Introduction to Heegaard Floer homology
Motoo Tange
University of Tsukuba

1

Atiyah-Floer conjecture and Ozsváth-Szabó’s motivation

In [25, 26] Ozsváth and Szabó defined topological 3-manifold invariants by using Floer
homology theory:d (Y, s), HF ∞ (Y, s), HF + (Y, s), HF − (Y, s).
HF
Those invariants are new invariants in terms of the point that the invariants are categorifications of some topological invariants: Casson invariant (with correction term) Alexanderpolynomial and Turaev torsion invariant. The motivation of defining these invariants isthat it is a symplectic counterpart of Seiberg-Witten Floer homology via Atiyah-Floer
conjecture. Original Atiyah-Floer conjecture for Yang-Mills equation is the following:
Conjecture 1.1 (Atiyah-Floer conjecture) Let Y be a closed oriented 3-manifolds.
The instanton Floer homology on Y and the Lagrangian intersection Floer homology forflat connections are isomorphic each other:
HF Inst (MY ) ∼= HF Symp (MΣ ; L0 , L1 ).
The instanton Floer homology HF Inst (MY ) is a Morse homology on a moduli space MY
of Yang-Mills equation on a 3-manifold Y up to gauge action. The generators are the flat
connections on Y . The diﬀerentials are the counting of the moduli space of Yang-Mills
solutions on the cylinder Y × I.
On the other hand for a Heegaard decomposition Y = H0 ∪Σ H1 we consider the
symplectic moduli space MΣ on Σ of flat connections. Let L0 , L1 in MΣ be Lagrangiansubmanifolds extending to H0 and H1 . The generators of symplectic side are intersection
points of the two manifolds with a suitable general condition. The diﬀerentials are thecounting of the holomorphic disks connecting two points. The boundary of a holomorphic
disk lies in union of L0 and L1 .
Heegaard Floer theory corresponds to the opposite side of the Seiberg-Witten Floer
homology through analogy of Atiyah-Floer conjecture. Before Heegaard Floer homology appearing, no one would have succeeded a symplectic counterpart of Seiberg-Witten
theory. Ozsváth and Szabó attacked that natural and essential problem to attain such
Lagrangian intersection Floer homology. Instanton Floer homology had rediscovered topological invariants, Casson invariant [45]. Hence they should have expected applications1

of interesting topological invariants from the Heegaard Floer homology. More than expected, it turns out to be a powerful tool to study low-dimensional topology and be an
attractable object for low-dimensional topologists. As a result, a whole lot of results havebeen built by Heegaard Floer homology, some new phenomena are discovered.
This article is a simple overview of Heegaard Floer homology up to now. But someinteresting topics had to skip because of space limitation. If this article arouses yourinterests in your mind, as present Heegaard Floer topologists were so in the past, thenyou should begin with one in a pile of papers.
One can also read some excellent lecture notes [30] [31] [32] as the first study of Heegaard Floer homology.

2

Heegaard Floer homology

2.1

Definition of Heegaard Floer homologies

We define Heegaard Floer homology in this section. Let Y = H0 ∪Σg H1 be a Heegaard
decomposition of a 3-manifold Y . Since H0 , H1 are two genus g handlebodies, there exist
two tuples of g simple closed disjoint curves α1 , · · · , αg ⊂ ∂H0 and β1 , · · · , βg ⊂ ∂H1each of which is compressing in each handlebody. Hence, on Σ = ∂H0 = ∂H1 we can see
the diagram (Heegaard diagram) consisting of 2g curves α1 , · · · , αg , β1 , · · · , βg (these arecalled α-curves and β-curves). Here we assume that all the intersection points between
the α-curves and β-curves are transversal in the general position.
In this situation ‘our moduli space’ is the g-th symmetric productSymg (Σg ) := Σg /Sg .
It is a smooth complex manifold by resolving the singular set by the Sg -action. Let Tα
and Tβ denote the image of
α1 × α2 × · · · × αg ,β1 × β2 × · · · × βg

by the quotient map Σg → Σg /Sg . The images in Symg (Σg ) are g-dimensional tori.
We denote the g curves by α = {α1 , · · · , αg } and β = {β1 , · · · , βg }. Let z be a basepoint in Σg in the complement of α and β. Naturally these are totally real submanifold
def(⇔ Y ⊂ X and i · Tp Y ∩ Tp Y = {0}). Floer’s theory works for the case of a pair of
totally real submanifolds in a complex (or almost complex) manifold. This tuple of thesedata (Σg , α, β, z) is called pointed Heegaard diagram. Since 2 dim Y = dim X holds, theintersection points Tα ∩ Tβ are finite points.
Here we verify the homology of the symmetric product is isomorphic to the homology
of Y .
H1 (Symg (Σg )) ∼= H1 (Y, Z)
H1 (Tα ) ⊕ H1 (Tβ )This means that the Symg (Σg ) is regarded as an ‘expounded body’ of a 3-manifold. One
direction is the ordinary homology and another sharpener direction is the Heegaard Floer
homology.
2

The intersection point x ∈ Tα ∩ Tβ and base point z determine a spinc structure.
The point x implies g Morse trajectories from index 2 to 1 with respect to the Heegaard
splitting. The point z is regarded as the trajectory from index 3 to 0 naturally. Thus
removing the neighborhoods of the g + 1 trajectories, we obtain a non-zero vector field ina holed 3-manifold. The homotopy classes of such fields up to homologous coincide withthe spinc structures on Y . The ‘homologous’ means that the two fields are homotopic
after deleting several balls. Hence, we get the map:sz : Tα ∩ Tβ → Spinc (Y ).
Let x, y be intersection points in Tα ∩ Tβ . Let π2 (x, y) be the homotopy classes
connecting x and y. The class [u] ∈ π2 (x, y) is represented by a continuous map u : D →Symg (Σg ) satisfyingu({Re ≥ 0} ∩ ∂D) ⊂ Tαu(i) = yu({Re ≤ 0} ∩ ∂D) ⊂ Tβu(−i) = x,where D is the unit disk in the complex plane. For any ϕ ∈ π2 (x, y) let nz (ϕ) be thecalgebraic count #((z × Symg−1 (Σg ) ∩ u(D)). M(ϕ)= M(ϕ)/R is the moduli space ofholomorphic disk which is divided by natural R-action. Then the diﬀerential ∂ˆ are defined
to be
∑
∑
ˆ =
c
M(ϕ)
· y.
∂x
y∈Tα ∩Tβ ϕ∈π2 (x,y),nz (ϕ)=0,µ(ϕ)=1d (Σg , α, β, z) = Z⟨x|x ∈ Tα ∩ Tβ ⟩. Then OzsváthHere µ is the Maslov index. We put CF
and Szabó proved the following:b is a chain complex and its homology is a topologicald (Σ, α, β, z), ∂)Theorem 2.1 (CF
3-manifold invariant.
d (Y ) and HF
d (Y ) respectively. The
We denote the chain complex and homology by CF
proof of this theorem is supported by an analytical argument on the moduli space ofholomorphic disks. The argument is important, however, skip here. Actually, to hold this
topological invariance, we need some general conditions: admissibility in Section 4.2.2 in[25].
d (Y ) is decomposed as follows:
The homology HF
d (Y ) = ⊕s∈Spinc (Y ) HF
d (Y, s),HF
because the diﬀerential keep the map sz , i.e., sz (x) = sz (y) when π2 (x, y) ̸= ∅. The Eulerd (Y, s) is
number of HF{
d (Y )) = |H1 (Y, Z)| b1 (Y ) = 0
χ(HF0b1 (Y ) ̸= 0.
d (Y ) is a categorification of the ordinary homology H1 (Y ) = H 2 (Y ).
Namely, HF
3

Let (Y, s) be a spinc 3-manifold with b1 > 0. Then the Euler characteristic of HF +agrees with the Turaev torsion.
χ(HF + (Y, s)) = −τ (Y, s)This equality makes sense in non-torsion spinc case, but this equality holds even for any
torsion spinc structure by taking truncated Euler characteristic. In particular if Y is 0surgery of a knot K, then the i-th Turaev torsion ti (K) is computed by the coeﬃcientsof the Alexander polynomial of K as in [33].
Furthermore by counting intersection numbers with z × Symg−1 (Σg ), we define thefollowing chain complex
CF ∞ (Σg , α, β, z) = Z⟨U n · x|x ∈ Tα ∩ Tβ , n ∈ Z⟩.
The diﬀerential ∂ ∞ is a U -equivariant map and
∑
∑
∂∞x =

nz (ϕ)c#M(ϕ)Uy.

y∈Tα ∩Tβ ϕ∈π2 (x,y),µ(ϕ)=1There exists a subchain complex CF − (Σ, α, β, z) = Z⟨U n · x|x ∈ Tα ∩ Tβ , n ∈ Z<0 ⟩. The
quotient complex CF ∞ /CF − denotes by CF + (Σ, α, β, z). Then we have the following:
Theorem 2.2 ([25]) These chain complexes give topological invariants
HF ∞ (Y ), HF − (Y ) and HF + (Y ).
Furthermore, these homology are Z[U ] ⊗ Λ∗ H1 (Y, Z)/T ors-module.
By the definition we have the following short exact sequence:
0 → CF − (Y ) → CF ∞ (Y ) → CF + (Y ) → 0.
iπThus we induce the long exact sequence:· · · → HF − (Y ) → HF ∞ (Y ) → HF + (Y ) → HF − (Y ) → · · ·d (Y ) is embedded
The cokernel of π is denoted by HFred (Y ). Since the chain complex CF
in the n = 0 level in CF ∞ (Y ), the short exact sequence:d (Y ) → CF + (Y ) → CF + (Y ) → 0
0 → CF
×U

Here the second is the U -multiple map. Thus we induce the long exact sequence:d (Y ) → HF + (Y ) → HF + (Y ) → HF
d (Y ) → · · ·· · · → HF
2.2

Holomorphic disks

Next step of Heegaard Floer homology is a dimensional reduction of holomorphic disks
in Symg (Σg ). One of most remarkable points of Heegaard Floer theory is computable byusing words of low-dimensional topology.
4Proposition 2.3 ([25]) For a continuous map u : D → Symg (Σg ), there exists a conb → Σg satisfying the following commutative
tinuous map from a branched cover û : D
diagram:b −−−→ {(x, v) ∈ Σ × Symg (Σg )|x ∈ v} −−−→ ΣDg-fold branched coveryyD −−−→

Symg (Σg )This proposition asserts the holomorphic disk in Symg (Σg ) is understood as a surface
mapped in Σg . The condition ∂D ⊂ Tα ∪ Tβ corresponds to û ⊂ α ∪ β. The Maslov
index in the definition of the Heegaard Floer homology is analytical information data ofholomorphic disk.
Proposition 2.4 ([13]) The Maslov index of a disk u in [u] = ϕ ∈ π2 (x, y) is determinedby a topological information of the region mapped in Σg .
2.3

The TQFT viewpoint and absolute grading

Floer theory is defined in the framework of topological quantum field theory (TQFT).
Any element in Floer homology can be delivered by the cobordism to an element in Floer
homology of another side. Let (W, s) be a spinc 4-dimensional cobordism from Y1 to Y2 .
Then there exists a U -equivariant map:∞FW,s
: HF ∞ (Y1 , t1 ) → HF ∞ (Y2 , t2 ),d in thewhere s|Yi = ti . This map is induced to other homologies HF + , HF − and HF
sameof two cobordisms X = X1 ∪ X2 we have FX◦ 1 ,s1 ◦ FX◦ 2 ,s2 =∑ way. For the composition◦{s∈Spinc (X)|s|Xi =si } FX,s , where ◦ = ∞, ±, ∧.
Let Y be a rational homology sphere. In general, (Y, s) is a torsion spinc 3-manifold,where c1 (s) is a torsion element. Then the Heegaard Floer homology admits an absoluted (S 3 )Q-grading gr. This grading is defined as follows. First for a generator x0 ∈ HF◦
gr(x0 ) = 0 holds. The diﬀerence of the gradings by FW,s (◦ = ∞, ±, ∧) is computed byc21 (s) − 2χ(W ) − 3σ(W ).
4The U -action lowers the degree by −2. These properties determine the absolute Q-gradingon the Heegaard Floer homology of (Y, s) uniquely.
Heegaard Floer homology relates to smooth 4-manifold invariant. Ozsváth and Szabó
mix
in [37] defined the mixed invariant as a map FW,s
: HF − (Y1 , t1 ) → HF + (Y2 , t2 ) by using
3+1 dimensional TQFT. First, we cut a 4-dimensional cobordism into 2 pieces V1 , V2 alonga suitable 3-manifold N . Suppose that each of V1 , V2 has at least one positive eigenvalue
in the intersection form and δH1 (N ) is 0 in H 2 (W, ∂W ). The spinc 4-manifolds (V1 , s|V1 )and (V2 , s|V2 ) are spinc cobordisms from (Y1 , t1 ) to (N, s|N ) and from (N, s|N ) to (Y2 , t2 )mix
respectively. Then FW,s
is a composition as follows:◦gr(FW,s(x)) − gr(x) =

mix
FW,t
: HF − (Y1 , t1 )1

FV− ,s|1→

V1

HF (N, s|N ) → HFred (N, s|N ) ⊂ HF (N, s|N )++5

FV+ ,s|2V→ 2 HF + (Y, t2 ).

The reduced part of HF + (N, s|N ) is also embedded in HF − (N, s|N ) naturally.
Let X be a closed smooth 4-manifold with b+
2 (X) > 1. Then deleting two 4-balls in
X we built a cobordism W from S 3 to S 3 . Applying U n · Θ− in HF − (S 3 ) to the mixed
invariant through this cobordism, we obtain an invariantmix
FW,s(U n · Θ− ) =: ΦX,s · Θ+ ∈ HF + (S 3 )Here Θ− (Θ+ ) is the top (bottom) generator in HF − (S 3 ) (HF + (S 3 )) and n = (c1 (s)2 −2χ(X) − 3σ(X))/4 (it agrees with the dimension of the moduli space of Seiberg-Witten
solutions). The number ΦX,s ∈ Z is a smooth 4-manifold invariant (smooth OS-invariant)and conjecturely, the number coincides with the Seiberg-Witten invariant SWX,s .
Definition 2.5 (correction term [33]) Let (Y, s) be a spinc 3-manifold. Then d(Y, s)
is defined to be the minimal grading in the image π : HF ∞ (Y, s) → HF + (Y, s). Thisinvariant is called correction term (or d-invariant).
In the case of integral homology sphere the invariant is denoted by d(Y ) ∈ 2Z. From theproperty of gr the correction term d is a spinc rational homology cobordism invariant.
Suppose that W is a rational homology cobordism between Y1 , Y2 . Let s be a spincstructure on W with s|Yi = ti . Then we haved(Y1 , t1 ) = d(Y2 , t2 ).
Let (Y, s) be a spinc rational homology sphere. Then+
HF + (Y, s) ∼⊕ HFred (Y, s).
= T(d(Y,s))+Here T(d)is isomorphic to T + and the minimal grading is d. The d-invariant and Euler
number of HFred compute the Casson invariant.

Theorem 2.6 Let Y be an integral homology sphere. Then the Casson invariant λ(Y ) is
computed by1
λ(Y ) = χ(HFred (Y )) − d(Y ).
2
The Heegaard Floer homology of (branched) covering space Σn (K) is less known so far.
Casson invariant formula of double branched cover is known by Mullins [18] as follows:1
1
λ(Σ2 (K)) = sign(K) − VK′ (−1)/VK (−1),
812where Σ2 (K) is a homology sphere and VK is the Jones polynomial. On the other handOzsváth and Szabó show the following in [35]:Theorem 2.7 Let L a link in S 3 . There exists a spectral sequence with E2 -term the
Khovanov homology of L such that the sequence converges to the F2 -coeﬃcient homologyd (Σ2 (L), F). Here F is a field isomorphic to Z/2Z.
HF
d and Kh have
The Khovanov homology is a categorification of Jones polynomial, and HF
surgery exact sequence. Does this spectral sequence interpret Mullins’ formula? Definitely,
6

covering space is a weak point for Casson invariant and Heegaard Floer homology. Dehn
surgery formula for these invariants have been studied ever, however there are not so muchresearch for covering space.
For a rational homolog sphere Y , the renormalized Euler characteristic is defined as1χ̂(Y, s) = χ(HFred (Y, s)) − d(Y, s).
2
Then due to [43] we have∑χ̂(Y, s) = |H1 (Y )| · λCW (Y ),s∈Spinc (Y )where λCW is the Casson-Walker invariant.
The correction term has the 4-dimensional information as follows:
Theorem 2.8 Let (Y, t) be a spinc rational homology sphere. If (Y, t) admits a negative
definite bounding (W, s) with s|Y = t, then we have the following inequality:
c1 (s)2 + b2 (W ) ≤ 4d(Y, t).
The extended inequality of this theorem to a 3-manifold with b1 > 0 can prove the Thom
conjecture again [33].
2.4

L-spacesd (Y, s) = Z for any spinc structures, then Y is
Let Y be a rational homology sphere. If HF
called an L-space. Examples of L-space are lens spaces. The genus one Heegaard diagramof a lens space L(p, q) consists of single α-curve and β-curve. Then the pointed Heegaard
digram is (T 2 , α, β, z). Thus the symmetric product is T 2 itself and we have Tα = α andTβ = β. Hence, the generators α ∩ β are p points. The diﬀerentials are all zero, because
the p points belong to distinct spinc structures. Thus we have the following:
Proposition 2.9 Let Y be a lens space and s any spinc structure on Y .
d (Y, s) ∼HF
= Z, HF ∞ (Y, s) = Z[U, U −1 ], HF + (Y, s) ∼= Z[U, U −1 ]/U Z[U ], HF − (Y, s) ∼= Z[U ]
We put T ∞ = Z[U, U −1 ], T + = Z[U, U −1 ]/U · Z[U ], T − = Z[U ]. Another component as a
Z[U ]-module is Z[U ]/U n , which is a finite rank Z-module and we denote it by T + (n).
Any elliptic manifolds (i.e., it is a finite fundamental group) are an L-space. In otherwords L-space is a generalization of such a manifold. In addition, L-spaces contain notelliptic 3-manifolds. Suppose that a hyperbolic knot K is an L-space knot (i.e., an integral
Dehn surgery is an L-space), e.g., the (−2, 3, 7)-pretzel knot. Then a suﬃciently large
Dehn surgery of the knot is also an L-space. This is an easy result due to the surgery exact
sequence of Heegaard Floer homology. In other examples, double cover of any alternatingknot is also L-space [35]. [35] shows that the double cover of quasi-alternating link is
L-space. The quasi-alternating knots Q is the smallest set defined inductively as follows:(1) unknot∈ Q; (2) if L0 , L1 ∈ Q, det(L0 ) det(L1 ) ̸= 0 and L0 , L1 are two types of theresolution of a crossing of L and det(L) = det(L0 ) + det(L1 ), then L ∈ Q.
L-space includes interesting geometric properties besides:
7

Theorem 2.10 ([29]) Any L-space does not admit co-oriented taut foliation.
This result is useful, because the nonexistence of taut foliation is hard to prove without L-space property. We do not find a method in Heegaard Floer theory to prove thenonexistence of tight contact structure, so far.
Recently developing noticeable property is left-orderbility of the fundamental group.
‘Left-orderable’ (LO) means that the existence of a total order on a group that any left
action of the group keeps the order relation between any two elements. A group is LO
if and only if it is a subgroup in Homeo+ (R). We believe that the following mysterious
relationship:
Conjecture 2.11 Let Y be a rational homology sphere. Then Y is not LO, if and only
if Y is an L-space.
It is well-known that LO group is a non-torsion group. Then any elliptic manifold is not
LO. Furthermore, Boyer, Gordon, and Watson [2] proved that for any Seifert manifold or
Sol manifold this conjecture is true. Any other hyperbolic manifold is not so known. For
example for not all double covers of (quasi-)alternating link this relationship is verified.
2.5

Surgery exact triangle

In general Floer theory an exact triangle among three homologies is inherent. In Heegaard
Floer theory, the three homologies correspond to replacements of α or β-curves. In thecontext of a low-dimensional topology, the replacements are 0-surgery, and p-surgery. Let
Yn be a n-surgery of a 3-manifold Y along a null-homologous knot K. Then the following
exact triangle holds:· · · → HF + (Y ) → HF + (Y0 , [i]) → HF + (Yp , i) → HF + (Y ) → · · · .
Read [26] for surgery exact sequence. The decategorification of this exact triangle is the
Dehn surgery formula of Casson invariant as follows:
λCW (Yp ) = λ(Y ) − s(1, p) +1 ′′∆ (1),
2p K

where s(1, p) is the Dedekind sum, which is equivalent to the Casson-Walker invariant ofL(p, 1). Unlike Casson invariant, generally, for determining Floer homology of Yp we needhomologies of Y and Y0 , exact triangle and a more extra information. The other triads
of surgery exact sequence are {Y, Yn , Yn+1 } or {Y, Y0 , Y1/q }. These correspond to surgery
formulas for Casson invariant. The surgery triangle is applied on the variable situation.
2.6

Contact structure

Let (Y, ξ) be a contact 3-manifold, which the 2-plane filed ξ is nowhere integrable. Any
contact 3-manifold is decomposed into two types of non-isotopic contact structures: tightand overtwisted. The classification of tight contact structures is more diﬃcult than the oneof overtwisted structures. The latter structures agrees with the classification of homotopy
types of 2-plane fields due to Eliashberg [4]. The situations which a tight contact structure8

naturally appear is the cases where it is the boundary of a symplectic 4-manifold withsome suitable boundary condition. This contact structure is called symplectic fillable. The
boundary of Stein manifold is also a tight contact structure, which is called Stein fillable.
Ozsváth and Szabó define in [36] an isotopy invariant of contact structure. The classd (−Y, s(ξ))/ ± 1 and has the following property:c(ξ) is an element in HF
Theorem 2.12 If ξ is an overtwisted contact structure, then c(ξ) = 0. If ξ is Stein
fillable contact structure, then c(ξ) ̸= 0.
The converse of this theorem does not holds. This theorem implies that the invariantc(ξ) detects the tight-ness and Stein fillability. Since the detection of tight-ness by using
contact topology is diﬃcult in general, after this theorem, the computation of c(ξ) has
been a reasonable method. The most simple example in [15] is the contact +1-surgery
of a Legendrian right-handed trefoil, which is topologically −Σ(2, 3, 4). It admits nosymplectic fillable but a tight contact structure. The former is due to an application of
the Seiberg-Witten invariant and the latter part is due to the computation of c(ξ).
In general, invariants of some tight contact structures are vanishing (structures withsome symplectic fillable or positive Giroux torsion etc.) However if one take a suitable
twisted system, one can sometimes give a non-vanishing class [29]. For other examples
related to contact structures, symplectic topology, Lefschetz fibration, and Stein manifold
readers should take a look at [22].
2.7Graph manifolds

In [34], Ozsváth and Szabó gave a method to compute HF + for plumbed 3-manifolds.
In short, this method is a machinery put surgery exact sequence together. Némethi[19] gives a systematical algorithm (the tau function and the graded root) to compute
Heegaard Floer homology for any plumbed 3-manifold with at most one bad vertex. This
computation gives the module structure of HF + completely, and it is used as a first usefultrial to explore Heegaard Floer behavior for some topological phenomenon.
The more general 3-manifolds are hyperbolic 3-manifolds. Does Heegaard Floer homology capture hyperbolic structure? This is a natural and challenging question.

3

Knot Floer homology

3.1

Definition and an example

The most attractable point of Heegaard Floer theory is what it is able to build somevariations of Floer homology. One of variation is knot Floer homology. Knot Floer
homology was defined by Ozsváth and Szabó [27] and independently by Rasmussen [42].
Let K be a knot in S 3 . We take a (g, 1)-decomposition S 3 = H0 ∪Σg H1 of (S 3 , K).
In other words K is transversal about Σg and the union of the two arcs Ai = K ∩ Hi
(i = 0, 1). Furthermore, we assume that Ai does not intersect compressing disks of αcurves and β-curves. Such a Heegaard splitting always exists for any knot K. From theassumption K intersects two points ∂A0 = ∂A1 in Σg with α-curves and β-curves disjoint.
We denote the points by z, w. We call such a diagram double pointed Heegaard diagram
9

Then the chain complex of knot Floer homology CF K ∞ (S 3 , K, i) is isomorphic to
CF ∞ (Σg , α, β, w) as a Z[U ]-module. We denote by M as(x) the absolute grading on
CF ∞ (Y ) which is defined in the previous section. The grading satisfies:
M as(x) − M as(y) = µ(ϕ) − 2nw (ϕ),
where ϕ ∈ π2 (x, y).
We will introduce a filtration (Alexander filtration Alex(x)) on the module by using aknot in the S 3 . The filtration is defined as follows. For x, y ∈ Tα ∩ TβAlex(x) − Alex(y) = nz (ϕ) − nw (ϕ),
where ϕ is a disk with ϕ ∈ π2 (x, y). This definition is a relative Z-filtration only. To
make an absolute filtration we impose a symmetry.
#{x|Alex(x) = i} = #{x|Alex(x) = −i} mod 2.
We denote such a filtered chain complex of CF ∞ (Σg , α, β, z) by CF K ∞ (Σ, α, β, z, w).
Here we give an example of double pointed Heegaard diagram and the filtered chain
complex. Consider the trefoil K = 31 . In Figure 1 we describe two digging arcs and two

Figure 1: The trefoil knot and the Heegaard decomposition of S 3 .

2-handles on a 3-ball. The circles β1 , β2 are the attaching circles. The circle β2 is themeridian disk of K. Moving the circles and holes on the 3-ball we obtain the first picturein Figure 2. The dashed arc presents a longitude of a knot K. In this case the double
points z, w are two points close to the meridian circle β2 . The position of curves α2 , β2
can be made cancel the two critical points in terms of Morse theory. Hence, the diagram
moves to the next picture and it is isotopic to the last diagram in Figure 2.
From the last diagram we compute the Alexander filtration of the trefoil. The intersection points are 3 points: x1 , x2 , x3 . The non-trivial disks on the diagram are D1 , D2 .
D1 is a disk connecting from x2 to x1 and D2 is a disk connecting from x2 to x3 . Thus,
the filtration definition implies
Alex(x2 ) − Alex(x1 ) = −1, Alex(x2 ) − Alex(x3 ) = 1.
From the symmetric condition we haveAlex(x1 ) = 1, Alex(x2 ) = 0, Alex(x3 ) = −1.
10Figure 2: A double pointed Heegaard diagram of the trefoil knot and a move.

The Maslov grading is
M as(x2 ) − M as(x1 ) = −1, M as(x2 ) − M as(x3 ) = 1
The diﬀerential ∂ ∞ works as follows:∂ ∞ x1 = 0, ∂ ∞ x2 = U · x1 + x3 , ∂ ∞ x3 = 0
Here we compute hat Heegaard Floer homology. Since the version is embedded in the
0-level in CF ∞ with U = 0, then we have
ˆ 1 = 0, ∂x
ˆ 2 = x3 , ∂x
ˆ 3 = 0,∂xd (S 3 ) ∼HF= Z · x1 .
By the definition of the absolute grading, gr(x1 ) = 0. Since U -action decreases by 2, theMaslov gradings are computed as M as(x1 ) = 0, M as(x2 ) = −1, M as(x3 ) = −2.
\
Here we define diﬀerential ∂ˆK of knot Floer homology CF
K(S 3 , K) as follows:∑∑c#M(ϕ)y.
∂ˆK x =y∈Tα ∩Tβ ϕ∈π2 (x,y),nz (ϕ)=nw (ϕ)=0d (Σg , α, β, w) and the dif\ (Σg , α, β, z, w) is isomorphic to CF
The chain complex CKF
Kferentials are ∂b .
11

\ (Σg , α, β, z), ∂ˆK ) is an
Theorem 3.1 (Knot Floer homology) The homology of (CKF
isotopy invariant of a knot K. This homology decomposes as follows:\\
HF
K(S 3 , K) = ⊕i HF
K(S 3 , K, i),\where HF
K(S 3 , K, i) is the homology with the Alexander grading i.
\ (S 3 , K) is the i-th Alexander polynomial
Furthermore, the Euler characteristic of HKF\
χ(HF
K(S 3 , K, j)) = ai .
In the case of our trefoil, the diﬀerentials ∂ˆK are computed as follows:∂ˆK x1 = 0, ∂ˆK x2 = 0, ∂ˆK x3 = 0.
Hence, we obtain



Z(0)
3
\
HF K(S , K, j) = Z(−1)

Z
(−2)

j=1
j=0
j = −2.

Taking the Euler characteristic of these complexes, we have
∑

g(K)

\
χ(HF
K(S 3 , K, j)) · tj = t − 1 + t−1 = ∆K (t).

j=−g(K)We denote for any x ∈ Tα ∩ Tβ the map CF K ∞ (S 3 , K) ∋ U n · x 7→ −n ∈ Z by Alg.
Then one can visualize the double complex Alex-filtration and Alg-filtration on the plane.
In the case of K = 31 , the double complex is described as Figure 3.
|ai |\Generally, due to [40] if K is alternating, then HF
K(K, j) ∼= Z(s+
σ . In this case the)2homology is determined by the Alexander polynomial and the knot signature. However,for non-alternating knot, it is diﬀerent in general, see Theorem 3.7.
Other important variations of definitions of knot Floer homology are the ones by grid
diagram [17] and Kauﬀman state [40]. By the former, knot Floer homology is computedfrom a purely combinatorial information of a knot. By the latter, on the knot Floer
homology, a geometric meaning of a state sum is given.
3.2

Genus and fiberness.

In this section we give the characterizations of Seifert genus and fiberness on knot Floer
homology. The knot Floer homology detects the Seifert genus g(K).
Theorem 3.2 ([29]) Let K be a knot in S 3 . Then we have\g(K) = max{s|HF
K(S 3 , K, s) ̸= {0}}.
Furthermore, Heegaard Floer homology detects the fibered-ness of knot.
Theorem 3.3 ([20]) Let K be a knot in S 3 . Then K is a fibered knot if and only if\
HF
K(S 3 , K, g(K)) ∼
= Z.
12

\
Figure 3: Diﬀerentials of CF K ∞ (S 3 , K) and CF
K(S 3 , K) in it.

3.3

Legendrian and transverse knot.

Legendrian knot (or transverse knot) in a contact 3-manifold is a curve tangent (or transverse) on any plane of the standard contact S 3 at any point on the knot. Isotopy invariants
for those knots are classically the Thurston-Bennequin invariant (tb), and the rotation
number (rot) for Legendrian knots, and the self-linking number (sl) for transverse knots.
These and smooth isotopy type are called classical invariants. Classical invariants distinguish these knots but not complete. For example Chekanov [3] and Eliashberg foundnon-isotopic Legendrian knots (CE-pairs) with the same classical invariants.
In the Heegaard Floer theory, (Legendrian or transverse) (unclassical) isotopy invariants in (S 3 , ξstd ) are included. Let L be a Legendrian knot in the standard contact
S 3 . Ozsváth, Szabó and Thurston defined in [41] defined Legendrian isotopy invariantλ(L) ∈ HF K − (S 3 , L) using the grid diagram. After that, in [14] Lisca, Ozsváth, Stipsicz,
and Szabó defined another Legendrian invariant L(L) ∈ HF K − (S 3 , L) using open bookdecomposition compatible with the contact structure, which it is proved these invariants
are equivalent to each other in [1]. For the transverse knot invariant T, which is definedby Legendrian approximation, the same history has been traced. This equivalence is so
fruitful in terms of what some deeper information may be appeared. The image of theb\
hat version homology HF
K(L) is referred to as L(L).
The invariants reprove CE-pair2
is not isotopic and a link 63 is not simple (i.e., the knot is determined uniquely fromclassical invariants), and transverse CE-pair, which is trivial DGA invariant are also not(transverse) isotopic [23]. The DGA invariant is sometimes not useful for Legendrian
approximation of transverse knot because it vanishes for a stabilized knot. However theinvariant L is sensitive for stabilized knots (it is multiplied by U or identity depending13

on stabilization). By using T, it is proven that the CE’s twist knots are transverse notsimple. The Alexander and Maslov gradings of L and T compute as follows:1
Alex(L(L)) = (tb(L) − rot(L) + 1)
2
2Alex(L(L)) − M as(L(L)) = d3 (ξ).
M (T(T )) = 2A(T(T )) = sl(T ) + 1.
Here d3 (ξ) is the invariant of 2-plane field on 3-dimensional obstruction defined in [7].
3.4

L-space surgery

The lens space knot K is a knot yielding a lens space by an integral Dehn surgery. This
research field is the one which was remarkably developed by Heegaard Floer homology.
Ozsváth and Szabó obtained the Alexander polynomial restriction for lens space knot (orfor L-space knot more strongly).
Theorem 3.4 Let K be an L-space knot. Then the Alexander polynomial is of form:m∆K (t) = (−1) +m∑(−1)m−j (tnj + t−nj ),j=1

where the exponents nj give an increasing sequence:
0 < n1 < n2 < · · · < nm = d.
As a corollary, this theorem and Theorem 3.3 imply that any L-space knot must be afibered knot. By this theorem many knots are ruled out from L-space knot, in particularlens space knot. Besides, if K yields a lens space L(p, q), then due to [12], we have2g(K) − 1 ≤ p.

(1)

This inequality follows from HF + (S03 (K), i) ∼= T + (ti ) (i ̸= 0) and g(K) = d (degree of∆K (t)). Greene in [8] improved the inequality (1) in the case of lens space knot in S 3 asfollows:√
2g(K) − 1 ≤ p − 2 (4p + 1)/5
by using Theorem 3.4 essentially. [8] classifies all lens spaces obtained by Dehn surgeriesin S 3 by extracting some information of a lattice embedding coming from embeddingof the resolution of a lens space in a definite 4-manifold. This implies this property in
Theorem 3.4 is so strong among the other L-space knot restrictions.
3.5

Dehn surgery formula3
We introduce the Heegaard Floer homology of a Dehn surgery Sp/q(K) in [27], [38], [39].
Briefly speaking, the computation of the homology is the mapping cone technique of thechain complex. Let K be a knot in S 3 . We denote Alex and Alg by j- and i-coordinate.

14

+Let C be CF K ∞ (K). Let A+
s = C{min{i, j − s} ≥ 0} and B = C{i ≥ 0}. We define+
Ai , B+ , v + , h+ as follows:+++A+
i = ⊕s∈Z (s, A⌊(i+ps)/q⌋ ), Bi = ⊕s∈Z (s, B )++++v + : (s, A+⌊(i+ps)/q⌋ ) → (s, B ), h : (s, A⌊(i+ps)/q⌋ ) → (s + 1, B ).
+The second components vs+ and h+
and h+ on the level s are defined as follows:s of v+
The map vs is defined to be the quotient to i ≥ 0. The map h+s is defined to be the
⌋
and
the
identification
with i ≥ 0. Here wecomposition of the quotient to j ≥ ⌊ i+psq++
define Ai → B to be+
Di,p/q(s, as ) = {(s, v⌊+i+ps ⌋ (as ) + h+i+p(s−1) (as−1 )}.
⌊

q

q

⌋

3
Theorem 3.5 HF + (Sp/q(K), i) is isomorphic to the homology of the mapping cone X+
i,p/q with+respect to Di,p/q .
+The mapping cone with respect to Di,p/q
is the chain complex which the generators are++
Ai ⊕ Bi and the diﬀerential is
(
)
∂A+i
0
.
+
Di,p/q∂B+ i

In the case where q = 1 and p ≥ 2g(K) − 1, as in [27], the module of HF + of Sp3 (K)3or S−p(K) is isomorphic to
+
3
b +
∼
HF∗+ (Sp3 (K), [s]) ∼= H∗+ (2s−p)2 −p (A+s , [s]), HF∗ (S−p (K), [s]) = H∗+ p−(2s+p)2 ( As , [s])4p4pwhere b A+ = C{min{i, j − s} ≥ 0}. Further the homology of Âs := C{max{i, j − s} = 0}3
and b Âs := C{min{i, j − s} = 0} compute the hat version of Sp3 (K) and S−p(K) in thesame way.
Ni and Wu [21] gave the correction term formula of rational surgery of a knot. The
spinc structures of Dehn surgery of a knot are identified with Z/pZ naturally.
Theorem 3.6 Let K be a knot in S 3 and p, q positive integers. For 0 ≤ i < p we have3d(Sp/q(K), i) = d(L(p, q), i) − 2 max{V⌊ i ⌋ , V⌊ p+q−1−i ⌋ }.
qq

Here Vi is a concordance invariant will be defined as below.
3.6

Satellite knots\
Hedden computed HF
K of the (+)-Whitehead double D+ (K, n) of a knot K. The coefficient is F = Z/2Z. Let g be the genus of K.
Theorem 3.7 ([9]) Let K be a knot in S 3 . The case of n ≥ 2τ (K):
 n−2g−2 g⊕j=−g [H∗−1 (F(K, j))]2 i = 1

F(1)
\
⊕gj=−g [H∗ (F(K, j))]4
i=0
HF
K(D+ (K, n), i) = F2n−4g−3(0)
 n−2g−2 g2F(−1) ⊕j=−g [H∗+1 (F(K, j))] i = −1.
15The case of n < 2τ (K): 2τ (K)−2g−22τ (K)−n⊕ F(0)⊕gj=−g [H∗−1 (F(K, j))]2
F(1)4τ (K)−4g−44τ (K)−2n−1\
HF
K(D+ (K, n), i) = F(0)⊕ F(−1)⊕gj=−g [H∗ (F(K, j))]4 2τ (K)−2g−22τ (K)−nF(−1)⊕ F(−2)⊕gj=−g [H∗+1 (F(K, j))]2i=1
i=0
i = −1.

Here τ (K) (concordance invariant defined in the knot Floer theory) will be defined in thenext section. Since the Alexander polynomial of D+ (K, n) is −nt + (2n + 1) − nt−1 , thistheorem says that knot Floer homology is strictly stronger invariant than Alexander polynomial. For example, it follows immediately that if D+ (K, n) is fibered non-trivial knot,
then the double is figure-8 knot only (K= the unknot and n = 1) by using Theorem 3.3
and the reduced knot filtration formula of τ (K) in [44]. Thus, due to [9] the τ -invariantof D+ (K, n) is following:{
0 n ≥ 2τ (K)τ (D+ (K, n)) =1 n < 2τ (K).
Hence, untwisted (n = 0) Whitehead double with positive τ (K) is not smoothly slice. On
the other hand, Freedman’s result [5] says that any knot with trivial Alexander polynomial is topologically slice. Then, for example for any positive torus knot K, D+ (K, 0) is
topologically slice but not smoothly slice knot. Such a knot gives an exotic R4 by Freedman’s result [5] and Gompf’s result [6]. The result of 0-framed attachment of the knot
on the 4-ball is embeddable in a 4-manifold homeomorphic to R4 but not in R4 .

4

Concordance invariants

Two knots K0 , K1 are defined to be (knot) concordant if there exists a smoothly embedded
annulus f : S 1 × [0, 1] ,→ S 3 × [0, 1] such that f |S 1 ×{i} = Ki . The set C of the equivalentclasses of knots by knot concordance is an abelian group so that the connected-sum is the
addition. The zero element in C corresponds to the equivalent class of slice knot. We say aknot K to be slice, if there exists a proper smoothly embedded disk in 4-ball such that theboundary is isotopic to K. In the Heegaard Floer package there exist some concordance
invariants. We give the definitions of those. In this section we denote CF K ∞ (K) by C.
4.1

The τ -invariant and variations.

Let F(K, j) be C{Alex ≤ j, Alg = 0}. Then we have
· · · ⊂ F(K, i − 1) ⊂ F (K, i) ⊂ F (K, i + 1) ⊂ · · ·d (S 3 ). Knot Floer homology HF\and ∪i∈Z F(K, i) = CF
K(S 3 , K, j) is the graded homologyd (S 3 ) induces a map
H∗ (F(K, j)/F(K, j − 1)). The inclusion map ιm : F(K, m) ⊂ CFd 3 ∼ιm
∗ : H∗ (F(K, m)) → HF (S ) = Z on the homology. If m is suﬃciently small, thend (S 3 ).
F(K, m) is the 0-map. If m is suﬃciently large, then F(K, m) is isomorphic to CF
The τ -invariant is defined as the following minimal:τ (K) := min{m|ιm∗ is non-trivial}.
16

Theorem 4.1 ([28]) The τ -invariant gives a group homomorphism τ : C → Z.
This invariant τ gives a bound of the 4-ball genus g4 (K).
Theorem 4.2 ([28]) Let K be a knot. Then the following inequality holds:|τ (K)| ≤ g4 (K).
The knot signature σ(K) is also similar inequality |σ(K)| ≤ 2g4 (K). For any alternatingknot, σ(K) = −2τ (K). Generally, this inequality on τ is sharper than the one on σ.
For τ , various sharpener invariants exist. To compute τ of a cable knot, Hom in [10]introduced ϵ-invariant:d (S 3 ) → b Âτ (K) : trivial on the homology,CF1d (S 3 ): trivial on the homology,
ϵ = −1 Âτ (K) → CF0otherwise.
Theorem 4.3 ([10]) Let Kp,q be the (p, q)-cable knot of K with p > 0. Then, we have{pτ (K) + (p − 1)(q − ϵ(K))/2
ϵ(K) ̸= 0
τ (Kp,q ) =(p − 1)(q − sgn(q))/2 = τ (Tp,q ) ϵ(K) = 0
Here Tp,q is the (p, q)-torus knot.
Ni and Wu defined knot concordance invariant Vk to compute the correction term of Dehn+
surgery of a knot in [21]. Let vk+ be a map A+
k → B which is defined in the similar way
to the previous section (note that the degree of the map is slightly diﬀerent from theprevious section). The restriction to the T ∞ -part gives a U -power map. We define the++exponent to be Vk . The same invariant Hk replacing vk+ with h+k : Ak → B is given. Vk
is a decreasing Z≥0 -valued function on Z and Vk+1 = Vk or Vk − 1. The following minimalvalue ν(K) := min{k|Vk = 0} is a concordance invariant and τ (K) ≤ ν(K) ≤ g4 (K) holds
in [11]. This ν(K) is a sharpener invariant than τ (K).
4.2

Υ-invariant.

Recently, Ozsváth, Stipsicz and Szabó in [24] (also Livingston [16]) defined the concordance invariant ΥK (t), whose value is a continuous function on the interval [0, 2]. And
Υ is a group homomorphism C → Cont([0, 2]). ΥK (t) is a piecewise linear function withfinite non-smooth points, which is the number of the smooth points is also concordance
invariant. We have Υ′K (0) = −τ (K).
Here we define Υ according to [16]. Let Ft be an s-filtered chain complex with Ft,s =C{ 2t Alex + (1 − 2t )Alg ≤ s}. Then Υ is defined to be ΥK (t) = −2ν(K, Ft ), whereν(K, Ft ) = min{s|Image(H∗ (Ft,s ) → H∗ (C)) contains the non-trivial element of grading 0}.
ΥK (t) give a 4-ball genus bound:
Theorem 4.4 ([24]) For any 0 < t < 1,|ΥK (t)|/t ≤ g4 (K).
17

This invariant Υ is weaker than {Vk } invariant but is useful for the linear independence
on concordance group of a family of knots. For example, consider torus knots {Tp,p+1 }.
Let G0 be generators in C with grading 0. Any element in G0 lies on the boundary of theconvex hull of G0 in R2 . Since ΥTp,p+1 has a convex function with p singular points, these
are linearly independent in Cont([0, 2]) and the knots are also linearly independent.
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Institute of Mathematics
University of Tsukuba
Ibaraki 305-8571
JAPANE-mail address: tange@math.tsukuba.ac.jp
筑波大学
20

丹下

基生