69

The Heegaard Floer complexes of a trivalent graph defined on
two Heegaard diagrams
Yuanyuan Bao
Graduate School of Mathematical Sciences, the University of Tokyo

1

Introduction

In the report, let G denote an embedded oriented connected trivalent graph in S^{3} without
source or sink. Namely each vertex of G is one of the following two cases.

orFigure 1: Vertices of G.

Choose a generic point \delta in

G and assign a coloring

c

on

G.

Then we consider the

Heegaard Floer complex CFG(G, \delta, c) . The homology of CFG (G, \delta, c) , which is denotedby HFG(G, \delta, c) , is a topological invariant of G with the choice of \delta and c.
In this report, we consider two types of Heegaard diagrams, which have both beenstudied in [5] for singular knots. We extend the constructions to the case of trivalentgraphs and study the Heegaard Floer complexes CFG(G, \delta, c) defined on them.

For the Heegaard diagram of type one, which is given in Example 2.2, we generalize

the results in [5]. This part is a joint work with Zhongtao Wu.
Let \mathbb{F}=\mathbb{Z}/2\mathbb{Z} . Let C(G, \delta, c) be the \mathb {F}‐vector space\bigoplus_{x\inS(G\delta)},\bigotimes_{e\inE_{x}(\frac{\mathb {F}[U_{e}] {U_{e}^{c(e)}=0}) ,where S(G, \delta) is the set of Kauffman states, E_{x} is a subset of edges of G associatedwith x and U_{e} is the variable corresponding to an edge e . We define the \mathcal{A}‐grading and\mathcal{M} ‐grading of a Kauffman state by using the local contributions in Figures 6 and 7, andextend them to gradings of C(G, \delta, c) by adding the rules \mathcal{A}(U_{e})=-c(e) and \mathcal{M}(U_{e})=0.
Theorem 1.1 (B. and Wu). Let C_{d}(G, \delta, c;s) be the F ‐vector space generated by thegenerators of C(G, \delta, c) with \mathcal{A} ‐grading s and \mathcal{M} ‐grading d . Then there is a differential\partial :C(G, \delta, c;s)arrow C(G, \delta, c;s)that carries C_{d}(G, \delta, c;s) to C_{d-1}(G, \delta, c;s) satisfying
H_{d}(G, \delta, c;s)\cong HFG_{d}(G, \delta, c;s) .

70Note that \mathcal{A}‐grading and \mathcal{M} ‐grading for a Kauffman states correspond to the Alexander
grading and Maslov grading of the original chain complex.
As a corollary of Theorem 1.1, we get the following formula for the Euler characteristic.

Let \triangle_{(G,c)}(t) be the Alexander polynomial defined in [3], which is defined up to a factor

of \pm t^{k} for k\in \mathbb{Z} . We say f(t)=g(t) if f(t)=\pm t^{k}g(t) for some k\in \mathbb{Z}.

Corollary 1.2 (B. and Wu).

\chi(HFG(G, \delta, s))=[\delta]\triangle_{(G,c)}(t) ,(1)where [\delta] is a factor related to the position of \delta.

For a plane trivalent graph, we have the following result.

Corollary 1.3 (B. and Wu). Suppose G is a plane trivalent graph. Then the group
HFG(G, \delta, c) is determined by the Alexander polynomial \triangle_{(G,c)}(t) ; indeed, the homology
HFG (G, \delta, c) is supported on the zero Maslov grading level.
The Heegaard diagram of type two is given in Example 2.4.

The results on thisHeegaard diagram were written in [2], which has been submitted for publication, so we
briefly summarize the results in Section 4.

Acknowledgement: This work was partially supported by the Research Institute forMathematical Sciences, an International Joint Usage/Research Center located in Kyoto
University.
2

Preliminaries

2.1

Heegaard diagram

Suppose G is a connected trivalent graph. Let V be the set of vertices of G and E be theset of edges of G . Suppose |V|=n and |E|=m . Choose a generic point on G and callit \delta . It is convenient to regard \delta as a new bivalent vertex of G . The Heegaard diagram of(G, \delta) is defined as follows, which was proposed in [1].
Definition 2.1. A quintet (\Sigma, \alpha, \beta, w, z) is called a Heegaard diagram for (G, \delta) if itsatisfies the following conditions.
1. (\Sigma, \alpha, \beta, w) is an (n+1) ‐pointed Heegaard diagram for S^{3} , and z is a set of (m+1)
points in \Sigma\backslash (\alpha\cup\beta\cup w) .

2. For each vertex v\in V\cup\{\delta\} whose indegree is
s=1 or 2 ) ,l=1 or 2

there exists a smooth embedding \varphi_{v} :(\Sigma\backslash \alpha, w, z) (resp. \psi_{v} :(resp.

outdegree is(L_{1}ol2 , \{0\}, \{1,2, , l\})\simeq+(L_{1}o82 , \{0\}, \{1,2, \cdots , s\})\simeq+*(\Sigma\backslash \beta, w, z) ) so that the im‐ages of \varphi_{v} (resp. \psi_{v} ) are pairwisely disjoint and \bigcup_{v\in V\cup\{\delta\}}({\rm Im}(\varphi_{v})\cup{\rm Im}(\psi_{v}) recovers
G , when we push the interior of {\rm Im}(\varphi_{v}) (resp. {\rm Im}(\psi_{v}) ) slightly into U_{\alpha} (resp. U_{\beta} ).
Here U_{\alpha} (resp. U_{\beta} ) is obtained from \Sigma by attaching 2‐handles along \alpha ‐curves (resp.
\beta ‐curves).

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71
From the definition it is easy to see that on the Heegaard diagram \Sigma , we assign a basepoint w_{v} around a vertex and a basepoint z_{e} at an edge e of G , and the base points w_{\delta}
and z_{\delta} around \delta . In this report, we focus on two types of Heegaard diagrams. Here we
introduce them as examples.

Example 2.2 (Heegaard diagram of type one). This Heegaard diagram was introducedin [1], which extended the construction in [5].
Consider a graph diagram D\subset \mathbb{R}^{2} for a given graph G\subset S^{3}.

1. Regard D as a 1‐complex in S^{3} and take a tubular neighbourhood of it in S^{3} . It isa handlebody and its boundary is the Heegaard surface \Sigma.
2. The diagram D divides \mathb {R}^{2} into several regions. For each bounded region, introduce
an \alpha ‐curve on \Sigma which encloses the region.

3. For each crossing of D , introduce a \beta ‐curve as shown in Figure 2.
4. Place the base point w_{v} on each vertex v\in V.

5. For a vertex v\in V with indegree l=1 or 2, introduce l\beta ‐curves which are meridiansof the edges pointing to v and l base points of type z on the edges pointing to v.
Introduce an \alpha ‐curve \alpha_{v} which bounds a disk in \Sigma that contains w_{v} and all the basepoints of type z on the edges pointing to v.
6. For \delta , introduce a \beta ‐curve which is the meridian of the edge containing \delta and intro‐
duce two base points w_{\delta} and z_{\delta} around \delta.

a-c\iota m\cdot e\beta_{-Cl}n\cdot e
0-\bullet 1t’

Figure 2: The Heegaard diagram of type one associated with a graph diagram.

Remark 2.3. Example 2.2 was first proposed in [1], where we studied bipartite graphs.
A trivalent graph without source or sink can be regarded as a perticular bipartite graph.

Example 2.4 (Heegaard diagram of type two). We consider the second type of Heegaard
diagram only for plane graphs. It extends the construction in [5] for a singular knot.
G.

For a plane oriented graph G without source or sink, choose a point \delta on an edge of
We introduce a Heegaard diagram for (G, \delta) as follows.

72—

a‐curve—\beta‐curveozew

Figure 3: A Heegaard diagram for G with a point 6.

ı. The Heegaard surface is S^{2} where G is embedded as a plane graph.

2. At each vertex v\in V , introduce a base point w_{v} , and on each edge e of G introduce
a base point z_{e}.
3. For \delta , introduce two base points w_{\delta} and z_{\delta} around \delta.

4. Around each vertex v\in V , introduce a curve a_{v} which encloses the base point w_{v} at
v and the base point(s) z(' s) on the edge(s) pointing to v . Introduce a curve \beta_{v} which

encloses the base point w_{v} at v and the base point(s) z(' s) on the edge(s) pointing
out of v.

As a result, we get the following data H=(S^{2}, \{\alpha_{v}\}_{v\in v}, \{\beta_{v}\}_{v\in v}, w, z) , which is a Hee‐gaard diagram for (G, \delta) . See Fig. 3 for an example.
2.2

The Heegaard Floer complex

Let (\Sigma,a,\beta, w, z) be a Heegaard diagram for a graph (G,\delta) whose number of a‐curves(and also \beta‐curves) is d . Let\Gamma_{\alpha}=\alpha_{1}x\alpha_{1}\cross\cross\alpha_{d} and T_{\beta}=\beta_{1}\cross\beta_{1}\cross\cross\beta_{d}be the tori in the symmetric product Sym^{d}(\Sigma) . Given x,y\in\Gamma_{\alpha}\cap T_{\beta} , let \pi_{2}(x,y) be theset of relative homology classes of Whitney disks from x to y with boundary in \Gamma_{\alpha} and\Gamma_{\beta} . For \phi\in\pi_{2}(x, y) , let \mu(\phi) be its Maslov index and \hat{\mathcal{M} (\phi) be the moduli space ofpseud (\succ holomorphic disks in the class \phi modulo \mathbb{R}.
Let D_{1}, D_{2} ,, D_{h} denote the closures of the components of \Sigma\backslash (a\cup\beta). A domain isa 2‐chain on \Sigma of the form D= \sum_{i=1}^{h}a_{i}D_{i} , where a_{i}\in \mathbb{Z} is called the local multiplicity of
D at D_{i} . For a point p in the interior of D_{i} , let n_{p}(D) denote the local multiplicity of D
at the point p , which equals a_{*}\cdot . A domain D is a positive domain if a_{1}\geq 0 for 1\leq i\leq h.
A domain P= \sum_{i=1}^{h}a_{i}D_{i} is called a periodic domain if \partial P is a &‐linear combination of\beta‐curves and P\cap w=P\cap z=\emptyset.
To each base point z_{e} , we assign a variable U_{e} . We define the chain complex C\Gamma G(G, \delta)
to be the free \Gamma[\{U_{e}\}_{e\in E}] ‐module generated by \Gamma_{\alpha}\cap\Gamma_{\beta} , where \Gamma=\mathbb{Z}/2\mathbb{Z} . Note that thegenerating set \Gamma_{\alpha}\cap\Gamma_{\beta} has a one‐to‐one correspondence with the set
a ‐curves and

S=\{ (x_{1},x_{2}, \cdots , x_{d})|x_{i}\in\alpha_{i}\cap\beta_{\sigma(:)}, 1\leq i\leq d,\sigma\in S_{d}\}.

73
Therefore S is also regarded as the generating set.
Then

CFG(G, \delta) is endowed with the differential\partial(x)=\sum_{y\in\Gam a_{\alpha}\cap \mathb {T}_{\beta} \sum_{2\{\phi\ n\pi(x,y)|\mu(\phi)=1,n_{w}(\phi)=\{0\},n_{z_{\delta} (\phi)=0\} \#\hat{\mathcal{M} (\phi)\cdot(\prod_{e\in E}U_{e}^{n_{z_{e} (\phi)} \cdot y ,
where

(2)

n_{w}(\phi)=\{n_{w}(\phi)|w\in w\}.

To simplify the discussion, we further endow a coloring to G. A balanced coloring is
a map c:Earrow \mathbb{Z}_{\geq 0} such that for each vertex v of G,\sum c(e)= \sum c(e)eLet: pointing into ve.

: pointing out of v

c(v) be the value above.

Definition 2.5. The relative Alexander grading A and Maslov grading M are as follows:

A(x)-A(y)= \sum_{v\in V}c(v)\cdot n_{w_{v} (\phi)-\sum_{e\in E}c(e)\cdot n_{z_{e} (\phi) ,(3)

M(x)-M(y)= \mu(\phi)-2\sum_{v\in V\cup\{\delta\}}n_{w_{v} (\phi) ,(4)

for x, y\in \mathbb{T}_{\alpha}\cap \mathbb{T}_{\beta} , where \phi\in\pi_{2}(x, y) .

The functions A and N defined above induce on CFG(G, \delta) , the Alexander and Maslov
gradings, with the convention that

A(U_{e})=-c(e) , M(U_{e})=0.
The differential \partial drops the Maslov grading by one, and preserves the Alexander grading.
Proposition 2.6. The homology of CFG(G, \delta, c) is a topological invariant of G together
with the choice of \delta and c.

3

Heegaard Floer complex: type one

The purpose of this section is to prove Theorem 1.1 and Corollaries 1.2 and 1.3.
3.1

Kauffman states

We recall the definition of Kauffman states, which was described in detail in [3].
Definition 3.1. Starting from a trivalent graph diagram (D, \delta) , we can obtain a decorated
diagram (D, \delta) by drawing a circle around each vertex of D.
1. Cr(D) : denotes the set of crossings, including the types \near ow^{\backslash }\backslash and_{/}\backslash ' which are thedouble points of the diagram and the type \varphi which are the intersection points
around each vertex between the incoming edges with the circle.

74
2. {\rm Re}(D) : denotes the set of regions, including the regular regions of \mathb {R}^{2} separated by D
and the circle regions around the vertices. Marked regions are the regions adjacentto the base point \delta , and the others are called unmarked regions.
3. Corners: For a crossing of type \near ow_{\backslash }^{\backslash }or\backslash _{/}' , there are four corners around it, and we

c\ovalbox{\t \smal REJECT} them the north, south, west, and east corners of the crossing. Around a crossingof type \varphi there are three corners, and we call the one inside the circle region thenorth corner, the one on the left of the crossing the west corner and the one on
the right the east corner. Note also that every corner belongs to a unique region in{\rm Re}(D).

Calculating the Euler characteristic of \mathb {R}^{2} using D gives|{\rm Re}(D)|=|Cr(D)|+2.
The point \delta is chosen so that it is adjacent to two distinct regions, which will bedenoted by R_{w} and R.

Definition 3.2. A Kauffman state, or simply, a state for a decorated diagram (D, \delta) is abijective map

x:Cr(D)arrow{\rm Re}(D)\backslash \{R_{w}, R_{v}\},

which sends a crossing in Cr(D) to one of its corners. Denote S(D, \delta) the set of all states.
For each Kauffman state x , around each vertex v there is a unique edge the crossingon which is sent to the circle region. Let E_{x} be the set of such edges.
Consider the Heegaard diagram in Example 2.2. We see that each Kauffman state is
associated to 2^{n} different generators in \Gamma_{\alpha}\cap\Gamma_{\beta} , where n is the number of vertices. Moreprecisely, each intersection point in T_{\alpha}\cap\Gamma_{\beta} is represented by a pair (x, \epsilon_{x}) , where x is a

Kauffman state and \epsilon_{c} : V arrow { \pm ı}.

Figure 4: The internal curve a_{v} intersects the same \beta ‐curve at two points, which we label them as 1 and-1.

Suppose x+ and x_{-} are two generators which are identical except that around thevertex v, x_{+} is labeled by ı and x_{-} is labeled by -1 . There is a bigon connecting x_{-} tox_{+} which contains a base point z_{e} . Then we haveA(x_{-})-A(x_{+})=-c(e)\leq 0.

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Figure 5: A kauffman state and its corresponding Alexander grading maximizer.

Therefore among all the intersection points in \mathb {T}_{\alpha}\cap \mathb {T}_{\beta} which corresponds to the same
Kauffman state x , the one with \epsilon_{x}(v)=+1 is the unique intersection point that has
maximal Alexander grading. We call it the Alexander grading maximizer of the state x.

For each regular region in {\rm Re}(D) , we recall the definition of its index Ind(\cdot) in [3],which was defined as follows.

1. The index of the unbounded region is set to be 0.

2. The indices of the other regular regions are inductively determined by the rule asexhibited below: when an edge with color i points upward, let the difference of theindex of its left‐hand side region and that of its right‐hand side region be i.
in n-i

The definition of a balanced coloring guarantees that the above rules give rise to awell‐defined index for each regular region.

Definition 3.3. Suppose \delta is on an edge with color i , and the indices of the regions
adjacent to \delta are n and n-i . Define|\delta|=t^{n-i}-t^{n} and [\delta]=(t^{-1/2}-t^{1/2})^{-1}|\delta|.
3.2

Alexander grading

Lemma 3.4. Let x, y be two Kauffman states and let
maximizers that correspond to them. Then we have\mathcal{A}(x)-\mathcal{A}(y)=A(x)-A(y)where\mathcal{A}(x)=\sum_{c\in Cr(D)}\mathcal{A}_{c}^{x(c)}x, ybe the Alexander grading,

is calculated using the local contribution in Figure 6.

76

Figure 6: The local contribution \mathcal{A}_{c}^{\triangle}.

Proof. The proof is similar with the proof of Lemma 4.2 in [5]. For x and y , It suffices toshow that any homology class \phi\in\pi_{2}(x, y) satisfies\mathcal{A}(x)-\mathcal{A}(y)=\sum_{v\in V}c(v)\cdot n_{w_{v} (\phi)- \sum_{e\in E}c(e)\cdot n_{z_{e} (\phi).

Around each vertex v , let v\Leftrightar ow be the oriented simple arc from w_{v} to z_{e} on the Heegaard

surface for each edge

adjecent to , and \frac{}{e[j} be the inverse of v‐2. Suppose each arc v‐2intersects transversely with the boundary of \phi . For an intersection point p\in\overline{v}2\cap\partial\phi (orevarrow\cap\partial\phi) , define\# p=(-1)^{sign(p)}c(e)ev

to be the algebraic intersection number of \overline{v}2 (or \frac{}{elj} ) and \partial\phi at p , where sign(p) is +1
if the orientation of the intersection agrees with the orientation of the Heegaard surface
and -1 otherwise.

Around a vertex v of G , let \eta_{v} be the union of \frac{}{vt},s for those edges e pointing to v . Let\theta_{v} be the union of \frac{}{e\prime[j},s for all the edges pointing to v and \overline{v}2 ’s for all the edges leaving v.
Then we haveA(x)-A(y) = \sum_{v\in V}\#(\eta_{v}\cap\partial\phi)= \sum_{v\in V}\#((\eta_{v}+\theta_{v})\cap\partial\phi).

The first equality follows from the definition of A(x)-A(y) and the fact that \phi is a
2‐complex on the Heegaard surface. The second equality follows from the facts that U_{v}\theta_{v}replicates G and c is a balanced coloring defined on G . Namely \sum_{v\in V}\#(\theta_{v}\cap\partial\phi)=0.
Let \beta_{c} be the \beta ‐curve around a double crossing c , and \alpha_{v} be the internal \alpha ‐curvearound a vertex v . For \alpha ‐curves, the only ones that intersect (\eta_{v}+\theta_{v}) are the internal\alpha_{v} ’s around vertices, and the only \beta ‐curves that algebraically intersect (\eta_{v}+\theta_{v}) are those\beta_{c}' s . Note that \partial\phi are union of arcs on \alpha\cup\beta . Therefore we have\sum\#((\eta_{v}+\theta_{v})\cap\partial\phi)v\in V= \sum\#((\eta_{v}+\theta_{v})\cap\partial\phi|_{\alpha_{v} )+ \sum \#[(\bigcup_{v}\theta_{v})\cap\partial\phi|_{\beta_{c} ].
v\in Vc: doubıe crossing

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We claim that\#((\eta_{v}+\theta_{v})\cap\partial\phi|_{\alpha_{v}}) = \mathcal{A}_{v}^{x(v) }-\mathcal{A}_{v}^{y}(v)+[Ind(x(v))-Ind(y(v))] ,\#[(\bigcup_{v}\theta_{v})\cap\partial\phi|_{\beta_{c}}] = \mathcal{A}_{c} ^{x(c)}-\mathcal{A}_{c}^{y(c)}+[Ind(x(c))-Ind(x(c))] ,

(5)
(6)

where \mathcal{A}_{v}^{x(v)} is the sum of the local contributions around v for the state x , and Ind(x(v))
is the sum of the indices of the regular regions occupied by x around v . For each vertex
v , the intersection number of(\eta_{v}+\theta_{v}) with the internal \alpha_{v} is zero. For each \beta_{c} , theintersection number of ( \bigcup_{v}\theta_{v}) with it is also zero. Therefore it suffices to verify (5) and(6) by choosing any arc connecting x to y on the internal \alpha_{v} ’s and \beta_{c}' s.

We consider all the possible locations of x and y around a vertex and around a crossing

to verify (5) and (6). It is easy to see that all the possible locations can be obtained ascompositions of the six cases in Tables 1 and 2.

Now summing equations (5) and (6) we haveA(x)-A(y)=[ \mathcal{A}(x)+\sum_{c\in Cr(D)}Ind(x(c) ]-[\mathcal{A}(y)+ \sum_{c\in Cr(D)}Ind(y(c) ].
On the other hand, we see that\sum_{c\in Cr(D)}Ind(x(c) =\sum_{c\in Cr(D)}Ind(y(c),

which is the sum of indices over all unmarked regular regions of D.

Table 1: Verify Eq. (5) around a vertex.

3.3

Maslov grading

For the Maslov grading, we can also calculate it locally as below.

\square78

Table 2: Verify Eq. (6) around a double crossing.

Figure 7: The local contribution \mathcal{M}_{c}^{\triangle}.

Lemma 3.5. Let x be a Kauffman state and let x be the Alexander grading maximizer
that corresponds to it. Then we have\mathcal{M}(x)=M(x)where\mathcal{M}(x)=\sum_{c\in Cr(D)}\mathcal{M}_{c}^{x(c)},is calculated using the local contribution in Figure 7.

The proof in [5] can be generalized trivially to our case. In the proof of [5], only four‐valent vertices (singular crossings) are considered. However, the proof works for graphs
which contains both trivalent and four‐valent vertices.

For the readers’ convenience, we summarize the sketch of the proof and point out thenotational differences. Note that the definition of M(x) does not depend on the z basepoints and the coloring c . Suppressing the z base points, we get an (n+1) ‐pointed
Heegaard diagram for S^{3}.
Let G be a plane trivalent graph. Two Kauffman states are said to be equivalent if
they coincide with each other inside the circle regions. This definition is an adaption ofDefinition 4.3 [5] to our situation. By using exactly the same argument, we could provethat two Kauffman states are equivalent if and only if they are the same state.

Following the discussion in Proposition 4.6 of [5], we can show the following fact. Let(G, \delta) be a plane trivalent graph. Suppose x is a Kauffman state and x is the corresponding
Alexander maximizer. Then we have M(x)=0.
Now we consider a general trivalent graph (G, \delta) . Consider the Heegaard diagram

79
of type one H=(\Sigma, \alpha, \beta, w) for (G, \delta) , whereis suppressed. Consider a Kauffmanz

for (G, \delta) . Then following the recipe in the proof of Theorem 4.1 of [5], one canconstruction a new Heegaard diagram H_{x}=(\Sigma, \gamma, \beta, w) , which deeply depends on x . It
is an (n+1) ‐pointed Heegaard diagram for S^{3} . Then we can show that the generators
state

x

induced from x are the only generators for H_{x} , the number of which is 2^{n} . Namely
the differential of the chain complex defined on H_{x} is trivial. Let x be the Alexandermaximizer of x in H=(\Sigma, \alpha, \beta, w) , and let x' be the induced generator on H_{x} . Then wehaveM(x')=0.

On the other hand, H_{\alpha,\gamma}=(\Sigma, \alpha, \gamma, w) is a Heegaard diagram for (S^{1}\cross S^{2})^{\# d} , whered is the cardinality of \alpha . The differential defined on H_{\alpha,\gamma} is also trivial. Suppose \theta is thegenerator with maximal Maslov grading, which is M(\theta)=0.
Consider the homological triangle connecting x, x' and \theta . One can finally show that
M(x)=n-p , where n is the number of negative crossings whose images under x are
north corners and p is the number of positive crossings whose images under x are northcorners. It is indeed the conclusion we want to show in Lemma 3.5.
3.4

Proof of Theorem 1.1.

The proof follows the idea in the proofs of Proposition 4.6 and Theorem 4.1 in [5].
Since G is connected, we have the following lemma.

Lemma 3.6. Let P be a periodic domain in the Heegaard diagram H=(\Sigma, \alpha, \beta, w) of
type one. If n_{z}(P)=n_{w}(P)=\{0\}, P is a trivial domain.
Using the lemma above, we can define a filtration on CFG(G, \delta, c) . Consider all theregions of \Sigma\backslash (\alpha\cup\beta) , and place a point in each region disjoint from z_{e}' s . Let Q be theset of such points.

Definition 3.7. We define a partial order \mathcal{K} on \mathbb{F}[\{U_{e}\}_{e\in E}]\langle \mathbb{T}_{\alpha}\cap \mathbb{T} _{\beta} }. For two elements
a, b\in \mathbb{F}[\{U_{e}\}_{e\in E}]\langle \mathbb{T}_{\alpha}\cap \mathbb{T}_ {\beta}\rangle , if there is a domain \phi connecting a to b with n_{w}(\phi)=\{0\}and n_{z_{\delta}}(\phi)=0 , let\mathcal{K}(a)-\mathcal{K}(b)=\#_{alg}(\phi\cap Q),

where the right hand side denotes the algebraic intersection number of \phi and Q.
The partial order

K is well‐defined since by Lemma 3.6 if the domain\phi exists, it isunique.

Lemma 3.8. For a, b\in \mathbb{F}[\{U_{e}\}_{e\in E}]\{\mathbb{T}_{\alpha}\cap \mathbb{T}_{\beta} \} , suppose b is a summand in \partial(a) . Then wehave\mathcal{K}(b)\leq \mathcal{K}(a).

Namely \mathcal{K} induces a filtration on (CFG (G, \delta, c), \partial ).
Proof. The pseudo‐holomorphic disk \phi connecting a to b always has non‐negative inter‐\squaresection number with Q.

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Proof of Theorem 1.1. For a Kauffman state x , let x_{+} and x_{-} are two generators corre‐
sponding to x which are identical except around a base point z_{e} . Then the bigon containingz_{e} connects x_{-} to x+\cdot Its homology class \phi admits a single holomorphic representative up
to reparameterization. The differential on the E_{0} page of CFG(G, \delta, c) under the filtration\mathcal{K} counts only those bigons described above. Namely it only connects generators whichhave common underlying Kauffman state. Therefore the chain complex on the E_{0} page
splits along the set S(G, \delta) .
For each Kauffman state x , the chain complex on the E_{0} page is the tensor product of
the chain complexes of the form

\mathbb{F}[\{U_{e}\}_{e\in E}]ar ow^{U_{e}^{c(e)} \mathbb{F}[\{U_{e}\}_{e\in E} ],for e\in E_{x} . The cokernal of each such short differential is generated as a \mathb {F}‐vector spaceby \{x_{+}, U_{e}\cdot x_{+}, \cdot\cdot\cdot , U_{e}^{c(e)-1}\cdot x_{+}\}.
The E_{1} page is now a free module generated by Kauffman states. More precisely, as a\mathb {F} ‐vector space, it is\bigoplus_{x\inS(G\delta)},\bigotimes_{e\inE_{x}(\frac{\mathb {F}[U_{e}] {U_{e}^{c(e)}=0}).

\squareNow we prove Corollaries 1.2 and 1.3.

Proof of Corollary 1.2. We elaborate the proof here by looking back at the differential \partial_{0}in E_{0}|‐page in the proof of Theorem 1.1. The differential given by the bigon that passes
z_{e}is of the form

\partial_{0}x_{-}=U_{e}^{c(e)}\cdot x_{+},
from which we see that the cokernal is generated by \{x_{+}, U_{e}\cdot x_{+}, , U_{e}^{c(e)-1}\cdot x_{+}\} . There
are a total of c(e) number of generators with the same Maslov grading and different
Alexander gradings, which contribute a factor

1+t+ \cdots t^{c(e)-1}=[c(e)]\cdot t\frac{c(e)-i}{2}\dot{ \imatguresh} n[3],weobta\6dot{ \imath} ntheident\dot{ \imath} ficat\dot{ \imath} on.\square aroundz_{e}fions
ore\in E_{x},where[k]:=\frac{t^{k/2}- t^{-k/2} {t^{1/2}-t^{-1/2},tr\dot{ \imathocal} bu}.Final1contributions
ybycompar\dot{ \imath} ngthel\dot{ \imath} nF\dot{ \imath}and7andthe1oca1cont

Proof of Corollary 1.2. When G is a plane trivalent graph, all the generators in Theorem\square1.1 have zero \mathcal{M} ‐grading. Hence, the differential on the chain complex is trivial.

4

Heegaard Floer complex: type two

We consider a plane trivalent graph G . Viro in [6] proposed a vertex state sum formulafor the gl(1|1) ‐Alexander polynomial. Choose a point \delta on an outermost edge of G. A
Viro state is a map s : Earrow\{0,1\} which sends the edge containing \delta to zero and satisfies
the condition that at each vertex, the sum of s(e) for all the edges e pointing toward the8181vertex equals that for the edges pointing out of the vertex. Let S be the set of states. By
definition, around each vertex we have the following six possibilities under a state, where
the dotted edges are those which are sent to zero and the solid edges are those which aresent to one.

r. . .\{\near ow \nwarow_{t}.\cdot\cdot\cdot\cdot\cdotN\backsla h\cdot\cdot(\nwarow \near ow 1_{r}. .

Consider the intersection points of \alpha ‐curves and \beta ‐curves. Let
T:= { \{x_{v}\}_{v\in V}|x_{v}\in\alpha_{\sigma(v)}\cap\beta_{v} for a bijection \sigma of V }.

Let \sigma_{x} be the bijection that defines x\in T . It is easy to see that the intersection points
always appear in pairs around a base point. Two elements x=\{x_{v}\}_{v\in V} and y=\{y_{v}\}_{v\in V}
are said to be equivalent (x\sim y) if x_{v} and y_{v} are around the same base point for any
v\in V . Let [x] be the equivalence class of x . It is obvious that \sigma_{x} keeps invariant within
the equivalence class of x.
Our first result is:

Proposition 4.1. There is an identification between T/\sim and S.
The gl(1|1) ‐Alexander polynomial is defined by assigning Boltzmann weights to eachvertex under each state as follows. The Boltzmann weights were originally obtained byscaling the Clebsch‐Gordan morphisms for irreducible U_{q}(gl(1|1)) ‐modules of dimension(1 | 1).

Our second result is:

Proposition 4.2. The Boltzmann weights around trivalent vertices, which determine

the gl(lll)‐Alexander polynomial, can be obtained from the Fox calculus on the Heegaard
diagram of type two.
References[1] Y. BAO, Heegaard floer homology for embedded bipartite graphs, arXiv:1401.6608v3(2016).
[2] —, A topological interpretation of Viro’s gl (l |1) ‐Alexander polynomial of a graph,
arXiv:1801.06301 (2018).

82[3] Y. BAO AND Z. WU, An Alexander polynomial for MOY graphs, arXiv: 1708.09092(2018).
[4] L. H. KAUFFMAN, Formal knot theory, vol. 30 of Mathematical Notes, Princeton
University Press, Princeton, NJ, (1983).

[5] P. OZSVÁTH, A. STIPSICZ AND Z. SZABÓ, Floer homology and singular knots, J.
Topology 2 (2009) 380‐404.
[6] O. Viro, Quantum relatives of the Alexander polynomial, Algebra i Analiz, 18 (2006)
63‐157.

Graduate School of Mathematical Sciences

the University of Tokyo
3‐8‐1 Komaba, Meguro‐ku, Tokyo 153‐8914
JAPAN

E‐‐mail address: bao@ms.u‐tokyo.ac.jp