stabilized. Recall that $R^{\prime }_0$ is constructed as a cornered neighborhood of the sorting arcs together with $A^{\prime }_0$ . Observe that $P^{\prime }_0:=S_0'\setminus R_0'$ consists of two bigons, each with an edge on B and an edge on the annular $\partial R^{\prime }_0$ . It follows that $S^{\prime }_0$ cannot be built up from $R^{\prime }_0$ by adding onehandles, which is the topological requirement for the surface/subsurface pair to define a partial open book.
Figure 2.9 shows a sufficiently stabilized foliated open book for the same manifold, built by stabilizing along the dotted arc which connected the two components of the binding. The new monodromy is the composition of the original monodromy with a positive Dehn twist along the circle formed by the core of the added handle and the stabilizing arc.
The new $P^{\prime \prime }_0:=S_0''\setminus R_0''$ is a single disk whose boundary intersects $\partial R^{\prime \prime }_0$ along two intervals; this ensures that $(\widetilde {S}'', \widetilde {P}'', \widetilde {h}'')$ is a partial open book. In order to understand $\widetilde {h}''$ , we first examine the flow of $P_0''$ through the manifold. Recall from Example 2.24 that the only nontrivial identification between successive pages is a righthanded Dehn twist at the $S_2''$ page before the cut yielding $S_3''$ . The twist is along the core of the annular $R_2''$ , so $P_4''$ is isotopic, relative to $A_4''=A_0''$ , to $P_0''$ . Thus, the partial open book monodromy is simply the restriction of the foliated open book monodromy to $P_4''$ .
As shown in [Reference Licata and VértesiLV20, Lemma 8.12], the cornered diffeomorphism type of the subsurfaces $R_i\subset S_i$ depends only on the foliation $\mathcal {F}$ , rather than on the pages $S_i$ . Construct the corresponding contact threemanifold $(M,\xi ,\mathcal {F})$ for a sorted abstract foliated open book $(\{S_i\},h,\{\gamma _i^\pm \})$ . Recall from Definition 2.10 that the subsurface $R_+(\mathcal {F})$ (respectively $R_(\mathcal {F})$ ) of $\partial M$ is a closed neighborhood of the (un)stable separatrices corresponding to positive (negative) hyperbolic points of $\mathcal {F}$ .
As noted in Section 2.3.4, the graph of positive separatrices of $\nabla {\widetilde {\pi }}$ from positive hyperbolic points on $\partial M$ is (nonsmoothly) isotopic to the graph of positive separatrices of $\mathcal {F}$ from positive hyperbolic points on $\partial M$ . On the other hand, the former is isotopic through the stable critical submanifolds to a neighborhood of the sorting arcs $\cup _{H_+}\gamma ^+_i$ on $S_0$ . This yields an identification between $R_+(\mathcal {F})\subset \partial M$ and $R_0\subset S_0$ . Similarly, studying negative separatrices of negative hyperbolic points and unstable critical submanifolds yields an identification between $R_(\mathcal {F})\subset \partial M$ and $R_{2k}\subset S_{2k}$ .
Similarly, all the $R_i$ can be identified with surfaces described only using the foliation $\mathcal {F}$ , independently of the foliated open book, as follows. Recall that $R_i$ is the (cornered) neighborhood of the union of $A_i$ and the intersection of all stable and unstable submanifolds of $\nabla \pi $ with $S_i$ . Each of these intersections can be pushed up (or down) onto $\partial M$ along the corresponding stable (or unstable) submanifolds. While performing these isotopies we push the halfneighborhood of $A_i$ up along $I_+$ and down along $I_$ ; the surface obtained after the isotopy is not a submanifold of $\partial M$ , as it has a half twist in the middle of each component I of $A_i$ which extends into the interior of M. See Figure 2.10.
2.3.6 Gluing foliated open books
In order to glue foliated open books, we describe the matching conditions imposed upon their boundary foliations. Recall that a foliation $\mathcal {F}$ on a surface F always refers to the data of a signed foliation together with the function ${\widetilde {\pi }}\colon F\setminus E\to S^1$ whose level sets are the leaves of $\mathcal {F}$ .
Definition 2.28. The reverse of a foliation $\mathcal {F}$ on F is the foliation on $F$ which is equal to $\mathcal {F}$ pointwise but with the leaf orientations and signs of singular points reversed.
The main advantage of foliated open books is that they are well suited for gluing. Suppose that $(M^L,\xi ^L,\mathcal {F}^L)$ and $(M^R,\xi ^R,\mathcal {F}^R)$ are foliated contact threemanifolds such that there exists an orientationreversing diffeomorphism $\psi \colon \partial M^L \to \partial M^R$ that maps the reverse of the foliation $\mathcal {F}^L$ onto $\mathcal {F}^R$ .
Recall that by hypothesis, the boundary foliations are divided, so the boundaries $\partial M^{\bullet }$ are convex with respect to $\xi ^{\bullet }$ . (Here, and throughout this section, $\bullet $ will be an element of the set $\{R, L\}$ .) Since the contact structures are Iinvariant near the boundaries, $\psi $ determines a closed contact threemanifold
If the initial contact manifolds were supported by foliated open books $(\{S^L_i\},h^L)$ and $(\{S^R_i\},h^R)$ , respectively, then $(M,\xi )$ naturally inherits a supporting open book whose pages and binding are built by gluing the pages and bindings of the constituent pieces. More precisely, recall that $\partial S^{\bullet }_i=A_i^{\bullet }\cup B^{\bullet }$ , where all the $B^{\bullet }$ are identified when forming the manifolds $M^{\bullet }$ . The map $\psi $ identifies the intervals $A_i^L$ and $A_i^R$ for all i, forming the surfaces
with boundary $B=B^L\cup B^R$ . Observe that a cut to $S_i^R$ pairs with a handle addition to $S_i^L$ , and vice versa, so the surfaces $S_i$ are diffeomorphic for all i. This allows us to identify $S_0\cong S_1 \cong \cdots \cong S_{2k}$ , and we denote the composition of these identifications by $\iota \colon S_0\to S_{2k}$ . Note that $\iota $ fixes B, and it restricts to $P_{0}^\bullet $ as the identification $\iota ^\bullet :P_0^\bullet \rightarrow P_{2k}^\bullet $ defined in Section 2.3.5. Letting $S:=S_0$ , the monodromy $h\colon S\to S$ for the gluedup open book can be obtained as the composition of $\iota $ with $h^L\cup h^R\colon S_{2k}=S_{2k}^L\cup S_{2k}^R\to S_{0}^L\cup S_{0}^R=S_0$ . By construction, h fixes B, as required.
Theorem 2.29. [Reference Licata and VértesiLV20, Theorem 6.2] Suppose that the foliated open books $(\{S_i^L\},h^L)$ and $(\{S_i^R\},h^R)$ define the threemanifolds with foliated boundary $(M^L,\mathcal {F}^L)$ and $(M^R,\mathcal {F}^R)$ , and assume that there is an orientationreversing diffeomorphism $\varphi \colon \partial M^L\to \partial M^R$ that takes the reverse of the foliation $\mathcal {F}^L$ onto $\mathcal {F}^R$ .
Then there are contact structures $\xi ^L$ and $\xi ^R$ supported by $(\{S_i^L\},h^L)$ and $(\{S_i^R\},h^R)$ , respectively, so that $\xi =\xi ^L\cup _\varphi \xi ^R$ is a contact structure $\xi $ on the manifold $M=M^L\cup _\varphi M^R$ that is supported by the honest open book $(S,h)$ constructed by pagewise gluing, as defined above.
Since $\iota \colon S_0\to S_{2k}$ is given as a sequence of identifications, the monodromy h may be difficult to reconstruct in a complicated case, and the above construction does not automatically give a factorization in terms of Dehn twists. For sorted open books, however, there is a straightforward way to describe $\iota $ (and thus h) as follows. Suppose now, that $(\{S^L_i\},h^L, \{\gamma _i^{\pm ,L}\})$ and $(\{S^R_i\},h^R,\{\gamma _i^{\mp ,R}\})$ are sorted foliated open books compatible with $(M^L,\xi ^L,\mathcal {F}^L)$ and $(M^R,\xi ^R,\mathcal {F}^R)$ , respectively. Recall that each page $S_i^{\bullet }$ decomposes as the union of a ‘constant’ part $P_i^{\bullet }$ and an idependent part $R_i^{\bullet }$ which is determined by the foliation $\mathcal {F}^{\bullet }$ . By construction, $P_0^{\bullet }\cong P_1^{\bullet }\cong \cdots \cong P_{2k}^{\bullet }$ . These diffeomorphisms are explicit in the construction of the pages $S_i^{\bullet }$ , and together they determine $\iota \vert _{P_0^L\cup P_0^R}=\iota ^L\cup \iota ^R\colon P_0^L\cup P_0^R\to P_{2k}^L\cup P_{2k}^R$ .
As described earlier, sorted foliated open books have the property that $R_0^L$ is isotopic in $M^L$ to $R_+(\mathcal {F}^L)$ . This surface is mapped by $\psi $ onto $R_(\mathcal {F}^R)$ , which in turn is isotopic in $M^R$ to $R_{2k}^R$ . The composition of these three maps gives $\iota \vert _{R_0^L}\colon R_0^L\to R_{2k}^R$ , and we can similarly obtain $\iota \vert _{R_0^R}\colon R_0^R\to R_{2k}^L$ . This yields a concrete description of the entire map $\iota $ on the remaining parts $R_{2k}^L\cup R_{2k}^L$ . Together, the above maps determine h everywhere.
3 Construction of the contact invariant
3.1 Bordered manifold associated to a triple
Let $(M,\xi ,\mathcal {F})$ be a foliated contact threemanifold with signed singular foliation $\mathcal {F}=({\widetilde {\pi }}, H=H_\cup H_+,E = E_\cup E_+)$ . Set $\mathsf {\Gamma }:={\widetilde {\pi }}^{1}(0)$ so that $\mathsf {\Gamma }$ is a disjoint union of oriented leaves connecting positive to negative elliptic points. For each leaf I, let $e_+(I)$ and $e_(I)$ be the corresponding positive and negative elliptic points, respectively. Choose $\epsilon>0$ small enough so that $\widetilde {\pi }^{1}[\epsilon , \epsilon ]$ contains no critical points of ${\widetilde {\pi }}$ . Note that then ${\widetilde {\pi }}^{1}[0,\epsilon ]$ is a union of disks, one disk containing each leaf $I\subset \mathsf {\Gamma }$ and denoted by $R_+(I)$ . Similarly, denote the connected component of ${\widetilde {\pi }}^{1}[\epsilon ,0]$ containing I by $R_(I)$ . Write $R_+(\mathsf {\Gamma })= \bigcup _{I\subset \mathsf {\Gamma }} R_{+}(I)$ and $R_(\mathsf {\Gamma })= \bigcup _{I\subset \mathsf {\Gamma }} R_{}(I)$ . Set $F=\partial M\setminus (R_+(\mathsf {\Gamma })\cup R_(\mathsf {\Gamma }))$ .
Next, we use the foliation to define a natural parametrization of F via an arc diagram $\mathcal Z=(Z,a,m)$ . For each hyperbolic point $h_i$ , let $\delta _i$ be the union of the two stable separatrices at $h_i$ if $h_i$ is positive or the union of the two unstable separatrices at $h_i$ if $h_i$ is negative. Let $Z(I)$ be a pushoff of ${\widetilde {\pi }}^{1}(\epsilon )\cap R_(I)\subset \partial F$ into F satisfying the following:

• $\partial Z(I)=\left (e_(I)\right )\cup e_+(I)$ .

• If $\delta _i$ has an endpoint at $e_\pm (I)$ , then $Z(I)$ intersects $\delta _i$ in a unique point; otherwise, $\delta _i$ and $ Z(I)$ are disjoint.
Let $Z=\sqcup _{I\subset \mathsf {\Gamma }} Z(I)$ . Define a to be the set of all intersection points of Z with the union of the $\delta _i$ , and let m be the pairing induced on the points in a by $\delta _i$ . See Figure 3.1. Observe that Z divides F into two subsurfaces: a surface containing all hyperbolic points (shown in white on Figure 3.1) and a union of disks (shown in dark red on Figure 3.1). Let $e_i$ be the curve that is the intersection of $\delta _i$ with the white surface. Note that after removing the disks bounded by the circles $I\cup Z(I)$ from F, and decomposing the resulting surface along the union of $e_i$ , we get a disjoint union of disks so that each one of them contains exactly one of the leaves in $\pi ^{1}(\epsilon )$ on its boundary. Thus, $\mathcal Z=(Z,a,m)$ parametrizes F as a $\beta $ type arc diagram as defined in [Reference ZarevZar10, Definition 3.2]; that is, F can be identified with $F(\mathcal {Z})$ .
Definition 3.1. The triple $(M,\mathsf {\Gamma },\mathcal Z)$ described above is the bordered sutured manifold associated to the foliated contact threemanifold $(M,\xi ,\mathcal {F})$ .
3.2 Adapted bordered Heegaard diagram
Let $(\{S_i\}_{i=0}^{2k},h, \{\gamma _i^\pm \})$ be an abstract sorted foliated open book compatible with the foliated contact threemanifold $(M,\xi ,\mathcal {F})$ . In what follows, we will describe a generator in a bordered sutured Heegaard diagram $\mathcal {H} = (\Sigma , {\boldsymbol {\beta }},{\boldsymbol {\alpha }}, \mathcal Z)$ constructed from the data of $(\{S_i\},h, \{\gamma _i^\pm \})$ . Let $g_i$ be the genus of $S_i$ , and let $n_i$ be the number of boundary components of $S_i$ . Recall that the boundary of the cornered surface $S_i$ is $B\cup A_i$ , where B is a union of circles and arcs, and $A_i$ is a union of intervals only.
We let $\Sigma =S_{0}\cup _{B}S_{0}$ . In order to distinguish the two copies, we will write
but we emphasize that $S_\epsilon $ can be identified with $S_0$ . The surface $\Sigma $ has genus $2g_0+n_01$ and $A_0$ boundary components. For $i\in H_+$ , let $\gamma _i^+$ be the $S_{\epsilon }$ copy of the associated sorting arc. The endpoints of $\gamma _i^+$ lie near the $E_+$ end of intervals of $A_\epsilon $ . Isotope the arcs $\{\gamma _i^+\}$ (simultaneously, to preserve disjointness) near the endpoints by pushing them along $\partial \Sigma $ in the direction opposite the orientation of the boundary until the endpoints all lie in $I_+\subset A_0$ ; the isotopy stops after crossing $E_+$ and before encountering $\cup _{j\in H_}h(\gamma _j^)\subset S_0$ . Call the resulting arcs $\beta _i^a$ .
For $i\in H_$ , consider the $S_{2k}$ copies of the sorting arcs $\gamma _i^$ , and let $\beta _i^a = h(\gamma _i^)$ on $S_0$ . Write ${\boldsymbol {\beta }}^a = \{\beta _1^a, \ldots , \beta _{2k}^a\}$ . For convenience, instead of $\beta _i^a$ we will sometimes write $\beta _i^{\pm }$ if $i\in H_{\pm }$ .
For $i\in H_{+}$ , let $b_i^+ = \beta _i^a\cap S_{\epsilon }$ . That is, $b_i^+$ is the arc $\beta _i^a$ with the two end segments that lie on $S_0$ removed so that $\partial b_i^+$ lies on B. We claim that after cutting $S_{\epsilon }$ along the arcs $b_i^+$ , each connected component contains at least one interval of $A_{\epsilon }$ .
Lemma 3.2. Each component of $S_{\epsilon }\setminus \cup _{i\in H_+}b_i^+$ contains an interval component of $A_{\epsilon }$ , and hence at least one point in $E_+$ , and, equivalently, at least one point in $E_$ .
Proof. Recall that moving along any given interval of $A_{\epsilon }$ , we encounter curves $\gamma _i^+$ indexed in decreasing order. Thus, after the isotopy, moving along any given interval of B, we encounter curves $b_i^+$ indexed in decreasing order as well. Now, suppose there is a component C of $S_{\epsilon }\setminus \cup _{i\in H_+}b_i^+$ that does not intersect $A_{\epsilon }$ . Then the boundary of C consists of possibly some complete binding circles and at least one circle that alternates between intervals of B and entire arcs $b_i^+$ . Since C is a subsurface of $S_{\epsilon }$ , the orientation on B agrees with the orientation on $\partial C$ . When traversing this circle of $\partial C$ , it follows that each interval of B starts at an intersection point with some $b_i^+$ and ends at an intersection point with some $b_j^+$ , where $j<i$ , thus inducing a circular $<$ ordering on integral indices, which is a contradiction.
Let ${\boldsymbol b} = \{b_1, \ldots , b_{2g_0+n_0+A_0k2}\}$ be a set of cutting arcs for $P_{\epsilon }\subset S_{\epsilon }$ disjoint from ${\boldsymbol {\beta }}^a$ and with endpoints on B so that each connected component of $S_{\epsilon }\setminus ({\boldsymbol b}\cup {\boldsymbol {\beta }}^a) = S_{\epsilon }\setminus ({\boldsymbol b}\cup \{b_i^+\}_{i\in H_+})$ is a disk with exactly one interval of $A_{\epsilon }$ on its boundary. Note that Lemma 3.2 guarantees this can be achieved. In other words, ${\boldsymbol b}$ is a basis for $H_1(P_\epsilon , B)$ . Recalling the identification $S_\epsilon =S_0$ , we may push $b_i\subset S_0$ through M to lie on $S_0$ again and define
where $\iota $ is the identification of $P_0$ with $P_{2k}$ from Section 2.3.5. Write ${\boldsymbol {\beta }}^c=\{\beta _1, \dots , \beta _{2g_0+n_0+A_0k2}\}$ .
Note that by modifying the isotopy that transforms $\{\gamma _i^+\}$ into $\{\beta _i^+\}$ , possibly forcing it to happen in a smaller neighborhood of $E_+$ , we may assume that all curves in ${\boldsymbol {\beta }}^a \cup {\boldsymbol {\beta }}^c$ are pairwise disjoint.
For each cutting arc $b_i\in \boldsymbol b$ on $S_\epsilon $ , let $a_i$ be an isotopic curve formed by pushing the endpoints negatively along the boundary so that $a_i$ and $b_i$ intersect once transversely. Similarly, for each arc $b_j^+$ , let $\tilde a_j$ be an isotopic curve formed by pushing the endpoints negatively along the boundary so that $\tilde a_j$ and $b_j^+$ intersect once transversely. We ‘double’ each of these arcs to form the $\alpha $ circles which define the handlebody $S_0\times [0,\epsilon ]$ . Namely, define
and write ${\boldsymbol {\alpha }} = \{\widetilde \alpha _i\}_{i\in H_+}\cup \{\alpha _1, \dots , \alpha _{2g_0+n_0+A_0k2}\}$ . Finally, let
We obtain $\mathcal Z = (Z, \partial {\boldsymbol {\beta }}, m)$ , where m matches a pair of points if they belong to the same $\beta $ arc, and we get an identification of $G(\mathcal Z)$ with $Z\cup {\boldsymbol {\beta }}^a\subset \Sigma $ .
We say that a bordered sutured Heegaard diagram constructed as above is adapted to the sorted abstract foliated open book $(\{S_i\},h, \{\gamma _i^\pm \})$ .
Let $\mathcal {H} = (\Sigma , {{\boldsymbol {\alpha }}}, {{\boldsymbol {\beta }}},\mathcal Z)$ be a bordered sutured Heegaard diagram adapted to $(\{S_i\},h, \{\gamma _i^\pm \})$ . Using the notation introduced above, define the set
as the unique intersection points
Example 3.3. We show the stepbystep construction of the bordered sutured Heegaard diagram adapted to the sorted foliated open book from Example 2.24. To simplify notation, we write $S_i$ and h instead of the $S_i'$ and $h'$ labels used in Example 2.24.
In Figure 3.2, we see the sorting arcs $\gamma _i^+$ and their images $\beta _i^+$ after the isotopy taking their endpoints to $I_+\subset A_0$ . The arcs $b_i^+$ are the intersections $\beta _i^+ \cap S_\epsilon $ .
Figure 3.3 shows the sorting arcs $\gamma _{i}^$ on $S_0$ and, after mirroring, their images $\beta _i^ = h(\gamma _{i}^)$ on $S_0$ ; recall that in this example h is the identity. Recall that ${\boldsymbol {\beta }}^a = \{\beta _i^{+} \mid i\in H_+\}\cup \{\beta _i^{} \mid i\in H_\}$ .
Observe that in this example we do not need any cutting arcs $\boldsymbol b$ because $S_\epsilon \setminus \{b_i^+\}_{i=1,3}$ consists of two disks, each with exactly one interval of $A_\epsilon $ on its boundary; see Figure 3.4. Therefore, ${\boldsymbol {\beta }}^c$ is empty, and we have no $a_i$ arcs, either.
We next define $\tilde a_i$ by pushing the endpoints of $b_i^+$ negatively along the boundary so that $\tilde a_i$ and $b_i^+$ intersect once transversely. Figure 3.5 shows the arcs $\tilde a_i$ on $S_\epsilon $ and their mirror images on $S_0$ . Glued together, they form $\widetilde {\alpha }_i$ . Since this example has no $\alpha _i$ curves, we have ${\boldsymbol {\alpha }}= \{\widetilde \alpha _i\}_{i\in H_+}$ .
Finally, Figure 3.6 illustrates the bordered sutured Heegaard diagram adapted to the sorted foliated open book of Example 2.24. In this example, the set of intersection points $\mathbf {x}$ consists only of $x_i^+ = \tilde a_i\cap b_i^+\in S_{\epsilon }$ for $i\in H_+$ .
We will use $\mathbf {x}$ to define two contact invariants in bordered sutured Floer homology. In order to do that, we first show that $\mathcal {H}$ is an admissible diagram for the bordered sutured manifold associated to $(M,\xi ,\mathcal {F})$ .
Proposition 3.4. Let $(\{S_i\},h, \{\gamma _i^\pm \})$ be a sorted abstract foliated open book supporting a foliated contact threemanifold $(M,\xi ,\mathcal {F})$ . Suppose $\mathcal {H} = (\Sigma , {\boldsymbol {\alpha }},{\boldsymbol {\beta }},\mathcal Z)$ is a bordered sutured Heegaard diagram adapted to $(\{S_i\},h, \{\gamma _i^\pm \})$ . Then $\mathcal {H}$ gives an admissible bordered Heegaard diagram for the bordered sutured manifold $(M,\mathsf {\Gamma },\mathcal Z)$ associated to $(M,\xi ,\mathcal {F})$ .
Proof. Starting from a sorted foliated open book $(\{S_i\},h, \{\gamma _i^\pm \})$ , we may either first construct a foliated contact manifold $(M,\xi ,\mathcal {F})$ as in Section 2.3.1 and an associated bordered sutured manifold $(M,\mathsf {\Gamma },\mathcal Z)$ as in Section 3.1, or we may start by constructing a bordered sutured Heegaard diagram $\mathcal {H} = (\Sigma , {\boldsymbol {\alpha }},{\boldsymbol {\beta }},\mathcal Z)$ as above. To temporarily distinguish between the two constructions of the set of matched arcs, we will in fact temporarily replace this notation with $\mathcal {H} = (\Sigma , {\boldsymbol {\alpha }},{\boldsymbol {\beta }},\mathcal Z')$ , with $Z'$ the set of arcs in $\mathcal Z'$ . We may then use the construction of Section 2.2.2 to produce a bordered sutured manifold. We wish to verify that the bordered sutured manifold arising this way is $(M, \mathsf {\Gamma }, \mathcal Z)$ .
Recall that the foliated contact manifold $(M, \xi , \mathcal {F})$ is built via saddle cobordisms specified by the sorting arcs. We intend for the $\alpha $ curves to specify the handlebody from $S_0$ to $S_{\epsilon }$ and for the $\beta $ curves to specify the handlebody from $S_{\epsilon }$ to $S_0$ . We will start by modifying the graph $G(\mathcal Z')\subset \Sigma $ and the bordered sutured structure on $\partial M$ via isotopy, allowing us to identify the Heegaard surface $\Sigma $ with the surface $S_\epsilon \cup _BS_0$ in M in such a way that $\partial \Sigma \setminus Z'$ is identified with $\mathsf {\Gamma }$ . (This is equivalent to recovering $\mathsf {\Gamma }$ as $(\partial \Sigma \setminus Z') \times \{\frac {1}{2}\}$ in a thickened copy of the surface $\Sigma \times [0,1]$ .) Recall that we chose $Z'$ to be the union of $A_0$ with the arcs $\bigcup _{I\subset A_{\epsilon }}(I_+\cup I_)$ ; the purpose of this choice is to enable a simple discussion of gluing in Section 5. However, we may isotope $Z'$ inward along the arcs $I_{\pm }$ to $A_0$ . On $\partial M$ , we may isotope Z (resp. $\mathsf {\Gamma }$ ) through the disks bounded between Z and $A_0$ (resp. $A_0$ and $A_\epsilon $ ) and identify it with $A_0$ (resp. $A_\epsilon $ ). This isotopy can be chosen so that it carries the matched points on Z to the matched points on $Z'$ (as identified with $A_0$ ). Note that under this isotopy the edges $e_i$ of the graph