1

数理解析研究所講究録
第2004巻 2016年 1-12

Heegaard Floer homology for embedded bipartite graphs
Yuanyuan Bao
Graduate School of Mathematical Sciences, the University of Tokyo

Introduction

1

Heegaard Floer theory defines topological invariants for many low‐dimensional topolog‐
ical objects.

The word(Heegaard indicates that it is defined on a Heegaard diagram

associated with the given topological object, and the word Floer

comesfrom the factthat it is a special case of the Lagrangian intersection Floer homology, where the symplec‐tic manifold and its Lagrangian pair are constructed from the given Heegaard diagram.
In this note, we introduce a definition of the Heegaard Floer homology for an embeddedbalanced bipartite graph in a closed oriented 3‐manifold.

Details of the definition are

included in the preprint [2]. For simplicity, we assume that the ambient manifold M is arational homology 3‐sphere.
In Section 2, we review the definition of a Heegaard diagram for M and that for a link
in M

M

.

.

Then we show how to generalize the definition to a balanced bipartite graph in

In Section 3, we brief the definition of the Heegaard Floer complex for M and for alink in it.

Among various versions of the complex, we focus on the minus‐version since itis the most useful case and can be used to reconstruct the other versions. Then we showageneralization of the definition to a balanced bipartite graph in M

we.

In the last section,

discuss two combinatorial aspects of the theory, the definitions based on grid diagramand Kauffman state.
The topological invariance of the Heegaard Floer homology will not be discussed here.

Please refer to the original papers for the proofs.

Heegaard diagram for a 3 manifold, a link, or a balanced bi‐

2partite graph
2.1

For a 3‐manifold

Given a closed oriented 3‐manifold M let ( $\Sigma$, $\alpha$, $\beta$) be a Heegaard diagram for M where,,$\Sigma$ is a genus g closed oriented surface, and $\alpha$=\{$\alpha$_{1}, $\alpha$_{2}, \cdots , $\alpha$_{g}\} and $\beta$=\{$\beta$_{1}, $\beta$_{2}, \cdots, $\beta$_{g}\}2

Pa 1: A2‐pointed Heegaard diagram for the 3‐sphere (left figure), and a single‐pointed Heegaard diagram
for the right‐handed trefoil knot (right figure).

are

two sets of pairwisely

definedadisjoint simple closed curves on $\Sigma$

Ozsváth and Szabó.

[7]

topologicial invariant, called Heegaard Floer homology, for a 3‐manifold byusing its Heegaard diagram. In their definition, an extra datum w\in $\Sigma$\backslash ( $\alpha$\cup $\beta$) which,is called a base point, is needed to make the invariant non‐trivial. The data ( $\Sigma$, $\alpha$, $\beta$, w)

is called a single‐pointed Heegaard diagram for M.

It is usually convenient to define the chain complex on a multi‐pointed Heegaard dia‐gram, which can be obtained from a single‐pointed Heegaard diagram by applying(0,3)

stabilizations. The definition is as follows.
Definition 2.1. Let n\geq 1 be an integer.

An n ‐pointed Heegaard diagram for M is aquartet ( $\Sigma$, $\alpha$, $\beta$, w) which satisfies the following conditions.
,$\Sigma$ isaclosed oriented genus gsurface, which is called the Heegaard surface, and$\alpha$=\{$\alpha$_{1}, $\alpha$_{2}, \cdots , $\alpha$_{d}\} and $\beta$=\{$\beta$_{1}, $\beta$_{2},)$\beta$_{d}\} are two sets of d pairwisely disjoint

simple closed curves on $\Sigma$ where n=d-g+1.
,

Attaching 2‐handles to $\Sigma$ along curves in $\alpha$ (resp. $\beta$ ), we get an n‐punctured genus
g handlebody U_{ $\alpha$} (resp. U_{ $\beta$} ). The union U_{ $\alpha$}\displaystyle \bigcup_{ $\Sigma$}U_{ $\beta$} is the 3‐manifold \mathrm{M} with 2n‐punctures. The orientation of $\Sigma$ is induced from that of U_{ $\alpha$} which in turn coincides,

with that of M.

\{A_{i}\}_{i=1}^{n} (resp. \{B_{i}\}_{i=1}^{n} ) be the connected components of $\Sigma$\backslash $\alpha$ (resp. $\Sigma$\backslash $\beta$ ).
w=\{w_{1}, w_{2}, \cdots, w_{n}\} is a set of n points in $\Sigma$\backslash ( $\alpha$\cup $\beta$) so that w_{i}\in A_{i}\cap B_{i}(by relabelling \{B_{i}\}_{i=1}^{n} if necessary), which are called the base points.

Let

Then

See Figure 1(left) for an example of 2‐pointed Heegaard diagram for the 3‐sphere.

A convenient way to understand the construction of a Heegaard diagram is to consider
itscorresponding Morse function.

suppose f:

Chooseageneric Riemannian metric \mathfrak{g} on M andM\rightarrow \mathbb{R} is a self‐indexed Morse function. Then (f, \mathfrak{g}) induces a multi‐pointed

Heegaard diagram for M,

where the Heegaard surface $\Sigma$:=f^{-1}(3/2) and $\alpha$ (resp. $\beta$ )3(resp. descending disks)(resp. two) critical points. The base points in w are chosen in a waythat each component of $\Sigma$\backslash $\alpha$ or $\Sigma$\backslash $\beta$ contains exactly one base point. For any pointp\in $\Sigma$\backslash ( $\alpha$\cup $\beta$) there exists a path $\gamma$_{p}\subset M from an index zero critical point to an index
three one so that \mathfrak{g}(\dot{ $\gamma$}_{p}, \cdot)=df Namely it is the flow line passing through p (see Figure
2 for an illustration). Since the base points come from different components of $\Sigma$\backslash $\alpha$ or$\Sigma$\backslash $\beta$ it is easy to see that \{$\gamma$_{p}\}_{p\in w} are pairwisely disjoint simple paths.
is the set of intersection curves of $\Sigma$ with the ascending disksof the index one

,

.

,

2.2

For a link

Consider an oriented link L\subset M.

An n ‐pointed Heegaard diagram ( $\Sigma$, $\alpha$, $\beta$, w, z) for: M\rightarrow \mathbb{R} and a Riemannian metric \mathfrak{g} so that
(M, L)i) ( $\Sigma$, $\alpha$, $\beta$, w) is an n ‐pointed Heegaard diagram for M ii) z is a finite set of points in$\Sigma$\backslash ( $\alpha$\cup $\beta$\cup w) for which L=\displaystyle \bigcup_{p\in w\cup z}$\gamma$_{p} and iii) the orientation of L makes the paths

is defined by a Morse function f,,\{$\gamma$_{p}\}_{p\in w} direct downwards and \{$\gamma$_{p}\}_{p\in z} upwards. Note that in this case we always have|w|=|z| so there are totally 2n base points on the Heegaard surface. See Figure 2 for a,schematic illustration of the construction, and also see Figure 1 (right) for an example ofasingle‐pointed Heegaard diagram of the right‐handed trefoil knot in S^{3}.

Remark 2.2. Note that a single‐pointei Heegaard diagram of a knot K is nothing but a 1‐

bridge decomposition of the knot (see [6] for the definition). The minimal Heegaard genus
is a knot invariant which is closely related to the tunnel number of the knot.
asfar as the author knows,noNevertheless,research is known about the relation between Heegaard

Floer homology and these geometric invariants.

2.3

For a bipartite graph

A graph G with the vertex set V and the edge set E is called a bipartite graph if V is adisjoint union of two non‐empty sets V_{1} and V_{2} so that every edge in E is incident to bothV_{1} and V_{2}.

We use G_{V_{1},V_{2}} to denote the graph with the choice of (V_{1}, V_{2})the graph G_{V_{1},V_{2}} is called balanced.

.

If |V_{1}|=|V_{2}|,Furthermore, we call an orientation for G_{V_{1},V_{2}} balanced

if there are n :=|V_{1}| edges \{e_{i}\}_{i=1}^{n} directing from V_{1} to V_{2} and the endpoints of which
occupyV_{1} and V_{2} and the other edges direct from V_{2} to V_{1},example, where the solid line edges are \{e_{1}, e_{2}\}.

.

See Figure 4 (left) for an

We assume that all the graphs in thispaper have no isolated vertices and single‐valency vertices.

Two smooth embeddings f_{i} : G_{V_{1},V_{2}}\leftarrow+M are said to be ambient isotopic if there is ahomeomorphism of M isotopic to the identity map, sending f_{1}(G_{V_{1},V_{2}}) to f_{2}(G_{V_{1},V_{2}}) and4

3‐manifold

knot / link

bipartite graph((\}r\tiangle ft:\}.cdot\cdot.r\cdot\mathrm{} \} |\cdot\cdot\bullet\}^{:.cdot\ .cdot:_{\mahr{}|_:=^r{$\eta} :_{\ulcorne}$\ta.:cdo\bakslh\cdot. \cdot.sim=\cdot. \cdot.

2 : The schematic diagram for a Morse function of a 3‐manifold. The flow lines constitute a link or agraph embedded in the manifold.

f_{1}(V_{i}) to f_{2}(V_{i}) for i=1 2. The Heegaard Floer homology to be defined is an invariant,

under the ambient isotopy.
To define the homology, we consider a multi‐pointed Heegaard diagram ( $\Sigma$, $\alpha$, $\beta$, w :=\{w_{i}\}_{i=1}^{n} z ) for (M, G_{V_{1},V_{2}}) with a balanced orientation. It is defined by a Morse function
f : M\rightarrow \mathbb{R} and a Riemannian metric \mathfrak{g} so that i) ( $\Sigma$, $\alpha$, $\beta$, w) is an n‐pointed Heegaard
diagram for M ii) z is a finite set of points in $\Sigma$\backslash ( $\alpha$\cup $\beta$\cup w) for which G_{V_{1},V_{2} =\displaystyle \bigcup_{p\in w\cup z}$\gamma$_{p}and iii) the orientation of G_{V_{1},V_{2}} makes \{e_{i}\}_{i=1}^{n}=\{$\gamma$_{p}\}_{p\in w} direct downwards and \{$\gamma$_{p}\}_{p\in z}
upwards. Since G_{V_{1},V_{2}} has no isolated vertices and single‐valency vertices, we have |z|\geq|w| but the position of w and z can be complicated. See Figure 2 for an illustration.
,,,

The Heegaard Floer complex \mathrm{C}\mathrm{F}^{-}( $\Sigma$, $\alpha$, $\beta$, w, z)

3
3.1

Nearly symmetric almost complex structures

Let ( $\Sigma$, $\alpha$, $\beta$, w) be an n ‐pointed Heegaard diagram of M as in Def. 2.1. Consider thed‐fold symmetric product of $\Sigma$,

Sy \mathrm{m}^{}( $\Sigma$)=$\Sigma$^{\times d}/S_{d},where S_{d} is the symmetric group of degree d , and let\mathbb{T}_{ $\alpha$}=$\alpha$_{1}\times$\alpha$_{1}\times\cdots\times$\alpha$_{d} and \mathbb{T}_{ $\beta$}=$\beta$_{1}\times$\beta$_{1}\times\cdots\times$\beta$_{d}.
Then Sy \mathrm{m}^{} ( $\Sigma$ ) is a 2d‐dimensional smooth manifold, whose local coordinate chart canbe defined by the correspondence between coefficients and roots of a d‐variable complex

5polynomial. Then \mathb {T}_{ $\alpha$} and \mathb {T}_{ $\beta$} are two d‐dimensional submanifolds of Sy \mathrm{m}^{} ( $\Sigma$ ).

Supposethat \mathb {T}_{ $\alpha$} and \mathb {T}_{ $\beta$} are in general position.

Let( $\eta$,

be a Kähler form on $\Sigma$

Consider the quotient map $\pi$ :.

Then it induces$\Sigma$^{\times d}\rightar ow \mathrm{S}\mathrm{y}\mathrm{m}^{d}( $\Sigma$)aKähler form($\eta$^{\times d}, \mathfrak{j}^{\times d}) on $\Sigma$^{\times d}.

The complex structure \mathfrk{j} gives rise to a.

complex structure Sy \mathrm{m}^{} ( \mathfrk{j} ) on Sy \mathrm{m}^{} ( $\Sigma$ ) for which $\pi$ is a holomorphic map. Let
diagonal of Sy \mathrm{m}^{} ( $\Sigma$ ) Namely

D be the.

D := { \{x_{1}, x_{2}, \cdots,x_{n}\}\in \mathrm{S}\mathrm{y}\mathrm{m}^{d}( $\Sigma$)|x_{i}=x_{j} forsome

i\neq j }.

$\eta$^{\times d} is invariant under
the action of S_{d} we get a Kähler form (Sym ( $\eta$), Sym( \mathfrk{j} )) on Sy \mathrm{m}^{} ( $\Sigma$)\backslash D Note that
Then

$\pi$

induces a covering map away from the diagonal.

Since

.

,

the submanifolds \mathb {T}_{ $\alpha$} and \mathb {T}_{ $\beta$} keep away from the diagonal, and are two Lagrangian andtotally real tori with respects to (Sym ( $\eta$), \mathrm{S}\mathrm{y}\mathrm{m}^{d}(\mathfrak{j}) ).
For a finite set of points \{p_{ $\lambda$}\}_{ $\lambda$\in $\Lambda$}\subset $\Sigma$\backslash ( $\alpha$\cup $\beta$) find an open set V\subset \mathrm{S}\mathrm{y}\mathrm{m}^{d}( $\Sigma$) keeping,

away from \mathb {T}_{ $\alpha$} and \mathb {T}_{ $\beta$} so that\{p_{ $\lambda$}\}_{ $\lambda$\in $\Lambda$}\times \mathrm{S}\mathrm{y}\mathrm{m}^{d-1}( $\Sigma$)\cup D\subset V.
An almost complex structure J on Sy \mathrm{m}^{} ( $\Sigma$ ) is called (\mathrm{j}, $\eta$, V) ‐nearly symmetric if i) J
iscompatible with Sy \mathrm{m}^{} ( $\eta$ ) on Sy \mathrm{m}^{}( $\Sigma$)\backslash V and ii) J=\mathrm{S}\mathrm{y}\mathrm{m}^{d}(\mathfrak{j}) over Vof$\eta$, V) ‐nearly symmetric almost complex structures is denoted \mathcal{J}(\mathfrak{j}, $\eta$, V)\mathrm{S}\mathrm{y}\mathrm{m}^{d}3.2\in \mathcal{J}(\mathrm{j}, $\eta$, V)

The space.

Obviously.

.

The chain complex for a manifold

For an n ‐pointed Heegaard diagram ( $\Sigma$, $\alpha$, $\beta$, w) of M the definition of the chain com‐,plex (\mathrm{C}\mathrm{F}^{-}( $\Sigma$, $\alpha$, $\beta$, w), \partial^{-})

is an analogue of that of Lagrangian intersection Floer com‐

plex. It is a free \mathbb{F}[U_{1}, U_{2}, \cdots, U_{n}] ‐module generated by the intersection points \mathb {T}_{ $\alpha$}\cap \mathb {T}_{ $\beta$},where \mathbb{F}[U_{1}, U_{2}, \cdots, U_{n}] is the n ‐variable polynomial ring with coefficient \mathb {F} :=\mathbb{Z}/2\mathbb{Z}.

The

differential is defined by counting pseudo‐holomorphic disks connecting two generators.

Precisely, for x\in \mathbb{T}_{ $\alpha$}\cap \mathbb{T}_{ $\beta$},\displaystyle\partial^{-}(x)=\sum_{y\in\mathrm{T}_{$\alpha$}\cap\mathrm{T}_{$\beta$}\sum_{\ $\phi$\in$\pi$_{2}(x,y)|$\mu$($\phi$)=1\}\#\hat{\mathcal{M}_{J_{S}($\phi$)\cdotU_{1}^{n_{w_{1}($\phi$)}U_{2}^{n_{w_{2}($\phi$)}\cdotsU_{n}^{n_{w_{n}($\phi$)}y
The notations $\pi$_{2}(x, y)

,

.

(1)

$\mu$( $\phi$) \hat{\mathcal{M} _{J_{S} ( $\phi$) and n_{w_{i} ( $\phi$) are defined as follows. Let \mathbb{D} be the unit,

disk on the complex plane. A smooth map
u :(\mathbb{D}, -i, i)\rightarrow(\mathrm{S}\mathrm{y}\mathrm{m}^{d}( $\Sigma$), x, y)sending \{s+it\in\partial \mathbb{D}|s\geq 0\} (resp. \{s+it\in\partial \mathbb{D}|s\leq 0\} ) to \mathb {T}_{ $\alpha$} (resp. \mathb {T}_{ $\beta$} ) is called a
Whitney disk from x to y Let $\pi$_{2}(x, y) be the set of homotopy classes of all Whitney.

6(also called the formal dimension or
expected dimension in some literatures) of $\phi$ Fix a path J_{s}\subset \mathcal{J}(\mathfrak{j}, $\eta$, V) where \{p_{ $\lambda$}\}_{ $\lambda$\in $\Lambda$}
disks from x to y.

Then $\mu$( $\phi$) is the Maslov index.

in the definition of V is chosen to be w

.

,

Let

\mathcal{M}_{J_{\mathcal{S} }( $\phi$)= { u : awhitney disk from x to y|
The translation action of \mathbb{R} on \mathbb{D} induces\mathcal{M}_{J_{S} ( $\phi$)/\mathbb{R},an[u]= $\phi$, \partial_{s}u+J_{s}\partial_{t}u=0 }.

action of \mathbb{R}on\mathcal{M}_{J}.( $\phi$)

which is called the unparameterized moduli space of $\phi$\#$\phi$^{-1}(\{w_{i}\}\times \mathrm{S}\mathrm{y}\mathrm{m}^{d-1}( $\Sigma$))

is called the local

multiplicity of w_{i} in\hat{\mathcal{M} _{J_{S} ( $\phi$):=

Finally, n_{w_{i} ( $\phi$) :=.

$\phi$

Let

.

.

When

$\phi$ is a J_{s^{-}}

holomorphic representative, we have n_{w_{i}}( $\phi$)\geq 0 (Lemma 3.2 [7]).
Theorem 3.1
1.

(Theorems 3.4, 3.18 [7]). For a generic choice of J_{s}\subset \mathcal{J}(\mathrm{j}, $\eta$, V)\mathcal{M}_{J_{S} ( $\phi$) is an oriented manifold of dimension $\mu$( $\phi$)
ifold of dimension $\mu$( $\phi$)-1 and,and\hat{\mathcal{M} _{J_{S} ( $\phi$) is

an

,

we

have

oriented man‐,\hat{\mathcal{M} _{J_{s} ( $\phi$) is compact,

2. when $\mu$( $\phi$)=1,

for any x, y\in \mathbb{T}_{ $\alpha$}\cap \mathbb{T}_{ $\beta$} and $\phi$\in$\pi$_{2}(x, y).

\hat{\mathcal{M} _{J_{S} ( $\phi$) appeared in \partial^{-}(x) is
count its number \#\hat{\mathcal{M} _{J_{S} ( $\phi$) modulo two. Here

From this theorem, we see that for a generic J_{s} , the set

compact 0-\dim manifold. Therefore we canweaonly consider the number modulo two to avoid the discussion of coherent orientationof the moduli spaces.
In order to make theright hand side of (1) a finite sum for each y,thediagram is required to satisfy a technic condition, called weak admissibility.

Heegaard

For detailsof the definition, please refer to [8, Section 3.4]. For this reason, we hereafter assume thatthe Heegaard diagrams are weakly admissible,

Finally we state that the differential defined in (1) respects Spi \mathrm{n}^{} (M)structures of M.

,

the set of spinC‐

In the context of Heegaard Floer theory, a spinC‐structure indicates thehomology class of a nowhere vanishing vector field over M.

Two nowhere vanishing vector

fields are said to be homologous if they are homotopic through nowhere vanishing vector
fields on the complement of a 3‐ball in M.

The base points w provide a maps_{w}:\mathbb{T}_{ $\alpha$}\cap \mathbb{T}_{ $\beta$}\rightar ow \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}^{c}(M),for the definition of which one can refer to [7, Section 3.3]. We have s_{w}(x)=s_{w}(y) when,y appears in the differential of x.

In this way, we have the splitting\displaystyle\mathrm{C}\mathrm{F}^{-}($\Sigma$, $\alpha$, $\beta$,w)=\bigoplus_{s\in\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}^{c}(M)}\mathrm{C}\mathrm{F}^{-}($\Sigma$, $\alpha$, $\beta$,w;s)

.

7

3.3

The chain complex for a link

Consider a Heegaard diagram ( $\Sigma$, $\alpha$, $\beta$, w, z) for a link L\subset M.

The additional basepoints z are used here to construct a filtration to the chain complex (\mathrm{C}\mathrm{F}^{-}( $\Sigma$, $\alpha$, $\beta$, w), \partial
It is defined by the map

s_{w,z}:\mathbb{T}_{ $\alpha$}\cap \mathbb{T}_{ $\beta$}\rightar ow \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}^{c}(M, L),

where Spi \mathrm{n}^{}(M, L) is the set of relative spinC‐structures defined as follows.

Let

v

be

a

nowhere vanishing unit vector field over M\backslash \mathrm{i}\mathrm{n}\mathrm{t}(N(L) for which the restriction of v on\partial N(L) coincides with the canonical vector field, which is defined by the condition thatits orbit on each component of \partial N(L) is a frame zero longitude of the link component.

Let$\chi$(M, L) be the set of such vector fields.

is a closed orbit of the extension.

Then v can be extended to M so that L

Two elements in$\chi$(M, L) are said to be homologous

relative to L(\sim(M,L)) if they are homotopic through nowhere vanishing vector fields onthe complement of a 3‐ball in M\backslash int (N(L)).

Let

Spi \mathrm{n}^{}(M, L) := $\chi$(M, L)/\sim(M,L)
For the construction of s_{w,z} , please refer to [8, Section 3.6]. The extension of v\in $\chi$(M, L)
to M definesamap

null‐homologous$\xi$ : Spi \mathrm{n}^{}(M, L)\rightar ow \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}^{c}(M)in M ,

have

we

an

.

We have$\xi$\circ s_{w,z}=s_{w}.

When L isaffine identification Spi \mathrm{n}^{}(M, L)\cong \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}^{c}(M)\oplus \mathbb{Z}^{l},where l is the component number of L.

The restriction on the second factor gives rise to\mathrm{a}\mathb {Z}^{l} ‐filtration to (\mathrm{C}\mathrm{F}^{-}( $\Sigma$, $\alpha$, $\beta$, w), \partial
The associated graded chain complex is denoted by (\mathrm{C}\mathrm{F}\mathrm{L}^{-}( $\Sigma$, $\alpha$, $\beta$, w, z), \partial_{L}We seethat \mathrm{C}\mathrm{F}\mathrm{L}^{-}( $\Sigma$, $\alpha$, $\beta$, w, z) is the same as \mathrm{C}\mathrm{F}^{-}( $\Sigma$, $\alpha$, $\beta$, w) , a free \mathbb{F}[U_{1}, U_{2}, \cdots, U_{n}] ‐module

generated by the intersection points \mathb {T}_{ $\alpha$}\cap \mathb {T}_{ $\beta$}.

The differential is\displaystyle\partial_{L}^{-}(x)=\sum_{y\in\mathb {T}_{$\alpha$}\cap\mathb {T}_{$\beta$} \sum_{\ $\phi$\in$\pi$2(x,y)|$\mu$($\phi$)=1,n_{z}($\phi$)=\{0\} }\#\hat{\mathcal{M} _{J_{s} ($\phi$)\cdotU_{1}^{n_{w_{1} ($\phi$)}U_{2}^{n_{w_{2} ($\phi$)}\cdotsU_{n}^{n_{w_{n} ($\phi$)}y)
where n_{z}( $\phi$) is the set of local multiplicities at $\phi$ of points in

3.4

z.

The chain complex for a graphIn this section we givethe differences.
a

(2)

a

paralleled description as Section 3.3 while keeping in mind

For details about the contents in this section,

Heegaard diagram ( $\Sigma$, $\alpha$, $\beta$, w, z)forabalanced bipartite

additional base points z can be used here to constructcomplex (\mathrm{C}\mathrm{F}^{-}( $\Sigma$, $\alpha$, $\beta$, w), \partialaplease see [2]. Consider

graph G_{V_{1},V_{2}}\subset M.

The

relative grading to the chain

Similar with the case of link, it is defined by the map

s_{w,z}:\mathbb{T}_{ $\alpha$}\cap \mathbb{T}_{ $\beta$}\rightar ow \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}^{c}(M, G_{V_{1},V_{2} ),8where Spi \mathrm{n}^{}(M, G_{V_{1},V_{2}}) is the set of relative spinC‐structures. We give its definition below.
Let X :=M\backslash \mathrm{i}\mathrm{n}\mathrm{t}(N(G_{V_{1},V_{2} ) and consider the decomposition \partial X=\partial_{+}X\cup N(\mathfrak{m})\cup\partial_{-}X,

where N(\mathrm{m}) is\partial_{+}X (resp.

atubular neighborhood of the meridian set \mathrm{m} of the edges on \partial X and\partial_{-}X ) is the intersection of \partial X with a neighborhood of V_{1}Define v_{G} to be the unit vector field on TX|_{\partial X} as follows:

outwards; 2) v_{G}|_{\partial_{-X}}\perp T\partial_{-}X and points inwards; 3)
identification N(\mathfrak{m})=\mathfrak{m}\times[-1, 1]

.

field

1) v_{G}|_{\partial+X}\perp T\partial_{+}X and points

v_{G}|_{N(\mathfrak{m}) =\displaystyle \frac{\partial}{\partial t}(\mathfrak{m}\times\{t\})under theWe perturb v_{ $\gamma$} around \partial N(\mathrm{m}) to make it continuous.

Note that v_{G} only depends on the topology of G
over X(resp. V_{2} ) in M.

Let v be a nowhere vanishing unit vector.

for which v|_{\partial X}=v_{G} , and we denote the set of such vector fields $\chi$(M, G_{V_{1},V_{2}})Two elements in.

$\chi$(M, G_{V_{1},V_{2}}) are said to be homologous relative to G_{V_{1},V_{2}}(\sim(M,G_{V_{1},V_{2}}))

if they are homotopic through nowhere vanishing vector fields on the complement of a

3‐ball in X.

Let

Spi \mathrm{n}^{}(M, G_{V_{1},V_{2}}) := $\chi$(M, G_{V_{1},V_{2}})/\sim(M,G_{V_{1},V_{2}})Note that there isSpi \mathrm{n}^{}(M, G_{V_{1},V_{2}})

.

a

.

free and transitive action of H_{1}(X;\mathbb{Z})\cong H^{2}(X, \partial X;\mathbb{Z})For [v], [w]\in \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}^{c}(M, G_{V_{1)}V_{2} ),the setonlet [v]-[w]\in H_{1}(X;\mathbb{Z}) be their differ‐ence.

Define the Alexander grading on \mathrm{C}\mathrm{F}^{-}( $\Sigma$, $\alpha$, $\beta$, w)

.

It is

a

relative

H_{1}(X;\mathbb{Z}) ‐grading

defined by the map A:(\mathbb{T}_{ $\alpha$}\cap \mathbb{T}_{ $\beta$})\cup\{U_{i}\}_{i=1}^{n}\rightarrow H_{1}(X;\mathbb{Z}) which satisfies the relations

A(x)-A(y)s_{w,z}(x)-s_{w,z}(y)

=

,

and

A(U_{i}) = [m_{w_{i}}].
for any x, y\in \mathbb{T}_{ $\alpha$}\cap \mathbb{T}_{ $\beta$} , where m_{w_{i}} is the meridian of the edge that contains w_{i}.
The map \partial^{-} does not preserve thegrading in general.

We define

a

chain

complex(\mathrm{C}\mathrm{F}\mathrm{G}^{-}( $\Sigma$, $\alpha$, $\beta$, w, z), \partial_{G}^{-}) that preserves the grading, where \mathrm{C}\mathrm{F}\mathrm{G}^{-}( $\Sigma$, $\alpha$, $\beta$, w, z) is thesame as\mathrm{C}\mathrm{F}^{-}( $\Sigma$, $\alpha$, $\beta$, w) The differential is.

\displaystyle\partial_{G}^{-}(x)=\sum_{y\in\mathb {T}_{$\alpha$}\cap\mathb {T}_{$\beta$} \sum_{\ $\phi$\in$\pi$2(x,y)|$\mu$($\phi$)=1,n_{z}($\phi$)=\{0\} }\#\hat{\mathcal{M} _{J_{s} ($\phi$)\cdotU_{1}^{n_{w_{1} ($\phi$)}U_{2}^{n_{w_{2} ($\phi$)}\cdotsU_{n}^{n_{w_{n} ($\phi$)}y
There

are

mainly two algebraic differences with the case of link.

relative grading takes value on H_{1}(X;\mathbb{Z})

,

on

For

a

.

(3)

graph thewhich there is no canonical order in general.

Therefore it is hard to discuss a non‐trivial filtration on it. Another difference is that the
chain complex (\mathrm{C}\mathrm{F}\mathrm{G}^{-}( $\Sigma$, $\alpha$, $\beta$, w, z), \partial_{L}^{-}) in general is not the associated graded complexof (\mathrm{C}\mathrm{F}^{-}( $\Sigma$, $\alpha$, $\beta$, w), \partial

Precisely we mean that there might be some terms in \partial^{-}(x)-\partial_{G}^{-}(x) having the same H_{1}(X;\mathbb{Z}) ‐grading with x for given x\in \mathbb{T}_{ $\alpha$}\cap \mathbb{T}_{ $\beta$}.
Remark 3.2. The chain complex considered here can be regarded as a special case of thedefinition in [1].

Here we choose the sutured manifold to be the graphcomplement and9

EZ 3: Two grid diagrams separately from [5] and [3] for a link and for a transverse graph. The dottedcurves are

the corresponding link and graph.

define the algebra for the boundary so that each edge e_{i} for 1\leq i\leq n is associated with avariable U_{i} and the other edges zero.

Then the algebra becomes \mathbb{F}[U_{1}, U_{2}, \cdots, U_{n}].

Combinatorial aspects of the homology

4
4.1

The

grid diagram

Heegaard Floer homology for a link in the 3‐sphere S^{3} has a completely combi‐

natorial definition, which was introduced in [5]. The chain complex is defined on a grid

diagram of the given link. The generators are the bijections between the set of horizontal
circles and that of the vertical circles, and the differential counts empty rectangles between
two generators. In a similar vein, Harvey and ODonnol [3] defined the grid diagram for atransverse graph (the definition of which was first introduced there) and constructed the

Heegaard Floer homology for it. See Figure 3 for the examples of both cases.
Note that an n\times n grid diagram of a link is in particular an n ‐pointed Heegaard diagram

for the link.

For this

Heegaard diagram, the $\alpha$- and $\beta$ ‐curves are the horizental andvertical circles respectively, and the pesudo‐holomorphic disks are the empty rectangles.

Therefore the combinatorial definition coincides with the definition in Section 3.3.
We remark that a balanced bipartite graph with a balanced orientation naturally gives
rise to a transverse graph, and vise versa.

See Figure 4 for an example. As in the caseof link, if we regard the grid diagram of a transverse graph as a Heegaard diagram of its

corresponding bipartite graph, the definition in [3] and ours coincide with each other.

10\mathrm{H}^{\backslash }4 : shrinking the solid line edges, whose orientation reverses that of the others, of the bipartite graphonthe left, one gets a transverse graph on the right.

Conversely, inserting an edge at each bar of thetransverse graph on the right, we get the bipartite one.

J4yt\mathrm{t}\backslash\backslash_{$\lambda$}\backslash- $\alpha$‐curve—$\beta$ ‐curve\dot{1}^/imes.f_\backlh}*nwro\mat{x}^Jnerow$\bta}p^{xi_nwro\pme}$bta_{v\lmd$sibacklh-\prme^{wdg \simwedg}.naro'\simewpr\cdotsim- weg\alphbcks$\eta_{ lsh\bcka}nwro_{\mthL},(,}

5 : A knot diagram for the right‐handed trefoil knot and its associted Heegaard diagram. The squareintersection point appears in every generator and therefore is omitted on the right.

4.2

Kauffman state

For a knot in S^{3} , its knot diagram in S^{2} defines a Heegaard diagram [10]. See Figure
5 for an example.

[4] and also corresponds
to a generator of the Heegaard Floer chain complex. For an alternating diagram, [10]The set of dots provides a Kauffman state

showed that the differential of the complex is trivial, and the Heegaard Floer homology
is determined by the Alexander polynomial and the signature of the given knot. But for

general diagram, a combinatorial description of the differential is unknown.
For a bipartite graph in S^{3}

diagram in S^{2}

.

,

we

also constructed in[2] a Heegaard diagram from its

We proved that the generators of the Heegaard Floer chain complex forthe graph are the states of the diagram (see Figure 6).

Compare with the case of knot,

there are still many questions to be solved.

Question 4.1 (Y. Bao). Can any two states be connected by transpositions of type I
and II^{(}? Is it possible to calculate the (relative) Alexander grading combinatorially ? For

alternating (there is no standard definition) bipartite graphs,

is theHeegaard Floer

complex completely determined by the Alexander polynomial (up to overall shifts of thegradings) /?

11type 1\mathrm{t}_{\mathrm{V}\upar ow\mathrm{j} \mathrm{e}\mathrm{I}\mathrm{I}
6 : The set of dots represents a generator of the Heegaard Floer complex biult on the diagram (leftfigure). Transpositions of type I and II between two generators (right figure).

\mathrm{g}7 : Slice a knot diagram.
In a recent paper [9], from a knot diagram, Ozsváth and Szabó constructed a bigradedchain complex over \mathbb{F}[U] , the homology of which is shown to be isomorphic to the Heegaard

Floer homology (minus version) of the given knot. It is freely generated by the Kauffman

states, and its differential is defined algebraically, built on bordered Floer homology.
The brief idea is as follows.

Slice the knot diagram at different heights so that eachpiece contains exactly one cap, cup, or crossing. See Figure 7 (left). Ozsváth and Szabó
announced that for each piece they constructed a differential graded algebra for the top

boundary and an A^{\infty} algebra for the bottom boundary and a bimodule for the piece inbetween and proved that the tensor product of all pieces reproduces the Heegaard Floer

homology of the given knot. For the case of graph, we may ask the following question.

Question 4.2. For a bipartite graph, or a general graph, is it possible to construct such acomplex? The first step might be how to define a module for the piece in Figure 7(right).

12References[1] A. S. Alishahiand E. Eftekhary, A refinement of sutured Floer homology, J. Symplec‐
tic Geom. 13 no. 3 (2015) pp. 609‐743.
[2] Y. Bao, Heegaard Floer homology for embedded bipartite graphs, arXiv:1401.6608v2,(2016).

[3] S. Harvey

and

D.

ODonnol,Heegaard

Floer

homologyofspatial graphs,

arXiv:1506.04785v1, (2015).

[4] L. H. Kauffman, Formal Knot Theory, Mathematical Notes, 30. Princeton University
Press, Princeton, NJ, (1983).
[5] C. Manolescu, P. Ozsváth, Z. Szabó and D. Thurston, On combinatorial link Floer
homology, Geometry & Topology 11 (2007) pp. 2339‐2412.
[6] K. Morimoto, Tunnel number, 1‐bridge genus and \mathrm{h} ‐genus of knots, Topology and its
Applications, Vol. 146‐147 (2005), pp. 149‐158.

[7] P. Ozsváth and Z. Szabó, Holomorphic disks and topological invariants for closed
three‐manifolds, Ann. of Math. (2), 159 (2004), pp. 1027‐1158.
[8] P. Ozsváth and Z. Szabó, Holomorphic disks, link invariants and the multi‐variable
Alexander polynomial, Algebraic & Geometric Topology 8 (2008) pp. 615‐692.
[9] P. Ozsváth and Z. Szabó, Kauffman states, bordered algebras, and a bigraded knot
invariant, arXiv: 1603. 06559vl, (2016).
[10] P. Ozsváth and Z. Szabó, Heegaard Floer homology and alternating knots, Geometry& Topology 7 (2003) pp. 225‐254.
Graduate School of Mathematical Sciences
the University of Tokyo
3‐8‐1 Komaba Meguro‐ku Tokyo 153‐8914

JAPAN\mathrm{E} ‐mail address:

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