DEFORMATIONS OF LATTICE COHOMOLOGY AND THE UPSILON INVARIANT ANTONIO ALFIERI Abstract. We introduce deformations of lattice cohomology corresponding to the knot homologies found by Ozsv´ ath, Stipsicz and Szab´ o in [12]. By means of holo- morphic triangles counting, we prove equivalence with the analytic theory for a wide class of knots. This yields combinatorial formulae for the upsilon invariant. 1. Introduction Based on the methods used by Floer [4] in Symplectic Topology to study the intersec- tion properties of Lagrangian submanifolds, in [19, 18] Ozsv´ ath and Szab´o introduced a package of three-manifold invariants called Heegaard Floer homology. This eventu- ally led to the definition of Knot Floer homology [17, 23], a related package of knot invariants. In the last two decades Knot Floer homology proved to be an extremely powerful tool for the study of knots in S 3 [14, 22, 10]. In particular, in the realm of knot concordance, invariants like the upsilon invariant Υ K (t) introduced by Ozsv´ ath, Stipsicz, and Szab´ o [12] turned out to be extremely efficient to decide certain questions about independence in the knot concordance group C . In [2], extending the definition of the upsilon invariant to knots inside rational homology spheres, new invariants for knots in S 3 were introduced. Indeed, given a knot K ⊂ S 3 one can consider its branched double cover Σ(K ). This is a rational homology sphere carrying a unique spin structure s 0 . By considering the pull-back e K ⊂ Σ(K ) of K to Σ(K ) we get a null-homologous knot e K ⊂ Σ(K ). One can see that the upsilon invariant Υ K,s 0 (t) of (Σ(K ), e K ) with respect to s 0 yields a knot concordance invariant of K ⊂ S 3 . Further invariants carrying information about the concordance type of K can be obtained by considering the invariants Υ K,s (t) of e K associated to the other Spin c structures of the double branched cover Σ(K ). More specifically in [2] the following theorem, reminiscent of the results of Grigsby, Ruberman, and Strle [6], was proved. Theorem 1.1 (Alfieri, Celoria & Stipsicz). If K is a slice knot then there exists a subgroup G<H 2 (Σ(K ); Z) of cardinality p |H 2 (Σ(K ), Z)| such that Υ K,s 0 +ξ (t)=0 for all ξ ∈ G. In [2] we performed computations of the invariants Υ K,s 0 +ξ (t), ξ ∈ H 2 (Σ m (K ), Z) for some families of knots having genus one doubly-pointed Heegaard diagrams. These are known as (1, 1) knots, and were first studied by Rasmussen [24]. In this paper we address the issue of computations in the case of graph knots. In what follows a graph knot is a knot that can be described by means of a pluming tree Γ with one unframed vertex v 0 , see Figure 1 below. Note that the knot of an irreducible plane curve singularity f (x, y) = 0 is a graph knot. Indeed, so is its lift to the branched double cover Σ(K ). (See Proposition 2.6 below.) The study of graph knots was initiated by Ozsv´ ath, Stipsicz, and Szab´ o in [11]. arXiv:2010.07511v1 [math.GT] 15 Oct 2020
Transcript
DEFORMATIONS OF LATTICE COHOMOLOGYAND THE UPSILON INVARIANT
ANTONIO ALFIERI
Abstract. We introduce deformations of lattice cohomology corresponding to theknot homologies found by Ozsvath, Stipsicz and Szabo in [12]. By means of holo-morphic triangles counting, we prove equivalence with the analytic theory for a wideclass of knots. This yields combinatorial formulae for the upsilon invariant.
1. Introduction
Based on the methods used by Floer [4] in Symplectic Topology to study the intersec-tion properties of Lagrangian submanifolds, in [19, 18] Ozsvath and Szabo introduceda package of three-manifold invariants called Heegaard Floer homology. This eventu-ally led to the definition of Knot Floer homology [17, 23], a related package of knotinvariants.
In the last two decades Knot Floer homology proved to be an extremely powerful toolfor the study of knots in S3 [14, 22, 10]. In particular, in the realm of knot concordance,invariants like the upsilon invariant ΥK(t) introduced by Ozsvath, Stipsicz, and Szabo[12] turned out to be extremely efficient to decide certain questions about independencein the knot concordance group C.
In [2], extending the definition of the upsilon invariant to knots inside rationalhomology spheres, new invariants for knots in S3 were introduced. Indeed, given aknot K ⊂ S3 one can consider its branched double cover Σ(K). This is a rationalhomology sphere carrying a unique spin structure s0. By considering the pull-back
K ⊂ Σ(K) of K to Σ(K) we get a null-homologous knot K ⊂ Σ(K). One can see that
the upsilon invariant ΥK,s0(t) of (Σ(K), K) with respect to s0 yields a knot concordanceinvariant of K ⊂ S3.
Further invariants carrying information about the concordance type of K can be
obtained by considering the invariants ΥK,s(t) of K associated to the other Spinc
structures of the double branched cover Σ(K). More specifically in [2] the followingtheorem, reminiscent of the results of Grigsby, Ruberman, and Strle [6], was proved.
Theorem 1.1 (Alfieri, Celoria & Stipsicz). If K is a slice knot then there exists a
subgroup G < H2(Σ(K);Z) of cardinality√|H2(Σ(K),Z)| such that ΥK,s0+ξ(t) = 0
for all ξ ∈ G.
In [2] we performed computations of the invariants ΥK,s0+ξ(t), ξ ∈ H2(Σm(K),Z)for some families of knots having genus one doubly-pointed Heegaard diagrams. Theseare known as (1, 1) knots, and were first studied by Rasmussen [24].
In this paper we address the issue of computations in the case of graph knots. Inwhat follows a graph knot is a knot that can be described by means of a plumingtree Γ with one unframed vertex v0, see Figure 1 below. Note that the knot of anirreducible plane curve singularity f(x, y) = 0 is a graph knot. Indeed, so is its liftto the branched double cover Σ(K). (See Proposition 2.6 below.) The study of graphknots was initiated by Ozsvath, Stipsicz, and Szabo in [11].
arX
iv:2
010.
0751
1v1
[m
ath.
GT
] 1
5 O
ct 2
020
2 ANTONIO ALFIERI
Theorem 1.2. Let K be a null-homologous graph knot associated to a plumbing Γ withunframed vertex v0. Suppose that G = Γ− v0 is a negative-definite plumbing tree withat most two bad vertices. Let s be a Spinc structure of Y (G) and k a characteristicvector of the intersection form of the associated plumbing of spheres X(G) representingthe spinc structure s on the boundary. Then
ΥK,s(t) = −2 minx∈Zs
χt(x) +
(k2 + |G|
4− t k · F − F
2
2
),
where χt denotes the twisted Riemann-Roch function
χt(x) = −1
2
((k + tv∗0) · x+ x2
),
and F ∈ H2(X(G),Q) a homology class representing −v∗0 ∈ H2(X(G),Z).
This leads to related formulae for the τ -invariants introduced by Grigsby, Ruber-man, and Strle [6]. These are related to the Υ-invariant via the identity τs(K) =− limt→0+ ΥK,s(t)/t.
The proof of Theorem 1.2 we outline presently is an adaptation of the argumentpresented in [15]. (A similar type of work was carried on in [1] where the same techniquewas employed to perform computations in the setting of Instanton Floer homology.)There are two main ingredients. The first is a surgery exact triangle involving the”t-modified” knot homologies introduced by Ozsvath, Stipsicz, and Szabo in [12].Compare with the knot Floer exact triangle of Ozsvath and Szabo [17, Theorem 8.2].
Theorem 1.3. Let K ⊂ Y be a knot in a rational homology sphere, and C ⊂ Y aframed knot in its complement. Let λ denotes the framing of C, and µ its meridian.Suppose that the surgery three-manifolds Yλ(C) and Yλ+µ(C) are rational homologyspheres. Then there is an exact triangle
tHFK−(Y,K) tHFK−(Yλ(C), K)
tHFK−(Yλ+µ(C), K)
. (1)
Secondly, in the spirit of [9, 25], we introduce a combinatorial invariant associated toalgebraic knots. This is a one-parameter family of knot homologies tHFK∗(Γ) with thesame formal structure as tHFK(Y,K). In Section 6 we establish a long exact sequenceplaying the role of the surgery exact triangle in the combinatorial theory.
Theorem 1.4. There is a long exact sequence of modules
// tHFKp(Γ− v)φ∗ // tHFKp(Γ)
ψ∗ // tHFKp(Γ+1(v))δ // tHFKp−1(Γ) //
Our main result is then obtained by comparing the two exact triangles as in [15].Note that the combinatorial theory we develop here builds on the work carried on in[11] by Ozsvath, Stipsicz, and Szabo.
Acknowledgements I would like to thank Peter Ozsvath for showing interest in this projects while
in its early stages, and for some useful advice. I would also like to thank Andras Stipsicz, Andras
Juhasz, Ian Zemke, Daniele Celoria, Andras Nemethi and Liam Watson for useful conversations.
Most of this work was carried on in the summer of 2018 when I was partially supported by the
NKFIH Grant Elvonal (Frontier) KKP 126683 and K112735.
3
2. Embedded resolutions of curves and branched double coverings
In what follows we will deal with knots in rational homology spheres. As a conse-quence we will adopt the following terminology.
Definition 2.1. A knot is a pair (Y,K) where Y is a smooth closed three-manifoldand K is the image of a C∞ embedding S1 → Y .
A knot (Y,K) is called null-homologous if [K] = 0 in H1(Y ;Z). This is the sameas asking that K has a Seifert surface, i.e. that there exists a surface with boundaryΣ ⊂ Y such that ∂Σ = K. Most of the time in what follows we will deal with null-homologous knots (Y,K) where the three manifolds Y is a rational homology sphere,that is H∗(Y,Q) ' H∗(S
3,Q). Furthermore, knots and three-manifolds are alwaysassumed to be oriented. Note that in a rational homology sphere knots are guaranteedto be rationally null-homologous, that is [K] = 0 in H1(Y ;Q).
Two knots (Y0, K0) and (Y1, K1) are called rationally homology concordant if there isa rational homology cobordism X : Y0 → Y1 containing a smoothly embedded cylinderC ⊂ X such that ∂C = C ∩ ∂X = K0 ∪ −K1. If Y0 and Y1 are equipped withspinc structures, say s0 and s1, and these extends over X we say that (Y0, K0, s0) and(Y1, K1, s1) are Spinc rationally homology concordant.
There are basically two ways one can use to represent a knot (Y,K):
(1) via a mixed diagram that is a pair (L,K) where L represents a framed link Lin the three-sphere S3, and K a knot lying in the link complement S3 − L,
(2) or via a doubly-pointed Heegaard diagram that is a Heegaard diagram
(Σ, α1, . . . , αg, β1, . . . , βg)together with a pair of base points z, w ∈ Σ − α1 − · · · − αg − β1 − · · · − βglying in the complement of the α- and the β-curves [17].
2.1. Algebraic knots. Let (C, 0) ⊂ (C2, 0) be the germ of an irreducible plane curvesingularity. By looking at the intersection of C ⊂ C2 with a small sphere Sε(0) ⊂ C2
centred at the origin of the axes we get a knot (S3, K). Knots of this kind are usuallycalled algebraic knots.
The topology of an algebraic knot (S3, K) = (Sε(0), Sε(0) ∩ C) can be understoodby means of an embedded resolution of the curve singularity (C, 0) ⊂ (C2, 0). By this
we mean a complex map ρ : C2#nCP 2 → C2 such that:
• ρ defines an isomorphism C2#nCP 2 \ ρ−1(0)→ C2 \ 0 away from the origin,• the exceptional divisor,
E := ρ−1(C) = E1 ∪ · · · ∪ En ∪ C ,
is an algebraic curve with smooth components E1, . . . , En, C. Furthermore,
Ei ' CP 1, i ∈ 1, . . . , n, and C = ρ−1(C \ 0)• no three components of the exceptional divisor pass through the same point,• the exceptional divisor E has only normal crossing singularities, that is: the
intersection of two components of E is locally modelled on
(x, y) ∈ C2 | xmyl = 0for some m, l ≥ 1 (the multiplicities).
The exceptional divisor E of an embedded resolution C2#nCP 2 → C2 is encoded inthe so called resolution graph. This is the graph Γ having as vertices the irreduciblecomponents of E and an edge connecting each pair of intersecting components. Note
4 ANTONIO ALFIERI
-
•§÷k-
,
K
Figure 1. A surgery diagram of the trefoil knot.
that the resolution graph comes with a distinguished vertex (the one corresponding
to the proper transform C of the curve C) and integers e1, . . . , en labelling the othervertices. These are defined as the self intersections ei = Ei ·Ei of the curves E1, . . . , Enin the blow-up C2#nCP 2.
Example 2.2. The curve C = (x, y) ∈ C2 : x2 +y3 = 0 has an isolated singularityat the origin 0 ∈ C2. After three consecutive blow-ups [5, Example 7.2.3 (a)] we get
an embedded resolution C2#3CP 2 → C2 with graph:
•• •
•
E3E1 E2
C
.
Here E21 = −3, E2
2 = −2, and E23 = −1. Furthermore the curves E1, E2, and E3 have
Note that once we erase the unframed vertex from the embedded resolution graphof a curve singularity we get a negative-definite plumbing tree representing S3. (This
is because an embedded resolution C2#nCP 2 → C2 is in particular a resolution of thetrivial surface singularity (C2, 0).)
The resolution graph Γ gives rise to a surgery diagram representing the algebraicknot (S3, K) associated to the curve singularity (C, 0) ⊂ (C2, 0). Indeed, given anyrooted tree (Γ, v0) and a weight assignment m : Γ \ v0 → Z, we can look at theplumbed three-manifold Y (G) associated to G = Γ \ v0 and consider the knot(Y (G), K) represented by the unframed vertex as in Figure 1. This gives rise toan interesting class of knots.
Definition 2.3. A knot (Y,K) that can be presented by means of a plumbing tree Γwith one unframed vertex v0 is called a graph knot.
Example 2.4. Let (X, 0) ⊂ CN be a complex surface singularity and (C, 0) ⊂ (X, 0)be a complex curve singularity. Then (Y,K) = (Sε(0) ∩X,Sε(0) ∩ C) is an algebraicknot in the sense of this new definition.
Example 2.5. Let (Y1, K1) and (Y2, K2) are algebraic knots with plumbing diagramsΓ1 and Γ2. Then the connected sum (Y1#Y2, K1#K2) is an algebraic knot. IndeedΓ = Γ1 ∗ Γ2, the graph obtained joining Γ1 and Γ2 along their unframed vertices, givesrise to a diagram for (Y1#Y2, K1#K2).
5
2.2. Branched coverings. A knot (S3, K) gives rise to a knot in a rational homology
sphere (Σ(K), K) where the ground three-manifold is given by the branched double
cover Σ(K) of S3 along K, and K = Fix(τ) by the fixed point set Fix(τ) of the coveringinvolution τ : Σ(K)→ Σ(K).
Proposition 2.6. The double branched cover (Σ(K), K) of an algebraic knot (S3, K)associated to a plane curve singularity (C, 0) ⊂ (C2, 0) is algebraic.
Proof. Let f(x, y) = 0 be an equation for C ⊂ C2, and ρ : C2#nCP 2 → C2 denote an
embedded resolution. Furthermore, let ρ−1(0) = E1 ∪ · · · ∪ En ∪ C = E denote theexceptional divisor of the embedded resolution.
First we note that φ = f ρ defines an equation for E. This assigns multiplicitiesm1, . . . ,mn ≥ 1 to the rational components of the exceptional divisor. By looking at
the branched double cover of C2#nCP 2 along the Weil divisor D = C +∑n
i=1miEi(see [5, pp. 239-241] ) we obtain a smooth surface S representing a resolution of thesurface singularity (S, 0) with equation z2 = f(x, y).
The various components of the exceptional divisor E ⊂ C2#nCP 2 lift to S as
explained in [5, p. 252] and form a configuration of curves inside S. (Note that for thepurposes of singularity theory one usually ignores the pull-back of the strict transform
C to S, while locating the latter here plays a crucial role.) The adjacency graph of
this configuration gives rise to a tree Γ describing (Σ(K), K).
Example 2.7. Following the procedure outlined in the proof of Proposition 2.6 we geta diagram for the branched double cover of the resolution graph of the trefoil knot:
•
•• •
•
−2
−1
−3 −3
K
.
This represents a knot in the lens space L(3, 2):
•
•• •
−1
−2−3 −3= •• •
−1−3 −3= ••
−2−2= L(3, 2) .
Similar computations can be run for all torus knots starting from the defining equa-tion xp + yq = 0.
3. t-Modified knot Floer homology, and the upsilon invariant ofknots in rational homology spheres
In [12] Ozsvath, Stipsicz and Szabo introduced a one-parameter family of knothomologies tHFK−(K) giving rise to knot invariants of knots in S3. In what followswe go through the straightforward generalisation of their construction taking intoaccount knots in rational homology spheres.
6 ANTONIO ALFIERI
3.1. Notation. In what follows we will work over the ring R of long power serieswith F coefficients. This is the commutative ring of infinite formal sums
∑α∈A q
α,with A ⊂ R≥0 well-ordered. One defines:(∑
α∈A
qα
)+
(∑β∈B
qβ
)=∑
γ∈A∪B
qγ
and (∑α∈A
qα
)·
(∑β∈B
qβ
)=∑
γ∈A+B
cγ · qγ ,
where A+B = α + β | α ∈ A, β ∈ B ⊂ R≥0 and
cγ = # (α, β) ∈ A×B | α + β = γ mod 2.
The ring R has the fundamental property that every finitely generated R-module Mis sum of cyclic modules [3, Section 11] i.e.
M ' Rk ⊕R/f1 ⊕ . . .R/fmfor some f1, . . . , fm ∈ R, and k ≥ 0 (the rank of M). Notice that the field of fractionof the ring R is given by
R∗ =
∑α∈A
qα
∣∣∣∣∣ A ⊂ R well-ordered
,
and that the rank of a finitely generated R-module M equals the dimension of MR∗ =M ⊗R R∗ as an R∗-vector space.
We will think R as a graded ring, with deg q = −1. Note that F[U ] → R via theidentification U = q2.
3.2. t-Modified Knot homologies. Let Y be a rational homology sphere. Recall[17] that a knot (Y,K) can be represented by a doubly-pointed Heegaard diagram(Σ,α,β, z, w). In the symmetric product Symg(Σ), the space of degree g divisorsover the genus g Riemann surface Σ, this specifies two half-dimensional, totally-realsubmanifolds Tα = α1×· · ·×αg and Tβ = β1×· · ·×βg, and two analytic submanifoldsVz = z × Symg−1(Σ) and Vw = w × Symg−1(Σ) of complex codimension one. Wedefine
CF (Tα,Tβ) =⊕
x∈Tα∩Tβ
R · x .
Given two intersection points x and y ∈ Tα∩Tβ consider the set π2(x,y) of homotopyclasses of topological disks u : D2 ' [0, 1]× R→ Symg(Σ) such that
• u (0× R) ⊆ Tα and u (1× R) ⊆ Tβ,• limt→−∞ u(s+ it) = x and limt→+∞ u(s+ it) = y.
For a generic choice of a path of almost-complex structures Js we can look at themoduli space M(φ) of solutions of the (perturbed) Cauchy-Riemann equation
∂u
∂s(s, t) + Js
(∂u
∂t(s, t)
)= 0 (2)
within a given homotopy class φ ∈ π2(x,y) as it was done in [19]. It turns out that ifwe restrict our attention to those classes with Maslov index µ(φ) = 1 then M(φ) is a
7
finite collection of lines (Gromov’s Compactness Theorem). In [19, Section 4] this ledto the definition of a differential
∂x =∑
y∈Tα∩Tβ
∑φ∈π2(x,y)µ(φ)=1
#
∣∣∣∣M(φ)
R
∣∣∣∣ q2nz(φ) · y
turning CF (Tα,Tβ) into a chain complex. Here nz(φ) = #|φ(D2) ∩ Vz| denotes theintersction with the divisor Vz. Notice that the differential of CF (Tα,Tβ) completelyignores the base point w. In fact, the chain homotopy type of CF (Tα,Tβ) only providesan invariant of the background three-manifold [19, Theorem 1.1].
In order to take into account the base point w and hence the knot K ⊂ Y , we canuse the following differential
∂tx =∑
y∈Tα∩Tβ
∑φ∈π2(x,y)µ(φ)=1
#
∣∣∣∣M(φ)
R
∣∣∣∣ qtnz(φ)+(2−t)nw(φ) · y for t ∈ [0, 2] ,
also recording the intersection with the divisor Vw. We will denote by tHFK−(Y,K)the homology of the resulting chain group CFt(Tα,Tβ) = (CF (Tα,Tβ), ∂t).
3.3. Spinc refinement. In [19, Section 2.6] Ozsvath and Szabo built a map sz :Tα∩Tβ → Spinc(Y ) associating to an intersection point a Spinc structure of Y . DefineCFt(Tα,Tβ, s) =
⊕sz(x)=sR · x . Since [19, Lemma 2.19] for any pair of intersection
points π2(x,y) is non-empty iff sz(x) = sz(y), we conclude that CFt(Tα,Tβ, s) is asub-complex of CFt(Tα,Tβ). We set tHFK−(Y,K, s) = H∗(CFt(Tα,Tβ, s)).
3.4. Gradings. Attached to an intersection point x ∈ Tα ∩ Tβ of a doubly pointedHeegaard diagram there are two rational numbers: the Alexander grading A(x) andthe Maslov grading M(x). These have the property that
A(x)− A(y) = nw(φ)− nz(φ) and M(x)−M(y) = µ(φ)− 2nz(φ)
where φ ∈ π2(x,y) is disk connecting x to y. We define a real-valued grading grt onCFt(Tα,Tβ) via the formula grt(x) = M(x) − tA(x). Note that ∂t drops the gradingby one [12, Lemma 3.3]. We set
ΥK,s(t) = maxgrt(ξ) | ξ ∈ tHFK−(Y,K, s) with qα · ξ 6= 0 for α > 0 .This is the upsilon invariant of the knot (Y,K) in the spinc structure s. In the sequelwe list some basic properties of the upsilon invariant.
Proposition 3.1. Let (Y,K) be a knot in a rational homology sphere and s a spinc
structure of Y , then
• ΥK,s(0) = d(Y, s) where d(Y, s) denotes the Heegaard Floer correction term ofthe pair (Y, s) as defined by Ozsvath and Szabo in [21],• ΥK,s(t) = τs(K) · t + d(Y, s) for all t > 0 close enough to zero, where τs(K)
denotes the τ -invariant defined by Grigsby, Ruberman, and Strle in [6],• the invariant ΥK,s(t) defined presently agrees with the one of [2], that is
ΥK,s(t) = −2 ·minξ
t
2A(ξ) +
(1− t
2
)j(ξ)
+ d(Y (G), s) ,
where ξ ranges between all cycles with Maslov grading d = d(Y (G), s)
Furthermore, ΥK,s(t) is an invariant of Spinc rational homology concordance.
8 ANTONIO ALFIERI
Proof. The first assertion follows from the fact that ∂t agrees with the differential ofCF−(Y, s) when t = 0. The fact that ΥK,s(t) = τs(K) · t + d(Y, s) for small values oft follows from the same argument of [12, Proposition 1.6]. While the last assertion isproved with the same argument presented in [8, Section 14.1].
The fact that ΥK,s(t) is an invariant of Spinc rational homology concordance wasproved in [2, Proposition 4.1].
3.5. Zemke’s inequality. A key feature of the Heegaard Floer correction terms is theOzsvath and Szabo inequality [21] relating the correction terms of two three-manifoldconnected by a negative-definite spinc cobordism. This asserts that given a spinc
cobordism (W, t) : (Y0, s0) → (Y1, s1) between two spinc rational homology spheres(Y0, s0) and (Y1, s1) with b1(W ) = b+
2 (W ) = 0 one has that:
d(Y1, s1) ≥ d(Y0, s0) +c1(t)2 + b2(W )
4.
In [26] Zemke proved that a similar inequality holds for the upsilon invariant.
Theorem 3.2 (Zemke [26]). Let (Y0, K0) and (Y1, K1) be two knots, and si ∈ Spinc(Yi)spinc structures. Suppose that there is a spinc cobordism (W, t) : (Y0, s0) → (Y1, s1)containing a properly embedded surface Σ → W with ∂Σ = K1 ∪ −K0. If b1(W ) =b+
Suppose that (Y,K) is an algebraic knot represented by a negative definite plumbingtree Γ. Let v0 ∈ Γ be the unframed vertex and G = Γ \ v0. Then K ⊂ Y = Y (G)bounds a smooth disk ∆ ⊂ X(G) properly embedded in the plumbing of spheresassociated to G. (If (Y,K) = (S3, K) is the link of a plane curve singularity (C, 0) ⊂(C2, 0) then X(G) is identified with the total space of a resolution ρ : C2#nCP 2 → C2
of C2 at the origin and ∆ = C is just the proper transform of C.)Since X(G) is simply connected, given a spinc structure s of Y (G) we can chose an
extension t to X(G). Then according to Theorem 3.2 one has that
ΥK,s(t) ≥k2 + |G|
4− t · k · F − F
2
2, (3)
where k = c1(t) denotes the first Chern class of t, and F ∈ H2(X(G),Q) a homologyclass representing the Poincare dual of φ(c) = −#(∆ ∩ c), c ∈ H2(X(G),Z). SeeSection 4.3 below.
In what follows we will show (Theorem 1.2) that the inequality displayed in Equa-tion (3) is sharp, that is ΥK,s(t) = (k2 +|G|)/4−t·(k ·F−F 2)/2 for some characteristicvector k, if the graph G satisfies suitable combinatorial hypothesis.
4. Deformations of lattice cohomology
4.1. A quick review of lattice cohomology. Let G be a negative-definite plumbingof spheres. Denote by X(G) the plumbing of spheres associated to G. Let
Char(G) = c1(s) : s ∈ Spinc(X(G))= k ∈ H2(X(G),Z) : k(x) ≡ x2 (mod 2) for every x ∈ H2(X(G),Z) .
9
be the set of characteristic vectors of the intersection form of X(G). Notice that, sinceX(G) is simply connected, a Spinc structure s of X(G) is uniquely determined by itsfirst Chern class c1(s). Thus, Char(G) ' Spinc(X(G)).
In what follows we will be interested in the Spinc structures of Y (G) = ∂X(G).These always extend over X(G), and two Spinc structures represented by characteristicvectors k and k′ induce the same Spinc structure on the boundary ∂X(G) = Y (G) ifand only if k − k′ ∈ 2 ·H2(X(G), Y (G)) ' 2 ·H2(X(G)).
In [16] a computational scheme for the Heegaard Floer homologies of graph manifoldswas described. This eventually lead to the definition of the lattice homology groups [9]whose construction we presently review. Let s denotes the number of vertices of G,and s a Spinc structure of Y . Think H2(X(G),Z) = Zs as a lattice in H2(X(G),R) =Rs. The points of Zs ⊂ Rs specify the vertices of a subdivision into hypercubes ofH2(X(G),R) = Rs, and hence a CW-complex decomposition of Rs. A p-cell of thisCW-complex decomposition is specified by a pair (`, I) where ` ∈ H2(X(G),Z) = Zs,and I ⊂ G with |I| = p. More specifically, we associate to such a pair (`, I) the|I|-cell corresponding to the convex hull of ` +
∑v∈J v | J ⊂ I. Fix a reference
characteristic vector k ∈ Char(G) representing s, and set χk(x) = −12(k(`) + `2). This
is the Riemann-Roch quadratic form associated to the characteristic vector. For ap-cell of the latter CW decomposition of Rs we set
wk() = maxvertices of
χk(v) .
For l ∈ Z consider the sub-level set Ml =⋃wk()≤l and form the chain complex
CF∗(G, s) =⊕l∈Z
C∗(Ml,F) ,
where C∗(−,F) denotes CW homology with F-coefficients. (Note that sub-level setsare sub-complexes of Rs since wk(i) ≤ wk() for every (p−1)-dimensional face i ofa given p-cell .) This is the lattice homology chain group associated to (G, s). Noticethat the inclusions . . .Ml−1 → Ml → Ml+1 . . . induce a chain map U : CF∗(G, s) →CF∗(G, s) turning CF∗(G, s) into a chain complex over the ring F[U ]. Notice thatCF∗(G, s) has a natural F[U ]-basis: B = ∈ C∗(Mwk(),Z) : p-face of Rs, 0 ≤p ≤ s. With respect to B the differential of CF∗(G, s) reads as:
∂ =∑i
Uw()−w(i) ·i , (4)
where the sum is extended to all (p− 1)-dimensional faces i of .
4.2. Gradings. In addition to the grading induced by the homological grading ofthe C∗(Mt,Z) summands, CF∗(G, s) has another grading corresponding to the Maslovgrading of the analytic theory:
gr() = dim()− 2wk() +k2 + |G|
4,
for a basis element ∈ B. This is then extended to F-generators via the identitygr(U j ·) = gr()− 2j.
4.3. The filtration of an algebraic knot. An algebraic knot K ⊂ Y (G) is a knotthat can be described by a plumbing tree Γ with one unframed vertex v0 such that
10 ANTONIO ALFIERI
G = Γ− v0. As in the analytic theory, the choice of such a knot K induces a filtrationon CF∗(G, s). We define the Alexander grading of a generator ∈ B by
A() = wk+2v∗0()− wk() +
k · F − F 2
2.
where F ∈ H2(X(G),Q) is a rational homology class representing the Poincare dual1
of −v∗0, i.e. such that F ·v = −v∗0 ·v for each v ∈ G. We define the Alexander grading ofa chain ξ =
∑mj=1 U
mjj as the maximum of the Alexander grading of its components
A(ξ) = maxj A(j) − mj. Note that the multiplication by U drops the Alexandergrading by one.
Proposition 4.1. A(∂) ≤ A().
Proof. We have to prove that for a p-cell we have
A(Uwk()−wk(i) ·i) ≤ A()
for every (p− 1)-face i of . Since multiplication by U drops the Alexander gradingby one everything boils down to prove that
A(i)− (wk()− wk(i)) ≤ A() .
Substituting the value of the Alexander filtration, and canceling (k ·F +F 2)/4 on bothsides we get
wk+2v∗0(i)−wk(i) − (wk() −wk(i)) ≤ wk+2v∗0
()−wk() ,
On the other hand the inequality wk+2v∗0(i) ≤ wk+2v∗0
() follows immediately fromthe definitions.
Remark 4.2. Exactly as in the analytic theory the group CF(G, s) has an algebraicfiltration j. For a q-cell ⊂Ml this is given by j() = wk()− l.
4.4. Proof of Theorem 1.2 for rational graphs. Recall that a vertex v of a neg-ative definite plumbing tree G is said to be bad if deg(v) > −v2. Using the numberof bad vertices we can partition algebraic knots into complexity classes: we say thata knot K ⊂ Y (G) is of type-k if it can be represented by a plumbing diagram Γ withunderlying plumbing tree G having no more than k bad points.
We now prove that (3) is sharp in the special case of algebraic knots that can berepresented by means of a plumbing diagram with no bad points.
Proposition 4.3. Suppose that K is an algebraic knot of type-0 (no bad vertices) then
ΥK,s(t) = −2 minx∈Zs
χt(x) +
(k2 + |G|
4− t k · F − F
2
2
), (5)
where χt denotes the twisted Riemann-Roch function
χt(x) = −1
2
((k + tv∗0) · x+ x2
).
1The class of v0 makes no sense in H2(X(G);Z) since v0 does not represent a closed surface inX(G). On the other hand, v0 represents a properly embedded disk ∆ ⊂ X(G) and hence a class inH2(X(G), ∂X(G);Z). Thus we can consider its Poincare dual v∗0 ∈ Hom(H2(X(G);Z),Z). This ischaracterized by the property that v∗0 · v = 1 if v0v is in Γ, and zero otherwise.
11
Proof. According to [11] CFK∞(K,Y (G), s) has the same filtered chain homotopytype of (CF−(G, s)⊗F[U ] F[U,U−1], ∂, A). Thus
ΥK,s(t) = −2 ·minξ
t
2A(ξ) +
(1− t
2
)j(ξ)
+ d(Y (G), s) (6)
where the minimum is taken over all cycles ξ with Maslov grading d = d(Y (G), s)representing the generator of Hd(CF−(G, s)⊗F[U ] F[U,U−1]) = F.
Since Hp(Rs,Z) = 0 for p > 0 any p-cycle ξ ⊂ Rs eventually bounds a (p+ 1)-chainin Ml. Hence the minimum in Equation (6) can be taken over all cycles of the formξ = U−j · x, with x representing a vertex of the CW-decomposition of Rs, and j ∈ Z.Imposing gr(U−j · x) = d, and substituting into Equation (6), we get
ΥK,s(t) = −2 ·minx∈Zs
t
2
(A(x) +
d− gr(x)
2
)+
(1− t
2
)(d− gr(x)
2
)+ d
= −2 ·minx∈Zs
t
2
(χk+2v0∗(x)− χk(x)
)+ χk(x) + t
k · F − F 2
4− k2 + |G|
8
= −2 ·min
x∈Zs
− t
2v∗0 · x+ χk(x)
+k2 + |G|
4− t k · F − F
2
2,
from where the claimed identity.
4.5. Constructions of the groups. Formula (5) suggests that the upsilon invariantof an algebraic knot can be expressed as the correction term of suitable lattice groups.
Let K ⊂ Y (G) be an algebraic knot (G negative-definite) presented by a tree Γ withone unframed vertex v0. For t ∈ [0, 2] we twist the Riemann-Roch function by meansof the real cohomology class tv∗0 ∈ H2(X(G),R)
χt(x) = −1
2
((k + tv∗0) · x+ x2
).
Again we stress the fact that the homology class of the vertex v0 does not exists whileits Poincare dual is always defined. With this said, we extend χt to p-cells (0 < p ≤ n)via the identity
wt() = (2− t) · wk() + t · wk+2v∗0() . (7)
Note that wt has the key property that wt(i) ≤ wt() for every face i of . Foreach real parameter l ∈ R we can now consider the sub-level set Ml =
⋃wt()≤l and
form the persistent homology module
CFt(Γ, s) =∏l∈R
C∗(Ml,F) .
This has an R-module structure: for α ≥ 0 we set qα · = ια#(), where ια : Ml →Ml+α denotes the inclusion, and we extend it to an R-action by linearity. (Note thatthe use of direct products in the definition of CFt is crucial.)
4.6. Gradings in the deformations. As in the analytic setting we define a gradingon CFt(Γ, s) setting grt() = gr()− tA().
Proposition 4.4. The differential of CFt(Γ, s) drops the grading by one.
Proof. Using B = ∈ C∗(Mwk(),Z) : p-face of Rs, 0 ≤ p ≤ s as F[U ]-basis thedifferential ∂t of CFt(Γ, s) writes as
∂t =∑i
qwt()−wt(i) ·i , (8)
12 ANTONIO ALFIERI
where the sum is extended to all faces i of . For a p-cell one computes
grt() = dim()− wt() +k2 + |G|
4− t k · F − F
2
2.
Thus, for a component ci = qwt()−wt(i) ·i of the differential ∂t of a p-cell we have
4.7. Correction terms. Here is a structure theorem for the module tHFK∗.
Lemma 4.5. The group tHFK∗(Γ, s) is an R-module of rank one. Furthermore, non-torsion elements are concentrated in lattice grading zero.
Proof. Let ξ ⊂ Ml be a p-chain, for some p > 0. Since Rs is contractible there existsl′ ≥ l such that ξ = ∂β in Ml′ . Thus ql
′−l · [ξ] = 0, showing that all cycles of this typeare torsion.
If p = 0 on the other hand, ξ = v1 + · · · + vm with either m even or odd. If m iseven then we can find l′ ≥ l and a 1-chain γ ⊂ Ml′ such that either ξ = ∂γ. Againthis proves that ql
′−l[ξ] = 0, hence that [ξ] is an element of R-torsion. If m is odd onthe other hand given any other 0-cycle ξ′ = v1 + · · ·+ v2n+1 ⊂Ml′ we can find 1-chainγ ⊂Ml′′ with l′′ > maxl, l′ such that ξ− ξ′ = ∂γ. In this case ql
′′−l[ξ] + ql′′−l′ [ξ′] = 0
and we are done showing that the rank is one.
As in the analytic theory we define
ΥΓ,s(t) = maxgrt(ξ) | ξ ∈ tHFK(Γ, s) with qα · ξ 6= 0 for α > 0 .The following proves that ΥΓ,s(t) can be computed combinatorially as in Equation (5).
Lemma 4.6. Let k be a characteristic vector representing the Spinc structure s. Then
ΥΓ,s(t) = −2 minx∈Zs
χt(x) +
(k2 + |G|
4− t k · F − F
2
2
).
Proof. As consequence of Lemma 4.5 we have that
ΥΓ,s(t) = maxx∈Zs
grt(x)
= −minx∈Zs
(2− t)χk(x) + tχk+2v∗0
(x)
+
(k2 + |G|
4− t k · F − F
2
2
)= −2 min
x∈Zsχt(x) +
(k2 + |G|
4− t k · F − F
2
2
),
and we are done.
5. The surgery exact triangle of the t-modified knot homologies
Let Σ be a genus g Riemann surface, and let η1, . . .ηk be collections of compressingcircles for some genus g solid handlebodies Uη1
, . . . , Uηk with boundary Σ. For i =1, . . . , k let Ti = η1
i × · · · × ηgi ⊂ Symg(Σ) denote the Lagrangian torus associated to
ηi = η1i , . . . , η
gi . Without loss of generality we can assume that the various η-curves
intersect transversely, hence Ti t Tj for i 6= j. Given intersection points xi ∈ Ti−1∩Tifor i = 1, . . . , k and y ∈ T1 ∩ Tk, denote by π2(x1, . . . ,xk,y) the set of homotopyclasses of continuous maps u : D2 → Symg(Σ) with domain the complex unit disk D2
13
with k + 1 marked points z0, z1, . . . zk ∈ ∂D2 (lying in the order on the unit circle)such that
• u(zi) = xi for i = 1, . . . , k and u(z0) = y• u(ai) ⊂ Ti where ai ⊂ ∂D2 denotes the boundary arc in between zi and zi+1
for i = 0, . . . , k (mod k).
We will be interested in the moduli spaces M(P ) of pseudo-holomorphic representa-tives of a given homotopy class P ∈ π2(x1, . . . ,xk,y), i.e. maps u : D2 → Symg(Σ) inP solving the Cauchy-Riemann equation on the interior of the unit disk. Note that fork ≥ 4 the source of these maps themselves have moduli: if M0,k denotes the modulispace of disks with k punctures on the boundary then dimM0,k = k − 3. As in thecase of pseudo-holomorphic strips discussed in Section 3, associated to an homotopyclass P ∈ π2(x1, . . . ,xk,y) there is a Maslov index µ(P ) ∈ Z. For a generic choiceof perturbations of the Cauchy-Riemann equation, the moduli space M(P ) forms asmooth finite-dimensional manifold of dimension
An inspection of the ends of moduli spaces of pseudo-holomorphic k-gons withMaslov index µ = 2−k shows that the maps fη1,...,ηk satisfy the so called A∞-relations:∑0≤i<j≤k
We will be interested in these relations for low values of k. If we set x·y = fη1,η2,η3(x⊗
y) then the A∞-relations for k = 4 read as
∂t(x · y) = ∂tx · y + x · ∂ty , (9)
proving that x · y satisfies the Leibeniz rule. Note that this product operation is notassociative. On the other hand, for k = 5 the A∞-relations say that so happens up tohomotopy. More precisely we have that:
(x · y) · z + x · (y · z) = ∂tfη1,η2,η3,η4(x⊗ y ⊗ z) + fη1,η2,η3,η4
(∂t(x⊗ y ⊗ z)) . (10)
Suppose now that K ⊂ Y is a knot in a rational homology sphere and that C ⊂ Y \Kis a framed loop in its complement. We wish to establish an exact triangle of the form
tHFK−(Y,K) tHFK−(Yλ(C), K)
tHFK−(Yλ+µ(C), K)
(11)
where λ denotes the chosen longitude of C, and µ a meridian. To this end we model thetriple (Y, Yλ(C), Yλ+µ(C)) by means of four collections of compressing circles α,β,γand δ on a genus g Riemann surface Σ. More precisely we choose:
• the α- and the β-curves so that (Σ,α,β) forms a Hegaard diagram of Y ,• the first β-curve β1 to be a meridian µ of C, the first γ-curve γ1 to be the
longitude λ, and the first of the δ-curves to be a curve of type λ+ µ,
14 ANTONIO ALFIERI
• the last g − 1 γ- and δ-curves to be small Hamiltonian translates of the corre-sponding last g − 1 β-curves.
Note that the base points z and w can be chosen so that Hα,β = (Σ,α,β, z, w) repre-sents (Y,K), Hα,γ = (Σ,α,γ, z, w) represents (Yλ(C), K), and Hα,δ = (Σ,α, δ, z, w)represents (Yλ+µ(C), K). Notice that we can assume: β2 to be a meridian of K, the twobase points z and w to lie near to the two sides of β2, and the Hamiltonian isotopiessending β2, . . . , βg onto γ2, . . . , γg and δ2, . . . , δg to not cross the two basepoints.
We now introduce a triangle of maps
CFt(Tα,Tβ) CFt(Tα,Tγ)
CFt(Tα,Tδ)
Fβ,γ
Fγ,δFδ,β
(12)
inducing in homology the maps appearing in (11). First we observe that, since thetwo basepoints z and w lie on the same connected component of Σ \ β ∪ γ, we havean identification H∗(CFt(Tβ,Tγ)) = Λ∗H1(T g−1) ⊗ R. In fact, the same equalityholds for H∗(CFt(Tγ,Tδ)) and H∗(CFt(Tδ,Tβ)). Denote by Θβ,γ, Θγ,δ and Θδ,β thecycles descending to the top-dimensional generator of Λ∗H1(T g−1)⊗R in CFt(Tβ,Tγ),CFt(Tγ,Tδ), and CFt(Tδ,Tβ) respectively. We define Fβ,γ : CFt(Tα,Tβ)→ CFt(Tα,Tγ)by Fβ,γ(x) = x ·Θβ,γ. Note that Fβ,γ is a chain map:
Analogously we define chain maps Fγ,δ : CFt(Tα,Tγ) → CFt(Tα,Tδ) and Fδ,β :CFt(Tα,Tδ)→ CFt(Tα,Tβ) using the top-dimensional generators Θγ,δ and Θδ,β.
We now consider the triangle of maps induced in homology by Fβ,γ , Fγ,δ and Fδ,β.We would like to show that the composition of two consecutive maps is zero. Tothis end define Hβ,γ,δ : CFt(Tα,Tβ) → CFt(Tα,Tδ) by counting pseudo hlomorphicrectangles: Hβ,γ,δ(x) = fα,β,γ,δ(x⊗Θβ,γ⊗Θγ,δ). In this case the A∞-relations prescribethe identity
On the other hand: Θβ,γ · Θγ,δ = 0 based on the very same neck stretching argumentof []. Hence,
Fγ,δ Fβ,γ = ∂t Hβ,γ,δ +Hβ,γ,δ ∂t ,showing that the composition Fγ,δ Fβ,γ is null-homotopic via the map Hβ,γ,δ. Sim-ilarly one defines homotopy equivalences Hγ,δ,β : CFt(Tα,Tγ) → CFt(Tα,Tβ) andHδ,β,γ : CFt(Tα,Tδ)→ CFt(Tα,Tγ) for Fδ,β Fγ,δ, and Fβ,γ Fδ,β.
Finally one would like to show that the triangle of maps induced by Fβ,γ , Fγ,δ andFδ,β has trivial homology (exactness). This will be based on the following algebraiclemma.
Lemma 5.1. Let Aii∈Z be a collection of chain complexes, and let fi : Ai →Ai+1i∈Z be a collection of chain maps satisfying the following two properties:
15
(1) fi+1 fi is chain homotopically trivial via a chain homotopy Hi : Ai → Ai+1,(2) the map ψi = fi+2 Hi +Hi+1 fi is a quasi-isomorphism.
Then H∗(Cone(fi)) ' H∗(Ai+2), where Cone(fi) denotes the mapping cone of fi.
Proof of Theorem 1.3. We proceed in analogy with the proof of the exact triangle
HFK(Y,K) HFK(Yλ(C), K)
HFK(Yλ+µ(C), K)
(13)
established by Ozsvath and Szabo [17, Theorem 8.2]. See also [20, Section 2].To meet precisely the hypothesis of Lemma 5.1 we choose a sequence of Hamilton-
ian translates β(i),γ(i) and δ(i) of the β-curves, the γ-curves, and the δ-curves. SetA3i = CFt(Tα,Tβ(i)), A3i+1 = CFt(Tα,Tγ(i)), and A3i = CFt(Tα,Tδ(i)), and notethat there are obvious identifications A3i = CFt(Tα,Tβ), A3i+1 = CFt(Tα,Tγ), andA3i = CFt(Tα,Tδ). Furthermore, we define f3i = Fβ(i),γ(i) , f3i+1 = Fγ(i),δ(i) , andf3i+2 = Fδ(i),β(i) . Similarly, we take H3i = Hβ(i),γ(i),δ(i) , H3i+1 = Hγ(i),δ(i),β(i) andH3i+2 : Hδ(i),β(i),γ(i) so that condition (1) of Lemma 5.1 is met.
Writing down the A∞-relations for k = 5 (the one coming from the count of pseudo-holomorphic pentagons) we get that
0 = fi+2(Hi(x)) +Hi+1(fi(x))
+ fα,γ(i),δ(i),γ(i+1)(x⊗Θγ(i),δ(i) ⊗((((
(((((((
Θδ(i),β(i) ·Θβ(i),γ(i+1) )
+ fα,γ(i),β(i),γ(i+1)(x⊗(((((((((
(Θγ(i),δ(i) ·Θδ(i),β(i) ⊗Θβ(i),γ(i+1))
+ x ·Hγ(i),δ(i),β(i),γ(i+1)(Θγ(i),δ(i) ⊗Θδ(i),β(i) ⊗Θβ(i),γ(i+1)) .
Hence, in order to show that the exact triangle holds, we must show that the map
ψi : x 7→ x ·Hγ(i),δ(i),β(i),γ(i+1)(Θγ(i),δ(i) ⊗Θδ(i),β(i) ⊗Θβ(i),γ(i+1))
induces an isomorphism in homology. To this end we observe that specialising q = 0
in the chain complex CFt we get the hat version of knot Floer homology CFK (withthe real-valued grt-grading instead of the usual bi-grading), and that ψi is a qusi-
isomorphism provided that ψi (its restriction to CFK) is a quasi-isomorphism. On
the other hand, the map ψi was shown to be a quasi-isomorphism in Ozsvath andSzabo’s proof of the exact triangle [17, Theorem 8.2].
6. A surgery exact sequence for deformations of lattice cohomology
Let Γ be a plumbing graph with one unframed vertex v0. Let G = Γ − v0 andv ∈ G be a vertex that is not directly connected to v0. We denote by G+1(v) thegraph obtained from G by increasing the weight of v by one, and by G′(v) the graphobtained from G by adding a (−1)-framed vertex e connected to v. Let Γ+1(v) andΓ′(v) the graphs obtained similarly from Γ. These represent knots in Y (G+1(v)),Y (G′(v)), and Y (G− v).
Theorem 6.1. There is a long exact sequence of R-modules
// tHFKp(Γ− v)φ∗ // tHFKp(Γ)
ψ∗ // tHFKp(Γ+1(v))δ // tHFKp−1(Γ) // .
16 ANTONIO ALFIERI
The exact sequence of Theorem 6.1 is obtained by applying the Snake Lemma to ashort exact sequence
0 // CFt(Γ− v)At // CFt(Γ)
Bt // CFt(Γ+1(v)) // 0 (14)
preserving the lattice grading. To describe the maps At and Bt appearing in (17)we resort to some convenient notation introduced by Ozsvath, Stipsicz, and Szabo[25]. This has the advantage that no choice of ground characteristic vector is neededin the definition of the differential. Passing from one notation to the other consistsin choosing an origin for the affine space of characteristic vectors associated to theplumbing.
Let G be a negative-definite plumbing tree. Let v1, . . . , vs denote the vertices of G.For every pair [K,E], with E ⊆ v1, . . . , vs, and K ∈ H2(X(G);Z) characteristic,define
2fG(K, I) =∑v∈I
K(v) +
(∑v∈I
v
)·
(∑v∈I
v
)= K · I + I2 ,
and take
gG[K,E] = minI⊆E
f(K, I) .
Then CF−(G) can be identified with the free F[U ]-module formally generated by allsuch pairs [K,E] with differential
∂[K,E] =∑v∈E
Uav [K,E] ⊗ [K,E − v] +∑v∈E
U bv [K,E] ⊗ [K + 2v∗, E − v] ,
where:
av[K,E] = g[K,E − v]− g[K,E] ,
bv[K,E] = g[K + 2v∗, E − v]− g[K,E] +K · v + v2
2.
If Γ is a negative-definite plumbing diagram with one unframed vertex v0 represent-ing a knot in Y (G), G = Γ \ v0, then the differential of CFt(Γ) = CF(G) ⊗ R isgiven by
∂t[K,E] =∑v∈E
qav(t) ⊗ [K,E − v] +∑v∈E
qbv(t) ⊗ [K + 2v∗, E − v] ,
where av(t) = (2− t)av[K,E] + tav[K+ 2v∗0, E], and similarly bv(t) = (2− t)bv[K,E] +tbv[K + 2v∗0, E]. In this context the grt-grading is given by
grt[K,E] = tgG[K,E] + |E|+ K2 + |G|4
− t K · F − F2
2,
where we set tgG[K,E] = (2− t)gG[K,E] + tgG[K + 2v∗0, E].Let now v be a vertex of G. We define ψv : CFt(Γ− v)→ CFt(Γ) by
ψv[K,E] =∑
p≡v2(mod 2)
[K, p,E]
where [K, p,E] stands for the generator of CFt(Γ) associated to the subset E, and thecharacteristic vector (K, p) ∈ Char(G) extending K ∈ Char(G−v) by K(v) = p. (Thefact p ≡ v2 mod 2 ensures that (K, p) is actually characteristic.)
Proposition 6.2. ∂t ψv = ψv ∂t.
17
Proof. One computes
ψv∂t[K,E] =∑w∈E
qaw(t) ⊗ ψv[K,E] +∑w∈E
qbw(t) ⊗ ψv[K,E]
=∑
p≡v2(mod 2), w∈E
qaw(t) ⊗ [K, p,E] +∑
p≡v2(mod 2), w∈E
qbw(t) ⊗ [K, p,E] ,
where aw(t) = (2− t)aw[K,E]+ taw[K+2v∗0, E], and bw(t) = (2− t)bw[K,E]+ tbw[K+2v∗0, E]. On the other hand,
∂tψv[K,E] =∑
p≡v2(mod 2)
∂t[K, p,E]
=∑
p≡v2(mod 2), w∈E
qa′w(t) ⊗ [K, p,E] +
∑p≡v2(mod 2), w∈E
qb′w(t) ⊗ [K, p,E] ,
where a′w(t) = (2− t)aw[K, p,E] + taw[K + 2v∗0, p, E], and b′w(t) = (2− t)bw[K, p,E] +tbw[K + 2v∗0, p, E]. Comparing the exponents the claim boils down to the followingidentities
when the vertex v does not belong to the vertex set E.
Thus, in (17) we can take At = ψv. Recall that the graph Γ′(v) is obtained fromΓ by adding a (−1)-framed vertex e. We define Bt : CFt(Γ) → CFt(Γ+1(v)) as thecomposition of the map ψe : CFt(Γ) → CFt(Γ′(v)) with the map Pt : CFt(Γ′(v)) →CFt(Γ+1(v)) defined as follows.
Let [K, p, 2m− 1, E] be the generator of CFt(Γ′(v)) associated to the vertex set E,and the characteristic vector (K, p, 2m−1) ∈ Char(Γ′(v)) extending K ∈ Char(G−v)to Γ′(v) so that K(v) = p, and K(e) = 2m− 1. Then we define:
where sm(t) ≡ m(m−1) since tgG+1(v)[K, p+2m−1, E] = tgG[K, p,E] in the eventualitythat v is not in E. Moreover the sum on the right hand side of (18) vanishes sincethe term corresponding to the parameter (p,m) cancels in pair with the one withparameters (p+4m−2,−m+1). (Here plays the fact that we are using F-coefficients.)Thus, Bt At = 0 proving that the sequence in (17) forms a five-term chain complex.Denote with H−∆(G, v) its homology.
If we prove that H−∆(G, v) = 0 we are done. A direct proof of this vanishing resultscan be lengthy and somehow confusing, we will argue along another line. First wenote that plugging in q = 0 in (17) we get a ”time independent” chain complex overthe base field F
0 // CFt(Γ− v)At // CFt(Γ)
Bt // CFt(Γ+1(v)) // 0 (19)
This can be verified to be a short exact sequence directly, adapting the computationsof [25, Proposition 6.6].
Thus the five-term chain complex in (19) is acyclic. If H∆(G, v) denotes its homol-
ogy, then we have that H−∆(G, v) ⊗R F = H∆(G, v) = 0 by the Universal CoefficientsTheorem. (A long power series f =
∑α∈A q
α ∈ R acts on x ∈ F by multiplicationby its constant term f(0).) Based on this we can argue for the vanishing of H−∆(G, v)as follows: suppose by contradiction that H−∆(G, v) 6= 0, then a non-trivial elementx ∈ H−∆(G, v) generates a graded R-module M ⊂ H−∆(G, v). Because of the specialnature of the ring R the module M can be decomposed as sums of cyclic modules ofthe form R/qα [12, Lemma 5.3], from where the contradiction since R/qα ⊗R F = F.
To see this note that R/qα is the vector space of long power series f =∑
γ∈Ω qγ with
γ ≤ α. These can be divided into two equivalence classes: those such that f(0) = 0and those such that f(0) = 1. If f =
∑γ∈Ω q
γ is of the first kind then
f ⊗ 1 = 1⊗ f(0) · 1 = 1⊗ 0 = 0 ,
otherwise f ⊗ 1 = 1⊗ f(0) · 1 = 1⊗ 1.
Remark 19.1. Note the perfect analogy of the argument presented in the proof ofLemma 6.4 and the one for the exact triangle in the analytic theory. A similar argu-ment appeared in [25, Theorem 6.4].
19
7. First consequences of the exact triangle
7.1. Floer simple knots. We briefly explore some immediate consequences of theexact triangle. First recall that given a knot (Y,K) there is a spectral sequence
starting with HFK(Y,K) and converging to HF (Y ), the Heegaard Floer homologyof the ambient three-manifold. This leads to a rank inequality:
rkHFK(Y,K) ≥ rkHF (Y ) .
If the equality holds then we say that (Y,K) is Floer simple. If Y is assumed to be anL-space this is the same as saying that tHFK−(Y,K) = R|H1(Y ;Z)|.
Example 7.1. The plumbing
•• K
−p.
represents a Floer simple knot in the lens space L(p, 1).
Floer simple knots played an important role in the work of Baker, Grigsby andHedden []. We use the exact triangle to produce some new examples of knots belongingto this class.
Theorem 7.2. Suppose that a knot (Y,K) can be represented by means of a negative-definite plumbing tree Γ with one unframed vertex v0 and no bad points. Then
tHFK−(Y (G), K) ' tHFK∗(Γ) = R|H1(Y (G);Z)| .
In particular, the knot (Y,K) is Floer simple.
Proof. First suppose that the strict inequality deg(w) < −w2 holds at every vertex wof G = Γ \ v0. Consider the long exact sequence of Theorem 6.1
and choose v to be a leaf. Then going by induction as in [16, Lemma 2.11] we canconclude that tHFKp(Γ) = 0 for p > 0, and tHFK0(Γ) = R| det(G)|.
If deg(w) = −w2 at some vertices vi1 , . . . , vik of G, we can proceed by inductionon k choosing v to be one of these vertices. This proves that
tHFK∗(Γ) = R| det(G)| = R|H1(Y (G);Z)|
when there are no bad points. On the other hand, in [11] Ozsvath, Stipsicz, and Szaboshowed that when there are no bad points there is a chain homotopy equivalenceCFK∞∗ (Γ) ' CFK∞(Y (G), K). Since tHFK−(Y (G), K) and tHFK∗(Γ) are obtainedby t-modification [12, Section 4] from CFK∞∗ (Γ) and CFK∞(Y (G), K) respectively,we can conclude that there is an isomorphism tHFK−(Y (G), K) ' tHFK∗(Γ).
Example 7.3. The trefoil knot (S3, K)is not Floer simple. Indeed, for the trefoil knot
rkHFK(S3, K) = 3, while rkHF (S3) = 1. Note that the trefoil knot can be representedby means of a plumbing tree with one bad vertex.
7.2. Reduced lattice groups. Given a CW -complex X one can consider the aug-mentation homomorphism ε : C0(X;F)→ F. This gives rise to the reduced homology
H∗(X;F) of X [7]. Similarly given a plumbing diagram Γ of an algebraic knot (Y,K)we can consider the augmentation homomorphism
CF0(G, s) =∏l∈R
C0(Ml;F) −→∏l≥γ(t)
F = R(γ(t)) ,
20 ANTONIO ALFIERI
where, in the notation of Section 4.5, γ(t) = minx∈Zs χt(x). This gives rise to the
reduced lattice group tHFKred,*(Γ, s) =∏
l∈R H∗(Ml;F) with the property that:
tHFK∗(Γ, s) = R(ΥΓ,s(t)) ⊕ tHFKred,*(Γ, s) .
In particular, we have that tHFKred,p(Γ, s) = tHFKp(Γ, s) for p ≥ 1.
Corollary 7.4. If Γ represents a type-n knot then tHFKred,p(Γ) = 0 for p ≥ n.
Proof. By induction on the number of bad points as in [25, Theorem 5.1]. The basestep here is provided by Theorem 7.2.
8. A map tHFK−0 (Γ)→ tHFK−(Y (G), K)
We start with an handy description of the group tHFK−0 (Γ).
Lemma 8.1. The group tHFK−0 (Γ) can be identified with the quotient
tHFK−0 (Γ) =
⊕k∈Char(G)
R⊗ k
/ ∼
where ∼ denotes the equivalence relation generated by the following elementary rela-tions. Let v be a vertex of G = Γ− v0, k ∈ Char(G) be a characteristic vector, and let2n = k(v) + v · v. If v is not connected to v0 in Γ then
• if n ≥ 0 then q2n+m ⊗ (k + 2v∗) ∼ qm ⊗ k,• while if n ≤ 0 then qm ⊗ (k + 2v∗) ∼ qm−2n ⊗ k.
If v is connected to v0 in Γ then
• if n ≥ 0 then qm+2n+t ⊗ (k + 2v∗) ∼ qm ⊗ k,• while if n ≤ −1 then qm ⊗ (k + 2v∗) ∼ qm−2n−t ⊗ k.
Proof. This is just a direct computation of the differential ∂t : CF1(G)→ CF0(G).
Based on this explicit description of the lattice group tHFK−0 (Γ) we build a modulehomomorphism tHFK−0 (Γ)→ tHFK−(Y (G), K).
First we observe that the algebraic knot associated to a plumbing diagram Γ comeswith an associated doubly pointed Heegaard diagram. This is obtained as follows. Fixa planar drawing Γ → R2 of the graph Γ and build the associated surgery diagram DΓ
suggested by Figure 1. Forgetting about the framings and the under-over conventionsthis gives a connected, 4-valent planar graph. Thickening DΓ ⊂ R2 × 0 ⊂ R3 we get agenus g = #(vertices) + #(edges) solid handlebody V ⊂ S3 providing, together with
its complement V c = S3 − V , a Heegaard splitting of S3. We choose as α-curves ofthe corresponding Heegaard diagram (supported on Σ = ∂V ) the boundary of thecompact complementary regions of DΓ ⊂ R2 (this provides a set of compressing circlesfor Uα = V c). As compressing circles of Uβ = V (β-curves) we take a meridian foreach link component of DΓ, and one further curve in correspondence of each edge ofΓ as suggested by Figure 2. By taking base points on the two sides of the β-curvecorresponding to the meridian of the unlabled vertex v0 ∈ Γ we get a doubly pointedHeegaard diagram Hαβ = (Σ, α, β, z, w) representing the unknot U ⊂ S3.
Notice that in the choice of the β-curves above we can substitute the meridiansof the framed components with the framings. By replacing the other curves withsmall Hamiltonian translates of the remaining β-curves we get another g-tuple ofcurves γ. This gives rise to other two doubly-pointed Heegaard diagrams: Hαγ =
21
-
•q÷;'-
K
i
Figure 2
(Σ, α, γ, z, w) representing K ⊂ Y (G), and Hβγ = (Σ, β, γ, z, w) representing theunknot in #g−`S1 × S2.
Let Θβγ ∈ Tβ ∩ Tγ denote the intersection point representing the top-dimensionalgenerator of H∗(CFt(Tβ,Tγ)) = Λ∗H1(T g−`) ⊗ R. Given intersection points x ∈Tα ∩ Tβ and y ∈ Tα ∩ Tγ Ozsvath and Szabo [19, Proposition 8.4] built a mapsz : π2(x,Θβγ,y) → Spinc(X(G)). Given a Spinc structure t of the plumbing ofspheres X(G) we define a map fΓ,t : CFt(Tα,Tβ)→ CFt(Tα,Tγ) setting
fΓ,t(x) =∑
y∈Tα∩Tγ
∑∆∈π2(x,Θβγ ,y)sz(∆)=t, µ(∆)=0
#M(∆) qtnz(∆)+(2−t)nw(∆) · y .
Of course, this is just a Spinc refinement of the holomorphic triangle count of Sec-tion 5. Inspecting the ends of moduli spaces of holomorphic triangles with Maslovindex µ = 1 one concludes that fΓ,t is a chain map. Thus given a characteristicvector k ∈ Chars(G) ⊂ Spinc(X(G)) we get a map FΓ,k : R = H∗(CFt(Tα,Tβ)) →tHFK−(K,Y (G), s), FΓ,k = (fΓ,k)∗.
Example 8.2 (Lens space cobordism). Let D−p denote the total space of the diskbundle D−p → S2 with Euler number e = −p, and let ∆ ⊂ Dp be a fibre disk. Then∂(D−p,∆) = (L(p, 1), K) is a Floer simple knot with plumbing
•• K
−p.
A doubly pointed Heegaard triple representing (D−p,∆) is given by (T 2, α, β, γ, z, w),where α and γ represent respectively a longitude and a meridian of the two torus T 2,and β a type (1, p) curve. The two base points z and w are chosen to lie on the twoopposite sides of the β-curve.
Let v represents the generator of H2(D−p,Z). If k is a characteristic vector suchthat 2n = k(v) + v · v then
• q2n+tFΓ, k+2v∗ = FΓ,k if n ≥ 0, and• FΓ, k+2v∗ = q−2n−tFΓ,k if n ≤ −1.
This is a direct computation along the line of [13, Section 4.1].
Lemma 8.3. ϕΓ descends to an R-module homomorphism
ΦΓ : tHFK−0 (Γ, s)→ tHFK−(Y (G), K, s) .
22 ANTONIO ALFIERI
Proof. A vertex v of G corresponds to a (−p)-framed sphere embedded in X(G). Thishas a tubular neighbourhood D−p diffeomorphic to a disk bundle with base the two-sphere S2, and Euler number e = −p. Let ∆ ⊂ Dp be a fibre disk.
There are two cases: the case when v is linked to the unframed vertex v0 and thecase when it is not. In the case when v is not linked to v0 in Γ one concludes that themap FΓ,k : R → tHFK−(K,Y (G), s) factors through the cobordism map associated toD−p : S3 → L(p, 1). This implies the first set of relations of Lemma 8.1. If v is linkedto v0 on the other hand, the map FΓ,k : R → tHFK−(K,Y (G), s) factors through thecobordism map associated to the pair (D−p,∆) we computed in Example 8.2. In thiscase we get the second set of relations.
Proposition 8.4. If Γ has at most one bad point then there is an isomorphism ofR-modules tHFK−(Y (G), K) ' tHFK−∗ (Γ).
Proof. Putting the pieces together we get a commutative diagram with exact rows:
provided that G+1(v) is negative definite and G− v has no bad points. To see this weacknowledge that tHFK∞(Y,K), the homology of the localisation CFt(Y,K) ⊗ R∗,agrees with HF∞(Y )⊗R∗ ' R∗ and that the map
HF∞(Y (G+1(v)))⊗R∗ // HF∞(Y (G− v))⊗R∗
tHFK∞(Y (G+1(v)), K ′) //
∼=
OO
tHFK∞(Y (G− v), K0)
∼=
OO
is trivial being the underlying cobordism indefinite.With the above said one can run the very same argument of [16, Theorem 2.1] to
show that there is an identification tHFK−(Y (G), K) ' tHFK−0 (Γ). On the otherhand, according to Lemma 7.4, for a diagram with at most one bad point we have thattHFK−p (Γ) = 0 for p > 0. Hence tHFK−∗ (Γ) = tHFK−0 (Γ), proving the claim.
8.1. Z/2Z-grading and extension to the case of two bad points. We now wantto extend our main result to the case of two bad points. To this end we introduce therelative Z/2Z-grading on some of the t-modified knot homologies.
First, note that tHFK−(Y,K) has a natural real-valued relative grt-grading
grt(x, y) = grt(x)− grt(y) ,
characterised by the property that for every pair of intersection points x and y
grt(x,y) = µ(φ) + t · nz(φ) + (2− t) · nw(φ) ,
23
where φ ∈ π2(x,y) denotes a Whitney disk connecting x to y. Analogously, in thecombinatorial theory tHFK∗ we have a relative grt-grading
Secondly, recall [12, Section 3] that if 0 ≤ t = mn≤ 2 is a rational number then in the
definition of tHFK−(Y,K) the ring R can be replaced with the ring of polynomialswith fractional exponents F[q1/n] ⊂ R∗. In this case the relative grt-grading is rationalvalued. More specifically, it takes values in Z[ 1
n] ⊂ Q, the ring of fractions associated
to the multiplicative set S = nk : k ≥ 0 ⊂ Z. Note that the ideals of Z[ 1n] are in one-
to-one correspondence with the ideals of Z that do not meet S. In particular if n is oddthen Z[ 1
n] contains a (unique) maximal ideal m with quotient field Z[ 1
n]/m ' Z/2Z.
Thus, if we choose 0 ≤ t ≤ 2 to be of the form t = m/(2b + 1) then there is a welldefined (mod 2) reduction of the relative grt-grading. The equivalence relation:
x ∼ y ⇐⇒ grt(x,y) = 0 (mod 2)
partitions the generators of tHFK−(Y,K) into two equivalence classes, and determinesa splitting of the chain group: CFt(Y,K) = CF even
t (Y,K) ⊕ CF oddt (Y,K). Since the
differential ∂t flips the two summands, we have a decomposition of the groups
tHFK−(Y,K) = tHFK−even(Y,K)⊕ tHFK−even(Y,K) .
Similarly one has a splitting in the combinatorial theory
tHFK∗(Γ) = tHFKeven(Γ)⊕ tHFKodd(Γ) .
Thirdly, we observe that if we choose t = m/(2b + 1) with m = 2a even thenthe situation greatly simplifies. In the analytic theory the relative Z/2Z-gradinggrt(x,y) = µ(φ) (mod 2) collapses on the ”classical” Maslov grading, while in thecombinatorial theory there are identifications:
tHFKeven(Γ) =⊕p even
tHFKp(Γ) , and tHFKodd(Γ) =⊕p odd
tHFKp(Γ) ,
since grt(,′) = dim()−dim(′) (mod 2). Note that in the case when Γ has at most
one bad point we have a vanishing result for the odd homologies: tHFK−odd(Y (G), K) 'tHFK−odd(Γ) = 0.
Proposition 8.5. Suppose that 0 ≤ t ≤ 2 is a rational number of the form t =2a/(2b + 1). If Γ has at most two bad points then there is an isomorphism of R-modules tHFK−even(Y (G), K) ' tHFK−0 (Γ).
Proof. If we choose v to be one of the two bad points we get a commutative diagramwith exact rows:
Furthermore the last map on the top row is surjective since the triangle map
tHFK−(Y (G+1(v)), K ′)→ tHFK−(Y (G− v), K0)
24 ANTONIO ALFIERI
flips the relative Maslov grading and tHFK−even(Y (G − v), K0) = 0. Like in [16,Theorem 2.2] this information is sufficient to conclude that the right-most map shouldbe an isomorphism provided that the other two are.
Proof of Theorem 1.2. First suppose that 0 ≤ t ≤ 2 is a rational number of the formt = 2a/2b + 1. According to the computations of the degree shifts of the trianglecounting maps performed by Zemke in [26] the isomorphism tHFK−even(Y (G), K) 'tHFK0(Γ) of Proposition 8.5 preserves the grt-grading. Since non-torsion elementsare concentrated in even gradings we have that ΥK,s(t) = ΥΓ,s(t) for all values of theparameter t in
C =
2a
2b+ 1: a, b ∈ Z≥0
∩ [0, 2] .
On the other hand C is dense in [0, 2], the upsilon function is continuous [12, Proposi-tion 1.4], and two continuous functions that agree on a dense set agree everywhere.
References
[1] A. Alfieri, J. Baldwin, I. Dai, and S. Sivek, Instanton floer homology of almost-rationalplumbings. To appear, 2020.
[2] A. Alfieri, D. Celoria, and A. Stipsicz, Upsilon invariants from cyclic branched covers.arXiv:1708.06389, 2018.
[3] W. Brandal, in Commutative rings whose finitely generated modules decompose, Lecture Notesin Mathematics, Springer, Berlin.
[4] A. Floer, Morse theory for Lagrangian intersections, J. Differential Geometry, 28 (1988),pp. 513–547.
[5] R. E. Gompf and A. I. Stipsicz, 4-Manifolds and Kirby Calculus, AMS Mathematical Surveysand Monographs, 1999.
[6] E. Grigsby, D. Ruberman, and S. Strle, Knot concordance and Heegaard Floer homologyinvariants in branched covers, Geometry & Topology, 12 (2008), pp. 2249–2275.
[7] A. Hatcher, Algebraic topology, Cambridge University Press, 2001.[8] C. Livingston, Notes on the knot concordance invariant upsilon, Algebr. Geom. Topol., 17
(2017), pp. 111–130.[9] A. Nemethi, Lattice cohomology of normal surface singularities, Publ. RIMS. Kyoto Univ., 44
(2008), pp. 507–543.[10] Y. Ni, Knot Floer homology detects fibred knots, Inventiones Mathematicae, 3 (2007), pp. 577–
608.[11] P. Ozsvath, A. Stipsicz, and Z. Szabo, Knot lattice homology in L-spaces, Journal of Knot
Theory and Its Ramificationsl, 25 (2016), pp. 1211–1299.[12] , Concordance homomorphisms from knot Floer homology, Advances in Mathematics, 315
(2017), pp. 366 – 426.[13] P. Ozsvath and Z. Szabo, Absolutely graded Floer homologies and intersection forms for
four-manifolds with boundary, Advances in Mathematics, 173 (2003), pp. 179–261.[14] , Knot Floer homology and the four-ball genus, Geometry and Topology, 7 (2003), pp. 615–
639.[15] P. Ozsvath and Z. Szabo, On the Floer homology of plumbed three-manifolds, Geom. Topol.,
7 (2003), pp. 185–224.[16] P. Ozsvath and Z. Szabo, On the floer homology of plumbed three-manifolds, Geometry and
Topology, 7 (2003), pp. 185–224.[17] , Holomorphic disks and knot invariants, Advances in Mathematics, 8 (2004), pp. 58–116.[18] , Holomorphic disks and three manifold invariants: properties and applications, Annals of
Mathematics, 159 (2004), pp. 1159–1245.[19] , Holomorphic disks and topological invariants for closed three manifolds, Annals of Math-
ematics, 159 (2004), pp. 1027–1158.[20] , Lectures on Heegaard Floer homology, in Floer Homology, Gauge Theory, and Low Di-
mensional Topology, Proceedings of the Clay Mathematics Institute, 2004.
25
[21] , Holomorphic triangles and invariants for smooth four-manifolds, Advances in Mathemat-ics, 202 (2006), pp. 326–400.
[22] , Knot Floer homology and integers surgeries. arXiv:math/0410300, 2007.[23] J. Rasmussen, Floer homology and knot complements. arXiv: math/0306378, 2003.[24] , Knot polynomials and knot homologies. arXiv:math/0504045, 2005.[25] P. O. A. Stipsicz and Z. Szabo, A spectral sequence on lattice homology. arXiv:1206.1654,
2012.[26] I. Zemke, Link cobordisms and absolute gradings on link floer homology, Quantum Topology, 10
