On the Ozsváth-Szabó invariant of negative definite plumbed 3-manifolds

András Némethi

Department of Mathematics Ohio State University Columbus, OH 43210 and Rényi Institute of Mathematics Budapest, Hungary

        Email: nemethi@math.ohio-state.edu and nemethi@renyi.hu
             URL: http://www.math.ohio-state.edu/~nemethi/

Abstract

The main goal of the present article is the computation of the Heegaard Floer homology introduced by Ozsváth and Szabó for a family of plumbed rational homology 3-spheres. The main motivation is the study of the Seiberg-Witten type invariants of links of normal surface singularities.

AMS Classification numbers Primary: 57M27, 57R57, 58Kxx, 14E15, 14B15

Secondary: 14J17, 32S25, 32S45

Keywords: 3-manifolds, Ozsváth-Szabó Heegaard-Floer homology, Seiberg-Witten invariants, Seifert manifolds, Reidemeister-Turaev torsion, ℚ-homology spheres, Lens spaces, Casson-Walker invariant, normal surface singularities, rational singularities, elliptic singularities

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1 Introduction

The main goal of the present article is the computation of the Ozsváth-Szabó ℤ[U]-module (Heegaard Floer homology) HF^{+}(M, [k]) [17, 18, 19, 20] for a family of plumbed rational homology 3-spheres M and any spin^{c}-structure [k] ∈ Spin^{c}(M). The main motivation (of the author) is the study of the Seiberg-Witten type invariants of links of normal surface singularities.

More precisely, we consider that family of connected, negative definite plumbing graphs – we call them AR-graphs (see section 8) – which satisfy the following property: there exists a vertex such that decreasing the decoration (Euler number) of that vertex we get a rational graph (in the sense of Artin). This class is surprisingly large: it contains the links of rational and weakly elliptic singularities, the graphs considered by Ozsváth and Szabó in [21], in particular, all the Seifert manifolds (associated with negative definite plumbing graphs). For such plumbing graphs Γ (and associated plumped 3-manifolds M(Γ)) we prove (extending the main result of [21]) the completely topological description HF^{+}(M(Γ), [k]) = ℍ^{+}(Γ, [k]), where ℍ^{+}(Γ, [k]) is the combinatorial ℤ[U]- module introduced in [21].

Moreover, for such graphs, we provide a precise combinatorial formula (algorithm) for ℍ^{+}(Γ, [k]). In order to do this, we define a “graded root” (R_{k}, χ_{k}) associated with any connected, negative definite plumbing graph Γ and characteristic element k. This object connects in a mysterious way two different types of properties, objects and invariants: those coming from the Ozsváth-Szabó (or Seiberg-Witten) theory with those coming from algebraic geometry and singularity theory. (E.g., its grading χ_{k} is in fact a Riemann-Roch formula, which guides – modulo a shift – the absolute grading of HF^{+}(M, [k]).) We believe that (R_{k}, χ_{k}) is the right object which guides the hierarchy of the topological types of links of normal surface singularities (cf. e.g. with section 6); and in the same time, it contains all the information about ℍ^{+}(Γ, [k]). In the body of the paper, for any AR graph, we determine all these “graded roots”, we read from them the combinatorial modules ℍ^{+}(Γ, [k]), and we compute the numerical invariants sw_{M,[k]}^{OSz} which are the candidates provided by the Ozsváth-Szabó theory for the Seiberg-Witten invariants.

We exemplify the theory with a detailed discussion of the lens spaces and Seifert manifolds. Additionally, in these two cases we verify that sw_{M,[k]}^{OSz} coincides with the Reidemeister-Turaev sign-refined torsion (function) normalized by the Casson-Walker invariant – a fact conjectured by Ozsváth and Szabó.

Examples show that the above results about ℍ^{+}(Γ, [k]) cannot be extended for a substantially larger class of plumbing graphs (than the AR graphs) – unless one modifies the definition of the combinatorial object ℍ^{+}.

Although the paper is more topological/combinatorial, in some places we emphasize the relevance of the results in the theory of normal surface singularities, in special in the light of [12] and [11].

The reader is invited to read subsection 2.5 of the preliminary section, which contains more details about the guiding problems (and the main results) of the present paper.

The author thanks Péter Ozsváth and Zoltán Szabó for their help, advises and encouragements.

The author is partially supported by NSF grant DMS-0304759 and OTKA grants 42769 and 46878.

2 Preliminaries

2.1 Plumbing graphs. We consider an oriented plumbed 3-manifold M = M(Γ) obtained from the graph Γ by plumbing construction (with all the edgesigns equal to +; see e.g. [14], section 8). We will assume that M is a rational homology sphere. This is equivalent to the fact that Γ is a disjoint union of trees, all the genera of the corresponding Riemann surfaces involved in the plumbing construction are zero, and the bilinear form B associated with Γ (see below) is non-degenerate. Additionally, we will assume that Γ is connected and B is negative definite (for motivation, see 2.1.1; although most of the arguments work for non-connected Γ as well). In fact, the plumbing construction provides an oriented 4-manifold X̃ whose oriented boundary ∂X̃ is exactly M. −M will denote M with the opposite orientation.

The vertices of Γ will be indexed by J . Each vertex j ∈ J has a decoration e_{j}, which is the Euler number of the corresponding disc-bundle (or S^{1}-bundle) used in the plumbing construction. Let L be the free ℤ-module of rank s := #J with a fixed basis {b_{j}}_{j∈J} .

In fact, Γ codifies by the bilinear form B = (b_{i}, b_{j})_{i,j∈J} , defined by (b_{j}, b_{j}) = e_{j}; and for i ≠ j one takes (b_{i}, b_{j}) = 1 if the corresponding vertices i, j are connected by an edge in Γ, otherwise = 0. For any vertex j ∈ J of Γ we denote by δ_{j} the number of adjacent vertices.

A cycle x = ∑_{j} n_{j}b_{j} ∈ L is called effective, denoted by x ≥ 0, if n_{j} ≥ 0 for any j. We write x ≥ y if x − y ≥ 0. x > 0 means x ≥ 0 but x ≠ 0. We define the support |x| of x by ∪_{j:n_{j}≠0}{b_{j}}. If x_{i} = ∑_{j} n_{j,i}b_{j} for i = 1, 2, then min{x_{1}, x_{2}} := ∑_{j} min{n_{j,1}, n_{j,2}}b_{j}. x^{2} means (x, x).

Set L′ = Hom_{ℤ}(L, ℤ), and consider the exact sequence 0 → L →^{i} L′ → H → 0, where i(x) = (x, ·). Then one has the identifications L ≈ H_{2}(X̃, ℤ), L′ ≈ H_{2}(X̃, M, ℤ) and H ≈ H_{1}(M, ℤ).

Sometimes it is convenient to identify the lattice L′ with a sub-lattice of L_{ℚ} = L ⊗ ℚ: any α ∈ L′ is identified with y_{α} ∈ L_{ℚ} which satisfies α(x) = (y_{α}, x) for any x ∈ L. Here (·, ·) is the natural extension of the form B to L_{ℚ} (denoted in the same way).

In sections 10 and 11 we will also use the dual base {g_{j}}_{j∈J} of L′ (i.e., (g_{j}, b_{i}) = δ_{ij} for any i, j ∈ J ). If L′ is identified with a sub-lattice of L_{ℚ} then g_{j} is the j^{th} column of B^{−1}. Since B is negative definite, −g_{j} ≥ 0 for any j ∈ J .

Clearly, the order |H| of the group H = H_{1}(M, ℤ) equals |det(B)|.

2.1.1 Example. (See e.g. [14].) Let M be the link of a normal surface singularity (X, 0). Assume that X̃ → X is a good resolution of (X, 0), (i.e. X is a sufficiently small Stein representative of (X, 0) and X̃ → X is a resolution with normal crossing exceptional divisor). Let Γ denote the dual resolution graph. Then ∂X̃ = M, and (X̃, ∂X̃) can be identified with the output of the plumbing construction provided by Γ. [In this case, the elements b_{j} of L = H_{2}(X̃, ℤ) correspond to the classes of the irreducible exceptional divisors.]

Recall, that a plumbed 3-manifold M can be realized as the link of some normal surface singularity if and only if its graph Γ is connected and the corresponding form B is negative definite [4].

2.2 ℤ[U]-modules. Later we will use the following notations. Consider the graded ℤ[U]-module ℤ[U, U^{−1}], and (following [21]) denote by T_{0}^{+} its quotient by the submodule U·ℤ[U]. This has a grading in such a way that deg(U^{−d}) = 2d (d ≥ 0). Similarly, for any n ≥ 1, define the graded module T_{0}(n) by the quotient of ℤ⟨U^{−(n−1)}, U^{−(n−2)}, …, 1, U, …⟩ by U · ℤ[U] (with the same grading). Hence, T_{0}(n), as a ℤ-module, is the free ℤ-module ℤ⟨1, U^{−1}, …, U^{−(n−1)}⟩ (generated by 1, U^{−1}, …, U^{−(n−1)}), and has finite ℤ-rank n.

More generally, for any graded ℤ[U]-module P with d-homogeneous elements P_{d}, and for any r ∈ ℚ, we denote by P [r] the same module graded in such a way that P [r]_{d+r} = P_{d}. Then set T_{r}^{+} := T_{0}^{+}[r] and T_{r}(n) := T_{0}(n)[r].

2.3 Characteristic elements. Spin^{c}-structures. Fix a plumbing graph Γ as above. The set of characteristic elements is defined by

Char = Char(Γ) := {k ∈ L′ : k(x) + (x, x) ∈ 2ℤ for every x ∈ L}. Clearly, for any fixed k_{0} ∈ Char, Char = k_{0} +2L′. There is a natural action of L on Char by x∗k := k+2i(x) whose orbits are of type k+2i(L). Obviously, H acts freely and transitively on the set of orbits by [l′]∗(k+2i(L)) := k+2l′+2i(L) (in particular, they have the same cardinality). If X̃ is as in 2.1, then the first Chern class (of the determinant line bundle associated with a given spin^{c}-structure) realizes an identification between the spin^{c}-structures Spin^{c}(X̃) on X̃ and Char ⊂ L′ ≈ H^{2}(X̃, ℤ) (see e.g. [3], 2.4.16). The restrictions to M defines an identification of the spin^{c}-structures Spin^{c}(M) of M with the set of orbits of Char modulo 2L; and this identification is compatible with the action of H on both sets. In the sequel, we think about Spin^{c}(M) by this identification, hence any spin^{c}-structure of M will be represented by an orbit [k] := k + 2i(L) ⊂ Char. The spin^{c}-structures Spin^{c}(M) and Spin^{c}(−M) are canonically identified.

2.4 Invariants of the 3-manifold M. In this note we will focus on the following set of invariants.

2.4.1 According to Turaev [24], a choice of a spin^{c}-structure on M is equivalent to a choice of an Euler structure. For every spin^{c}-structure [k], he defines the sign refined Reidemeister-Turaev torsion

𝒯_{M,[k]} = ∑_{h∈H} 𝒯_{M,[k]}(h)h ∈ ℚ[H]

determined by the Euler structure associated with [k] [24]. It is convenient to think about 𝒯_{M,[k]} as a function H → ℚ given by h ↦ 𝒯_{M,[k]}(h). We will be basically interested in 𝒯_{M,[k]}(1) where 1 denotes the neutral element of the group H (with the multiplicative notation). For plumbed 3-manifold M (as in 2.1), [12] (5.7) provides a combinatorial formula for 𝒯_{M,[k]}(1) in terms of Γ (involving regularized Fourier-Dedekind sums).

2.4.2 We will use the notation λ(M) for the Casson-Walker invariant of M (normalized as in [7] (4.7)). For plumbed manifolds M, it has a combinatorial formula from Γ (provided by A. Raţiu in his dissertation; and it can be deduced from the surgery formulas of [7] as well; see also (5.3) [12]):

−\frac{24}{|H|} λ(M) = ∑_{j} e_{j} + 3s + ∑_{j}(2 − δ_{j}) (B^{−1})_{jj}.

2.4.3 If M is a plumbed 3-manifold as in 2.1, it has a canonical characteristic element K defined by K(b_{j}) = −e_{j} − 2 for any j (see also 5.1.1 for more comments about these “adjunction formulas”). The rational number K^{2} + s does not depend on the (negative definite) plumbing graph, it is an invariant of M. It can be computed from Γ as follows (cf. 5.2 [12]):

K^{2} + s = ∑_{j} e_{j} + 3s + 2 + ∑_{i,j}(2 − δ_{i})(2 − δ_{j})(B^{−1})_{ij}.

2.4.4 The Heegaard-Floer homology HF^{+}(M, [k]). For any oriented rational homology 3-sphere M and [k] ∈ Spin^{c}(M), the Ozsváth-Szabó ℤ[U]- module (or, the absolutely graded Heegaard Floer homology) HF^{+}(M, [k]) was introduced in [17] (cf. also with the long list of recent articles of Ozsváth and Szabó). This has a ℚ-grading compatible with the ℤ[U]-action, where deg(U) = −2.

Additionally, HF^{+}(M, [k]) has an (absolute) ℤ_{2}-grading; HF_{even}^{+}(M, [k]), respectively HF_{odd}^{+}(M, [k]) denote the part of HF^{+}(M, [k]) with the corresponding parity.

By the general theory, for any spin^{c}-structure [k] and M as above, one has a graded ℤ[U]-module isomorphism

HF^{+}(M, [k]) = T_{d(M,[k])}^{+} ⊕ HF_{red}^{+}(M, [k]),

where HF_{red}^{+} has a finite ℤ-rank and an induced (absolute) ℤ_{2}-grading (and d(M, [k]) can also be defined via this isomorphism).

From the above data one can extract two crucial numerical invariants: d(M, [k]) and

χ(HF^{+}(M, [k])) := rank_{ℤ} HF_{red,even}^{+}(M, [k]) − rank_{ℤ} HF_{red,odd}^{+}(M, [k]).

With respect to the change of orientation they behave as follows:

d(M, [k]) = −d(−M, [k]) and χ(HF^{+}(M, [k])) = −χ(HF^{+}(−M, [k])).

Notice that one can recover HF^{+}(M, [k]) from HF^{+}(−M, [k]) by a standard procedure described by Oszváth and Szabó.

2.4.5 The combinatorial module ℍ^{+}(Γ). In the case of a plumbed 3manifold M associated with the graph Γ, one would like to obtain a completely combinatorial description of HF^{+}(−M, [k]) from Γ. In [21], Ozsváth and Szabó introduced the following combinatorial graded ℤ[U]-module as a candidate.

Let ℍ^{+}(Γ) be the set of those finitely supported functions ϕ : Char → T_{0}^{+} which satisfy the following property. For any characteristic element k and base vector b_{j} write 2n = k(b_{j}) + (b_{j}, b_{j}). Then if n ≥ 0 then one requires

U^{n} · ϕ(k + 2i(b_{j})) = ϕ(k);

while if n ≤ 0, then

ϕ(k + 2i(b_{j})) = U^{−n} · ϕ(k).

For any spin^{c}-structure [k] = k + 2i(L), let ℍ^{+}(Γ, [k]) be the subset of those maps ϕ which are supported on k + 2i(L). Then one has a direct sum splitting ℍ^{+}(Γ) = ⊕_{[k]} ℍ^{+}(Γ, [k]).

On ℍ^{+}(Γ) one defines the following (rational) grading. One says that a map ϕ ∈ ℍ^{+}(Γ) is homogeneous of degree d ∈ ℚ if for each characteristic vector k with ϕ(k) ≠ 0, ϕ(k) ∈ T_{0}^{+} is homogeneous element with

deg(ϕ(k)) − \frac{k^{2} + s}{4} = d.

Ozsváth and Szabó in [21] proved the following theorem.

2.4.6 Theorem. [21] Assume that Γ has at most one vertex j ∈ J where the inequality −e_{j} ≥ δ_{j} fails. Then for any spin^{c}-structure [k], HF_{odd}^{+}(−M, [k]) = 0 and HF_{even}^{+}(−M, [k]) = ℍ^{+}(Γ, [k]).

2.5 The main guiding problems of the article. We summarize in short those conjectures and problems which have determined our interest in this subject. This is an interesting mixture of the topology of 3-manifolds and theory of normal surface singularities.

I. First of all, as we already mentioned, for any plumbed manifold M one wishes to determine HF^{+}(M, [k]) in a combinatorial way from Γ.

Our goal (realized in 8.3) is to extend the isomorphism 2.4.6 for a larger class of plumbing graphs, namely for the AR graphs (introduced in section 8). This class includes all the links of rational and elliptic singularities, contains the class considered by Ozsváth and Szabó in 2.4.6, in particular, all the Seifert manifolds with negative orbifold Euler number. [Examples show that the statement of 2.4.6 cannot be extended for a substantially larger class of graphs, cf. 8.4.]

II. Even though ℍ^{+}(Γ, [k]) is combinatorial, its computation is not trivial at all. In the body of the paper we provide precise algorithm for this computation (valid for any AR graph).

The main idea comes from the technique of computation sequences used in singularity theory (see e.g. [5] and [6]). In fact, as a starting invariant, we will construct a graded root R_{k} for any plumbing graph Γ and for any characteristic element k. Surprisingly, this object is able to guide (and connect) two very different theories. First, from the point of view of singularity theory, it seems that it is the right object which controls the hierarchy and classification of surface singularities. E.g., it characterizes nicely the rational and elliptic singularities (see section 6). Also, it provides optimal topological upper bounds for some analytic invariants (see part IV below and 9.6). On the other hand, from R_{k} one can read easily ℍ^{+}(Γ, [k]).

Our discussion runs on three levels: we determine for any AR graph the graded root R_{k}, this determines automatically the combinatorial Ozsváth-Szabó graded ℤ[U]-modules ℍ^{+}(Γ, [k]), and finally we focus on the numerical invariants d(M, [k]) and χ(HF^{+}(M, [k]) as well.

One of the surprising byproducts of the discussion is 6.3, which shows that the links of rational singularities correspond exactly to the L-spaces in the sense of Ozsváth-Szabó.

III. One of the goals of Ozsváth-Szabó homology is to substitute the (modified) Seiberg-Witten topological invariant of (M, [k]). More precisely, it provides the numerical invariant

sw_{M,[k]}^{OSz} := χ(HF^{+}(M, [k])) − \frac{d(M, [k])}{2}

as a candidate for the Seiberg-Witten invariant.

On the other hand, the sign refined Reidemeister-Turaev torsion together with the Casson-Walker invariant also provides a candidate for the Seiberg-Witten invariant (cf. also with [16]):

sw_{M,[k]}^{T CW} := −𝒯_{M,[k]}(1) + \frac{λ(M)}{|H|},

where |H| is the order of H.

Our goal is to investigate the identity of these two candidates. Notice that in the presence of HF_{red,odd}^{+}(−M, [k]) = 0, this conjectured identity reads as

𝒯_{M,[k]}(1) − \frac{λ(M)}{|H|} = rank_{ℤ} HF_{red}^{+}(−M, [k]) − \frac{d(−M, [k])}{2}.

Recall that if M is an integral homology sphere (i.e. H = 0) then this identity holds by [20] (1.3).

Since for AR graphs the right hand side of the identity can be determined from ℍ^{+}(Γ, [k]), the above identity becomes a purely combinatorial property of Γ. In the body of the paper we verify it for any lens space and Seifert manifold (and any spin^{c}-structure).

IV. In [12], L. Nicolaescu and the author stated the following conjecture. Assume that (X, 0) is a normal surface singularity whose link M is a rational homology sphere. Let p_{g} be the geometric genus of (X, 0) (i.e. dim_{ℂ} H^{1}(X̃, OX̃), where X̃ is a resolution of X). Then the analytic invariant p_{g} has the following “optimal” topological upper bound:

p_{g} ≤ 𝒯_{M,[K]}(1) − \frac{λ(M)}{|H|} − \frac{K^{2} + s}{8}.

Moreover, if (X, 0) is ℚ-Gorenstein, then above one has equality.

Notice that in the presence of HF_{red,odd}^{+}(−M, [K]) = 0, and using the principle (III) above, the above inequality can be transformed onto another conjectured inequality

p_{g} ≤ rank_{ℤ}(HF_{red}^{+}(−M, [K])) + \frac{d(M, [K])}{2} − \frac{K^{2} + s}{8}

which conjecturally becomes equality for ℚ-Gorenstein singularities.

The point is that the computation algorithm of ℍ^{+}(Γ, [K]) of the present article will automatically provide this inequality for any singularity with AR-resolution graph. Moreover, we also prove the identity for rational and elliptic Gorenstein singularities and all singularities which admit a good ℂ^{∗}-action (all of these are ℚ-Gorenstein).

In fact, in 9.6, we extend the above inequality for any spin^{c} structure and any AR graph. In this way, the (candidates for) Seiberg-Witten invariants (provided by the Heegaard-Floer theory) provide a (conjecturaly optimal) topological upper bound for h^{1}(L), where L is a holomorphic line bundle on a resolution of (X, 0).

For the extension of the conjecture [12] to arbitrary spin^{c} structures, see [11]. For different definitions regarding surface singularities, the reader is invited to consult [10].

After the first version of the present article was completed, the author realized that the above inequality – and equality for Gorenstein singularities – is not true for non-AR singularities; for details see [8].

3 Graded roots

3.1 Preliminary remarks. Recall that the Ozsváth-Szabó ℤ[U]-module ℍ^{+}(Γ, [k]) is defined in a combinatorial way from the plumbing graph Γ. Our goal is to define an intermediate object, a graded root R_{k} associated with Γ (and a characteristic element k). This will contain all the needed information to determine the homological object ℍ^{+}, but it preserves also some additional, more subtle topological information about Γ (or, about M).

In this section we give the definition and first properties of the (abstract) graded roots. The next section contains the construction of the graded roots R_{k} from the plumbing graphs Γ. [Although both Γ (the plumbing graph) and the constructed graded root R_{k} are “connected trees”, they serve rather different roles. E.g., the edges of Γ codify the corresponding gluings in the plumbing, while the edges of R_{k} codify the ℤ[U]-action. We hope the terminology will not create any confusion.]

3.2 Definitions.

(1) Let R be an infinite tree with vertices V and edges E. We denote by [u, v] the edge with end-points u and v. We say that R is a graded root with grading χ : V → ℤ if

(a) χ(u) − χ(v) = ±1 for any [u, v] ∈ E;

(b) χ(u) > min{χ(v), χ(w)} for any [u, v], [u, w] ∈ E, v ≠ w;

(c) χ is bounded below, χ^{−1}(k) is finite for any k ∈ ℤ, and #χ^{−1}(k) = 1 if k is sufficiently large.

(2) v ∈ V is a local minimum point of the graded root (R, χ) if χ(v) < χ(w) for any edge [v, w].

(3) If (R, χ) is a graded root, and r ∈ ℤ, then we denote by (R, χ)[r] the same R with the new grading χ[r](v) := χ(v) + r. (This can be generalized for any r ∈ ℚ as well.)

3.3 Notations/Remarks. (1) For any vertex v set δ_{v} := #{[v, w] ∈ E}. One can verify that the set of vertices V_{1} := {v ∈ V : δ_{v} = 1} are exactly the local minimum points of χ, and #V_{1} < ∞.

(2) A geodesic path connecting two vertices is monotone if χ restricted to the set of vertices on the path is strict monotone. If a vertex v can be connected by another vertex w by a monotone geodesic and χ(v) > χ(w), then we say that v dominates w, and we write v ≻ w. ≻ is an ordering of V. For any pair v, w ∈ V there is a unique ≻-minimal vertex sup(v, w) which dominates both.

3.4 Examples. (1) For any integer n ∈ ℤ, let R_{n} be the tree with V = {v^{k}}_{k≥n} and E = {[v^{k}, v^{k+1}]}_{k≥n}. The grading is χ(v^{k}) = k.

(2) Let I be a finite index set. For each i ∈ I fix an integer n_{i} ∈ ℤ; and for each pair i, j ∈ I fix n_{ij} = n_{ji} ∈ ℤ with the next properties: (i) n_{ii} = n_{i}; (ii) n_{ij} ≥ max{n_{i}, n_{j}}; and (iii) n_{jk} ≤ max{n_{ij}, n_{ik}} for any i, j, k ∈ I.

For any i ∈ I consider R_{i} := R_{n_{i}} with vertices {v_{i}^{k}} and edges {[v_{i}^{k}, v_{i}^{k+1}]}, (k ≥ n_{i}). In the disjoint union ∐_{i} R_{i}, for any pair (i, j), identify v_{i}^{k} and v_{j}^{k}, resp. [v_{i}^{k}, v_{i}^{k+1}] and [v_{j}^{k}, v_{j}^{k+1}], whenever k ≥ n_{ij}. Write v̄_{i}^{k} for the class of v_{i}^{k}. Then ∐_{i} R_{i}/_{∼} is a graded root with χ(v̄_{i}^{k}) = k. It will be denoted by R = R({n_{i}}, {n_{ij}}).

Clearly V_{1}(R) is a subset of {v̄_{i}^{n_{i}}}_{i∈I}, and this last set can be identified with I. V_{1}(R) = I if in (ii) all the inequalities are strict. Otherwise all the indices I \ V_{1}(R) are superfluous, i.e. the corresponding R_{i}’s produce no additional vertices.

In fact, any graded root (R′, χ′) is isomorphic (in a natural sense) with some R({n_{i}}, {n_{ij}}). Indeed, set I := V_{1}(R′), n_{v} := χ′(v) and n_{uv} := χ′(sup(u, v)) for u, v ∈ I.

(3) Any map τ : {0, 1, …, l} → ℤ produces a starting data for construction (2). Indeed, set I = {0, …, l}, n_{i} := τ(i) (i ∈ I), and n_{ij} := max{n_{k} : i ≤ k ≤ j} for i ≤ j. Then ∐_{i} R_{i}/_{∼} constructed in (2) using this data will be denoted by (R_{τ} , χ_{τ}).

For example, for l = 4, take for the values of τ: −3, −1, −2, 0 and −2 (respectively −3, 0, −2, −1 and −2). Then the two graded roots are:

[figure: ⋮; χ = 0; R^{1} :; ⋮; χ = 0; R^{2} :]

In fact, we can even start with the infinite index set I = ℕ, and a function τ : ℕ → ℤ, provided that we assure that starting from a bound l, all the contributions R_{i} with i > l are superfluous. This happens, e.g., if for some l, τ(i + 1) ≥ τ(i) for any i ≥ l.

3.5 Definition. The associated ℤ[U]-modules. For any graded root (R, χ), let ℍ(R, χ) (shortly ℍ(R)) be the set of functions ϕ : V → T_{0}^{+} with the following property: whenever [v, w] ∈ E with χ(v) < χ(w), then

U · ϕ(v) = ϕ(w).

Or, equivalently, for any w ≻ v one requires

U^{χ(w)−χ(v)} · ϕ(v) = ϕ(w).  (∗)

Clearly ℍ(R) is a ℤ[U]-module via (Uϕ)(v) = U · ϕ(v). Moreover, ℍ(R) has a grading: ϕ ∈ ℍ(R) is homogeneous of degree d ∈ ℤ if for each v ∈ V with ϕ(v) ≠ 0, ϕ(v) ∈ T_{0}^{+} is homogeneous of degree d − 2χ(v). Notice that in (∗) one has 2χ(v) + deg ϕ(v) = 2χ(w) + deg ϕ(w), hence d is well-defined.

Notice that any ϕ as above is automatically finitely supported.

3.5.1 From the definitions, it is clear that ℍ((R, χ)[r]) = ℍ(R, χ)[2r] for any r ∈ ℤ.

3.6 Proposition. Let (R, χ) be a graded root. Set I := {v ∈ V : δ_{v} = 1}, and we order the set I as follows. The first element v_{1} is an arbitrary vertex with χ(v_{1}) = min_{v} χ(v). If v_{1}, …, v_{k} is already determined, and J := {v_{1}, …, v_{k}} ⊊ I, then let v_{k+1} be an arbitrary vertex in I \ J with χ(v_{k+1}) = min_{v∈I\J} χ(v). Let w_{k+1} ∈ V be the unique ≻-minimal vertex of R which dominates both v_{k+1}, and at least one vertex from J. Then one has the following isomorphism of ℤ[U]-modules

ℍ(R, χ) = T_{2χ(v_{1})}^{+} ⊕ ⊕_{k≥2}T_{2χ(v_{k})}(χ(w_{k}) − χ(v_{k})).

In particular, with the notations m := min_{v} χ(v) and

ℍ_{red}(R, χ) := ⊕_{k≥2}T_{2χ(v_{k})}(χ(w_{k}) − χ(v_{k})),

one has a canonical direct sum decomposition of graded ℤ[U]-modules: ℍ(R, χ) = T_{2m}^{+} ⊕ ℍ_{red}(R, χ). The ℤ[U]-module ℍ_{red}(R) has finite ℤ-rank; ℍ_{red}(R) = 0 if and only if #I = 1 and R = R_{min χ}.

Proof The proof is elementary and it is left to the reader.

3.7 Examples.(a) ℍ(R_{n}) = T_{2n}.

(b) The graded roots R^{1} and R^{2} constructed in 3.4(3) are not isomorphic but their ℤ[U]-modules are isomorphic: ℍ(R^{1}) = ℍ(R^{2}) = T_{−6}^{+} ⊕ T_{−4}(1) ⊕ T_{−4}(2). Hence, in general, a graded root carries more information than its ℤ[U]-module.

3.8 Corollary. Let (R_{τ} , χ_{τ}) be a graded root associated with some function τ : ℕ → ℤ, cf. 3.4(3), which satisfies τ(1) > τ(0). Then the ℤ-rank of ℍ_{red}(R_{τ} , χ_{τ}) is:

rank_{ℤ} ℍ_{red}(R_{τ}) = −τ(0) + min_{i≥0} τ(i) + ∑_{i≥0} max{τ(i) − τ(i + 1), 0}.

The summand T_{2m}^{+} of ℍ(T_{τ} , χ_{τ}) has index m = min_{i≥0} τ(i) = min_{v} χ_{τ} (v).

Proof Use induction over l (where τ : {0, …, l} → ℤ).

4 Graded roots associated with plumbing graphs

4.1 Fix a connected plumbing graph Γ whose bilinear form is negative definite (cf. 2.1). In this section we will construct a graded root (R_{k}, χ_{k}) associated with any characteristic element k.

4.2 The construction of (R_{k}, χ_{k}). Fix any k ∈ Char and define χ_{k} : L → ℤ by

χ_{k}(x) := −( k(x) + (x, x) )/2.

For any n ∈ ℤ, we define a finite 1-dimensional simplicial complex L̄_{k,≤n} as follows. Its 0-skeleton is L_{k,≤n} := {x ∈ L : χ_{k}(x) ≤ n}. For each x and j ∈ J with x, x + b_{j} ∈ L_{k,≤n}, we consider a unique 1-simplex with endpoints at x and x + b_{j} (e.g., the segment [x, x + b_{j}] in L ⊗ ℝ). We denote the set of connected components of L̄_{k,≤n} by π_{0}(L̄_{k,≤n}). For any v ∈ π_{0}(L̄_{k,≤n}), let C_{v} be the corresponding connected component of L̄_{k,≤n}.

Next, we define the graded root (R_{k}, χ_{k}) as follows. The vertices V(R_{k}) are ∪_{n∈ℤ}π_{0}(L̄_{k,≤n}). The grading V(R_{k}) → ℤ, still denoted by χ_{k}, is χ_{k}|π_{0}(L̄_{k,≤n}) = n.

If v_{n} ∈ π_{0}(L̄_{k,≤n}), and v_{n+1} ∈ π_{0}(L̄_{k,≤n+1}), and C_{v_{n}} ⊂ C_{v_{n+1}} , then [v_{n}, v_{n+1}] is an edge of R_{k}. All the edges E(R_{k}) of R_{k} are obtained in this way.

4.3 Proposition. For any k ∈ Char, (R_{k}, χ_{k}) is a graded root.

Proof Clearly, #V(R_{k}) is infinite. Notice that the validity of 3.2(1.b) guarantees that R_{k} has no closed cycles. Indeed, in the presence of a cycle v_{1}, v_{2}, …, v_{n}, v_{1} in R_{k}, χ_{k} restricted to the set {v_{1}, …, v_{n}} would have a local minimum which would contradict 3.2(1.b). Hence, it is enough to verify properties 3.2(1.ab-c). (a) is clear. With the notations of (b), assume that (b) is not true for some u, v, w, i.e. χ_{k}(u) < min{χ_{k}(v), χ_{k}(w)}. This would imply C_{u} ⊂ C_{v} ∩C_{w}. But this is impossible: two different connected components of a space cannot simultaneously contain a non-empty connected subset. The first two statements of (c) follow from the fact that B is negative definite. Finally, we have to show that L̄_{k,≤n} is connected for n sufficiently large. Set

Y := {x ∈ L : χ_{k}(x) ≤ χ_{k}(x + ϵ_{j}b_{j}) for every j ∈ J and both signs ϵ_{j} = ±1}.

Then, using the definition of χ_{k}, Y is {x ∈ L : −χ_{k}(−b_{j}) ≤ (x, b_{j}) ≤ χ_{k}(b_{j}) for every j ∈ J }. Hence Y is finite, and obviously non-empty since χ_{k} has a global minimum (which is in Y). Fix y_{∗} ∈ Y, and for each y ∈ Y \ {y_{∗}} fix a path which connects y_{∗} and y. By this we mean a sequence of elements x_{1}, x_{2}, …, x_{t} ∈ L so that x_{1} = y_{∗}, x_{t} = y, and x_{l+1} = x_{l} + ϵ_{l}b_{j(l)} for some ϵ_{l} ∈ {+1, −1}. Let n_{0} be the maximum of all the values of type χ_{k}(x_{l}), where x_{l} is an element on one of the above paths connecting y_{∗} with some y. Then L̄_{k,≤n} is connected provided that n ≥ n_{0}.

4.4 Hence, for any k ∈ Char we get a graded root (R_{k}, χ_{k}). Some of these graded roots are not very different. Indeed, assume that k and k′ determine the same spin^{c} structure, hence k′ = k + 2l for some l ∈ L. Then χ_{k′}(x − l) = χ_{k}(x)−χ_{k}(l) for any x ∈ L. This means that the transformation x ↦ x′ := x−l realizes an identification of L̄_{k,≤n} and L̄_{k′,≤n−χ_{k}(l)}. Hence, we get:

4.5 Proposition. If k′ = k + 2l for some l ∈ L, then: (R_{k′}, χ_{k′}) = (R_{k}, χ_{k})[−χ_{k}(l)].

In fact, there is an easy way to choose a graded root from the multitude {(R_{k}, χ_{k})}_{k∈[k]}. For any k ∈ Char we define

m_{k} := \frac{k^{2} − max_{k′∈[k]}(k′)^{2}}{8} ≤ 0.  (1)

Clearly, m_{k} is an integer (see the proof of the next lemma). Set also M_{[k]} := {k ∈ [k] : m_{k} = 0}.

4.6 Lemma. Fix a spin^{c} structure [k]. Then k_{0} ∈ M_{[k]} if and only if −χ_{k_{0}}(l) ≤ 0 for any l ∈ L. Moreover, if k_{0} and k_{0} +2l ∈ M_{[k]}, then −χ_{k_{0}}(l) = 0.

Proof Write (k_{0} + 2l, k_{0} + 2l) = (k_{0}, k_{0}) − 8χ_{k_{0}}(l).

4.6.1 In particular, using 4.5 and 4.6, for any fixed spin^{c} structure [k], any choice of k_{0} ∈ M_{[k]} provides the same graded root (R_{k_{0}}, χ_{k_{0}}), which will be denoted by (R_{[k]}, χ_{[k]}). Moreover, for any k ∈ [k]

(R_{k}, χ_{k}) = (R_{[k]}, χ_{[k]})[m_{k}], where m_{k} is defined in (1).  (2)

The notation is compatible with 3.6:

m_{k} = min χ_{k}  (3)

Indeed, if k_{0} ∈ M_{[k]}, then m_{k_{0}} = 0 from (1), and min χ_{k_{0}} = 0 from 4.6. Then (3) follows from (2). The relations (1) and (3) can be summarized in

k^{2} − 8 min χ_{k} = max_{k′∈[k]}(k′)^{2}.  (4)

Clearly, many different plumbing graphs can provide the same 3-manifold M. But all these plumbing graphs can be connected by each other by a finite sequence of blowups/downs (−1)-vertices, whose number of incident edges is ≤ 2. (This fact follows from the existence of an unique minimal resolution graph of a normal surface singularity.)

The next result shows that the set {(R_{[k]}, χ_{[k]})}_{[k]} depends only on M.

4.7 Proposition. The set (R_{[k]}, χ_{[k]}) (where [k] runs over the spin^{c} structures of M) depends only on M and is independent of the choice of the (negative definite) plumbing graph Γ which provides M. (Cf. also with [21](2.5).)

Proof By the above remark, it is enough to prove that the above set of graded roots is not modified during a blowup. First we assume that Γ′ is obtained from Γ by “blowing up a smooth point of the Riemann surface” which in the plumbing construction corresponds to a fixed index j_{0} ∈ J . More precisely, Γ′ denotes a graph with one more vertex and one more edge than Γ: we glue to the vertex j_{0} by the new edge the new vertex with decoration −1, the decoration of b_{j_{0}} is modified from e_{j_{0}} into e_{j_{0}} − 1, and we keep all the other decorations. We will use the notations L(Γ), L(Γ′), L′(Γ), L′(Γ′), B, B′ for the corresponding invariants. Set b_{new} for the new base element in L(Γ′) (with B′(b_{new}, b_{new}) = −1). The following facts can be verified:

• Consider the maps π_{∗} : L(Γ′) → L(Γ) defined by π_{∗}(∑ x_{j}b_{j} + x_{new}b_{new}) = ∑ x_{j}b_{j}, and π^{∗} : L(Γ) → L(Γ′) defined by π^{∗}(∑ x_{j}b_{j}) = ∑ x_{j}b_{j} + x_{j_{0}}b_{new}. Then B′(π^{∗}x, x′) = B(x, π_{∗}x′). This shows that B′(π^{∗}x, π^{∗}y) = B(x, y) and B′(π^{∗}x, b_{new}) = 0 for any x, y ∈ L(Γ).

• Identify (for both graphs) L′ with a sub-lattice of L_{ℚ} (cf. 2.1). Then consider the following (nonlinear) map: c : L′(Γ) → L′(Γ′) defined by ∑ r_{j}b_{j} ↦ ∑ r_{j}b_{j} + (r_{j_{0}} + 1)b_{new} (r_{j} ∈ ℚ), or equivalently, c(l′) := π^{∗}(l′) + b_{new}. Then c(Char(Γ)) ⊂ Char(Γ′) and c induces an isomorphism between the orbit spaces Char(Γ)/2L(Γ) and Char(Γ′)/2L(Γ′).

• Consider k ∈ Char(Γ) and write k′ := c(k) ∈ Char(Γ′). Then for any x ∈ L(Γ) one has: χ_{k}(x) = χ_{k′}(π^{∗}x). In particular, one has a well-defined injection

π^{∗} : {x ∈ L(Γ) : χ_{k}(x) ≤ n} ↪ {y ∈ L(Γ′) : χ_{k′}(y) ≤ n}.

• For any z ∈ L(Γ′) write z in the form π^{∗}π_{∗}z + ab_{new} for some a ∈ ℤ. Then χ_{k′}(z) = χ_{k′}(π^{∗}π_{∗}z) + χ_{k′}(ab_{new}). On the other hand, for any a′ ∈ ℤ (with |a′| ≤ |a|), one has χ_{k′}(a′b_{new}) = a′(a′ + 1)/2 ≥ 0. In particular, if χ_{k′}(z) = m, then z and π^{∗}π_{∗}z are in the same connected component of {y ∈ L(Γ′) : χ_{k′}(y) ≤ m}.

All these facts together imply R_{k} = R_{k′}. There is a similar verification in the case when one blows up “an intersection point” corresponding to two indices i_{0} and j_{0} with B(b_{i_{0}}, b_{j_{0}}) = 1. The details are left to the reader.

Finally, we verify the compatibility of the two combinatorial objects ℍ^{+}(Γ, [k]) (cf. 2.4.5) and ℍ(R_{[k]}, χ_{[k]}) (cf. 3.5). Recall that s denotes the number of vertices of Γ.

4.8 Proposition For any [k] ∈ Spin^{c}(M) one has:

ℍ^{+}(Γ, [k]) = ℍ(R_{[k]}, χ_{[k]})[ − max_{k′∈[k]} \frac{(k′)^{2} + s}{4}].

Proof Fix an arbitrary k_{0} ∈ M_{[k]}. Then [k] = k_{0} + 2L, hence ℍ^{+}(Γ, [k]) ⊂ Hom(Char, T_{0}^{+}) can be identified with a set of maps {ϕ_{L} : L → T_{0}^{+}} which satisfy the following: for any l ∈ L and b_{j} with 2n = (k_{0} + 2l + b_{j}, b_{j}) one has U^{n} · ϕ_{L}(l + b_{j}) = ϕ_{L}(l) if n ≥ 0, or ϕ_{L}(l + b_{j}) = U^{−n} · ϕ_{L}(l) if n ≤ 0. Notice that χ_{k_{0}}(l + b_{j}) − χ_{k_{0}}(l) = −n; hence the above property is equivalent with 3.5(∗), showing the compatibility of the two sets of restrictions. Still, one has to verify two more facts.

In order to formalize the first, let us consider the natural map θ : L → V(R_{k_{0}}), where we associate with any l ∈ L the component of L̄_{k_{0},≤χ_{k_{0}}(l)} containing l.

This induces a map Hom(V(R_{k_{0}}), T_{0}^{+}) → Hom(L, T_{0}^{+}) by composition ϕ ↦ ϕ_{L} := ϕ∘θ; and, by the above compatibility verifications, a map θ^{∗} : ℍ(R_{k_{0}}) → ℍ^{+}(Γ, [k]). Since (in general) θ is not onto, one has to verify that any ϕ_{L} has an extension to a ϕ with θ^{∗}(ϕ) = ϕ_{L}, and this extension is unique.

Assume that v ∈ V(R_{k_{0}}) \ Imθ corresponds to a component C_{v} in L̄_{k_{0},≤n}. Set m_{v} := max(χ_{k_{0}}(L ∩ C_{v}), and take some l_{v} ∈ L ∩ C_{v} with χ_{k_{0}}(l_{v}) = m_{v}. Then for any ϕ ∈ ℍ(R_{k_{0}}), the value ϕ(v) is uniquely determined by ϕ(v) = U^{n−m_{v}} · ϕ_{L}(l_{v}).

Finally, we have to fit the gradings. With the obvious notations, for any k = k_{0} + 2l, one has

deg(ϕ(k)) = \frac{(k_{0} + 2l)^{2} + s}{4} + d_{ℍ^{+}} = −2χ_{k_{0}}(l) + d_{ℍ},

or, equivalently,

d_{ℍ} − d_{ℍ^{+}} = \frac{k_{0}^{2} + s}{4} = max_{k′∈[k]} \frac{(k′)^{2} + s}{4}.

5 Distinguished characteristic elements

There is a more subtle way to choose a special characteristic element from each fixed orbit [k], a fact which will be crucial in the next discussions. The goal of the present section is its definition.

5.1 Definitions. The canonical graded root. Recall that the canonical characteristic element K ∈ Char is defined by the equations K(b_{j}) = −e_{j} − 2 (or equivalently, by χ_{K}(b_{j}) = 1) for any j ∈ J . We denote by (R_{can}, χ_{can}) the graded root associated with K. In order to simplify the notations, we also write χ_{K} = χ.

Sometimes it is preferable (motivated by the symplectic geometry) to define the “canonical” spin^{c} structure σ_{can} via c_{1}(σ_{can}) = [−K], i.e. identifying it by the anti-canonical bundle. Our choice above is motivated by the Riemann-Roch formulas, and the connection with singularity theory. But, in fact, from the point of view of the results of the present paper, the two choices are completely equivalent. Indeed, there is a natural involution x ↦ −x on L, L′ and H. This is compatible with the natural involution on Spin^{c}(M). All the formulas in this paper are stable with respect to these involutions (e.g., min χ_{K} = min χ_{−K}): the orbits [K] and [−K] share the same properties.

5.1.1 Example. If M is the link of a normal surface singularity, and X̃ is a resolution of (X, 0), then K is the first Chern class of the canonical bundle of the complex structure of X̃. b_{j} ∈ L denotes the fundamental class of the irreducible exceptional divisors E_{j}, hence x = ∑ n_{j}b_{j} ∈ L can be identified with the cycle Z = ∑ n_{j}E_{j} supported by the exceptional divisor. The linear equations defining K are the adjunction formulas for the canonical line bundle. Hence, by Riemann-Roch, χ(x) is exactly the Euler-characteristic of the sheafcohomology of O_{Z}.

5.2 Notations. We embed L′ into L_{ℚ} as in 2.1. We set S_{ℚ} := {x ∈ L_{ℚ} : (x, b_{j}) ≤ 0 for every j ∈ J } and S = S_{ℚ} ∩ L. Since B is negative definite, if x ∈ S_{ℚ}, then x ≥ 0, cf. 6.1(a).

5.3 Fix a characteristic element k, i.e. fix an l′ ∈ L′ with k = K + 2l′. Then the function χ_{k} : L → ℤ is equivalent to the rational valued extension of χ = χ_{K} restricted to the sub-lattice l′ + L of L_{ℚ}. Indeed, extend χ by the same formula χ : L_{ℚ} → ℚ, χ(x) := −(K + x, x/2). Then, for any x ∈ L one has:

χ_{k}(x) = χ(l′ + x) − χ(l′).

In fact, the equivalence class [k] = K + 2l′ + 2L can be identified with this sub-lattice l′ + L in L_{ℚ}. We wish to choose a distinguished element of l′ + L and list some of its properties.

5.4 Lemma. For any fixed [k] = K + 2(l′ + L), the intersection (l′ + L) ∩ S_{ℚ} in L_{ℚ} admits a unique minimal element l′_{[k]}. (Here minimality is considered with respect to the ordering ≤ in L_{ℚ}.)

The proof is similar to Artin’s proof of the existence of the fundamental cycle [1, 2].

Assume that l′ + l_{i} ∈ S_{ℚ}, l_{i} ∈ L for i = 1, 2. Set l := min{l_{1}, l_{2}}, and write x_{i} = l_{i} − l ≥ 0. We show that l′ + l ∈ S_{ℚ} as well. Consider any j ∈ J . Since |x_{1}| ∩ |x_{2}| = ∅, we can assume that either b_{j} ∉ |x_{1}| or b_{j} ∉ |x_{2}|. If b_{j} ∉ |x_{1}| then (l′ + l, b_{j}) = (l′ + l_{1}, b_{j}) − (x_{1}, b_{j}) ≤ 0.

5.5 Definition. In any fixed class [k] we fix the distinguished representative k_{r} = K + 2l′_{[k]} ∈ [k].

5.6 Remarks. (a) Since l′_{[K]} = 0, the distinguished representative in [K] is K.

(b) If e_{j} ≤ −2 for any j (e.g., if Γ is a minimal resolution graph of a normal surface singularity) then K(x) ≥ 0 for any x ≥ 0. This fails to be true for arbitrary plumbing graphs. Still, we have K(x) − (x, x) ≥ 0 for any x ≥ 0. This is a special case of the following general result.

5.7 Proposition. For any fixed [k] ∈ Spin^{c}(M) and the representative k_{r} ∈ [k] one has:

(a) (l′_{[k]}, b_{j}) ≥ b_{j}^{2} + 1 for any j ∈ J ;

(b) k_{r}(x) ≥ x^{2} for any x ≥ 0, x ∈ L;

(c) χ_{k_{r}}(−x) ≥ 0 for any x ≥ 0, x ∈ L.

Proof (a) Assume that (l′_{[k]}, b_{j_{0}}) ≤ b_{j_{0}}^{2} for some j_{0} ∈ J . Then by a computation l′_{[k]} − b_{j_{0}} ∈ (l′_{[k]} + L) ∩ S_{ℚ}, a fact which contradicts the minimality of l′_{[k]}, cf. 5.4.

(b) follows similarly. First notice that by (a) k_{r}(b_{j}) = 2(l′_{[k]}, b_{j})−b_{j}^{2} −2 ≥ b_{j}^{2} for any j ∈ J . Therefore, if there exists an effective cycle x > 0 with k_{r}(x) < x^{2}, then there exists a minimal one with this property. This means that that minimal x > 0 satisfies

x^{2} − k_{r}(b_{j}) > k_{r}(x) − k_{r}(b_{j}) ≥ (x − b_{j})^{2} for every b_{j} ∈ |x|.

Hence 0 ≥ (l′_{[k]} − x, b_{j}) for every b_{j} ∈ |x|. On the other hand, if b_{j} ∉ |x|, then 0 ≥ (l′_{[k]}, b_{j}) ≥ (l′_{[k]} − x, b_{j}). These two inequalities show that for this x > 0 one has l′_{[k]} − x ∈ (l′_{[k]} + L) ∩ S_{ℚ}, which contradicts the minimality of l′_{[k]}. For (c) use (b) and the identity 2χ_{k_{r}}(−x) = k_{r}(x) − x^{2}.

6 First properties of the graded roots (R_{k}, χ_{k})

In this section we focus on the distinguished representatives k_{r} = K + 2l′_{[k]} of the orbits [k]. [Then all their properties can be transformed into similar properties of arbitrary characteristic elements via 4.5.]

For any class [k], we define S_{[k]} := {x ∈ L : (x + l′_{[k]}, b_{j}) ≤ 0 for every j ∈ J }. Clearly, S_{[K]} = S.

6.1 Theorem. (a) For any [k], if x ∈ S_{[k]} then x ≥ 0.

(b) For any class [k], consider the distinguished representative k = k_{r} = K + 2l′_{[k]}. Let L̄_{k,≤n}^{s} be the subcomplex of L̄_{k,≤n} whose 0-skeleton consists of cycles x ∈ L_{k,≤n}^{s} := L_{k,≤n} ∩ S_{[k]}, and the 1-cells are exactly those 1-cells of L̄_{k,≤n} which have their endpoints in L_{k,≤n}^{s}. Then the natural inclusion L̄_{k,≤n}^{s} → L̄_{k,≤n} induces a surjection π_{0}(L̄_{k,≤n}^{s}) → π_{0}(L̄_{k,≤n}) for any n.

(c) Assume k = K. Then the component C_{0} in L̄_{K,≤0}, which contains the zero cycle, contains no non-zero effective cycle. In fact, if X denotes the set of non-zero cycles in the component C_{0}, then for any x ∈ X one has x < 0 and χ(x) = 0.

(d) For k = K, L̄_{K,≤n} is connected whenever n ≥ 1.

Proof (a) Notice that (l′_{[k]} + L) ∩ S_{ℚ} = l′_{[k]} + S_{[k]}, hence (a) follows from the minimality of l′_{[k]}, cf. 5.4. A direct argument goes as follows. Assume the contrary: x = x_{1}−x_{2} with x_{1} ≥ 0, x_{2} > 0 and |x_{1}|∩|x_{2}| = ∅. If b_{j} ∈ |x_{2}|, then (b_{j}, x_{1}) ≥ 0, hence by the definition of S_{[k]} one gets (l′_{[k]} − x_{2}, b_{j}) ≤ −(x_{1}, b_{j}) ≤ 0. If b_{j} ∉ |x_{2}| then (x_{2}, b_{j}) ≥ 0 ≥ (l′_{[k]}, b_{j}). Hence l′_{[k]} − x_{2} ∈ S_{ℚ}, which contradicts the minimality of l′_{[k]}, cf. 5.4.

(b) Assume that x ∈ L \ S_{[k]}. Then for some j ∈ J , (x + l′_{[k]}, b_{j}) ≥ 1. This is equivalent with χ_{k}(x + b_{j}) ≤ χ_{k}(x). If x + b_{j} ∈ S_{[k]} then we stop, otherwise we repeat the same algorithm for x + b_{j} instead of x. In this way we construct an increasing sequence along which χ_{k} is decreasing. Since B is negative definite, this procedure must stop.

(c) We prove that any x ∈ X satisfies (i) x < 0 and (ii) χ(x) = 0. First notice that b_{j} ∉ L_{K,≤0} and −b_{j} ∈ L_{K,≤0} if and only if χ(−b_{j}) = 0, or equivalently, e_{j} = −1.

Since X represents the (non-zero) cycles of the connected component C_{0}, it is enough to verify the following inductive step. If x ∈ X satisfies (i-ii), and x′ = x ± b_{j} ∈ X, then x′ satisfies (i-ii) as well. In order to verify (i) for x′, we have to verify that the situation x′ = x + b_{j} with b_{j} ∉ |x| is not possible. Indeed, in this case χ(x′) = χ(x) + χ(b_{j}) − (x, b_{j}) = 1 + (−x, b_{j}) ≥ 1 which would contradict x′ ∈ L_{K,≤0}. Now, using (i) and the fact χ(x′) ≤ 0, 5.7(c) implies χ(x′) = 0.

(d) Via (a-b), it is enough to show that any x_{0} ≥ 0, x_{0} ∈ L_{K,≤n}, can be connected to the zero cycle by a path inside of L̄_{K,≤n}. For this, it is enough to verify the following two facts:

(i) For any x > 0 with χ(x) > 0, there exists b_{j} ∈ |x| such that χ(x−b_{j}) ≤ χ(x).

Indeed, if this property would fail, then we would have (b_{j}, x − b_{j}) > 1 for any b_{j} ∈ |x|. This is equivalent with (b_{j}, x) ≥ −K(b_{j}), which implies K(x)+x^{2} ≥ 0, or χ(x) ≤ 0, a contradiction.

(ii) For any x > 0 with χ(x) ≤ 0, there exists b_{j} ∈ |x| such that χ(x − b_{j}) ≤ 1.

In order to prove this, denote χ(x) by c ≤ 0; write x = ∑_{j} n_{j}b_{j} and set n := ∑_{j} n_{j}. Clearly n ≥ 2 since χ(b_{j}) = 1 for any j. Now, assume that χ(x−b_{j}) > 1 for all b_{j} ∈ |x|. Then, in the identity −(b_{j}, x − b_{j}) = χ(x) − χ(x − b_{j}) − 1 the right hand side is < c − 2. This means that −(b_{j}, x) < c − 2 − e_{j} = c + K(b_{j}) for any b_{j} ∈ |x|. This implies that K(x) + x^{2} + cn > 0, or c(n − 2) > 0, which is a contradiction.

Then the construction of the wanted path connecting x_{0} with 0 is clear: in the path any x is succeeded by some x − b_{j} according to (i) or (ii).

6.2 Characterization of rational singularities. Recall that a normal surface singularity is called rational if its geometric genus p_{g} = 0. It is easy to verify that any resolution of a rational singularity is automatically good (i.e. any resolution graph can be considered as a plumbing graph), and the link M is a rational homology sphere. Artin characterized the rational singularities topologically in terms of any fixed resolution graph Γ [1, 2]. Namely,

p_{g} = 0 if and only if χ(x) ≥ 1 for any x > 0, x ∈ L.  (1)

A connected, negative definite plumbing graph with this property is called rational graph.

E.g., any plumbing graph with −e_{j} ≥ δ_{j} for every j ∈ J is rational. [This can be verified as follows: if −e_{j} ≥ δ_{j} then the Artin’s fundamental cycle x_{min} (cf. 7.5) is exactly ∑_{j} b_{j}, and by an easy computation one has χ(x_{min}) = 1, hence the rationality follows from Artin’s criterion, cf. 7.5 and [1, 2].] On the other hand, not any rational graph satisfies −e_{j} ≥ δ_{j}: consider e.g. the −E_{8} graph.

The class of rational graphs is closed while taking subgraphs and decreasing Euler numbers e_{j}.

6.3 Theorem. Let Γ be a connected, negative definite plumbing graph whose plumbed three-manifold is a rational homology sphere. Then the following facts are equivalent:

(a) Γ is rational; (a′) #χ_{can}^{−1}(0) = 1 (where (R_{can}, χ_{can}) is the canonical graded root associated with Γ);

(b) R_{can} = R_{0} (for the definition of R_{0}, or of any R_{m}, see 3.4(1));

(c) R_{can} = R_{m} for some m ∈ ℤ;

(d) For all characteristic elements k ∈ Char, R_{k} = R_{m_{k}} for some m_{k} ∈ ℤ;

(Hb) ℍ(R_{can}) = T_{0}^{+};

(Hc) ℍ(R_{can}) = T_{m}^{+} for some m ∈ ℤ; or equivalently, ℍ_{red}(R_{can}) = 0;

(Hd) For all k ∈ Char, ℍ(R_{k}) = T_{m_{k}}^{+} for some m_{k} ∈ ℤ; or equivalently, ⊕_{[k]}ℍ_{red}(R_{[k]}) = 0.

Moreover, if Γ is rational and k = K + 2l′, then

m_{k} = min χ_{k} = min_{x∈L} χ(l′ + x) − χ(l′) = χ(l′_{[k]}) − χ(l′) = χ_{k}(l′_{[k]} − l′) ≤ 0.

In particular, if Γ is rational and k_{r} = K + 2l′_{[k]}, then min χ_{k_{r}} = 0 (cf. also with 4.5 and 4.6.1).

It is instructive to compare (a′) with the property #χ_{can}^{−1}(n) = 1, valid for any graph Γ and integer n ≥ 1 (cf. 6.1(d)).

Proof a ⇒ b follows from 6.2(1) and 6.1 (since L̄_{K,≤0}^{s} consists of only one vertex, namely x = 0; and L̄_{K,≤n}^{s} = ∅ for n < 0). b ⇒ c ⇒ Hc and b ⇒ Hb ⇒ Hc are clear. Hc ⇒ c follows from 3.6. Next, we verify c ⇒ a′ ⇒ a. Indeed, if R_{can} = R_{m} for some m, then clearly m ≤ χ(0) = 0 and χ_{can}^{−1}(0) has only one element. This clearly corresponds to the connected componet C_{0} of L̄_{K,≤0} which contains the zero cycle. Hence L_{K,≤0} = C_{0}, and by 6.1(c) χ(x) > 0 for any x > 0.

Finally, d ⇒ Hd ⇒ Hc are clear. Hence it remains a ⇒ d. Using 4.5, it is enough to consider the characteristic elements of the form k = K + 2l′_{[k]}. Fix such a k.

Let L_{k,≤n}^{e} = L_{k,≤n} ∩ {effective cycles}, and define L̄_{k,≤n}^{e} similarly as L̄_{k,≤n}^{s} in 6.1(b). Then 6.1(a-b) shows that π_{0}(L̄_{k,≤n}^{e}) → π_{0}(L̄_{k,≤n}) is onto for any n. Hence it is enough to show that any x ∈ L_{k,≤n}^{e} can be connected with the zero cycle in L̄_{k,≤n}^{e} (a fact which automatically show that n ≥ 0 whenever L_{k,≤n}^{e} in non-empty). More precisely, we will show that for any x > 0 there exists at least one b_{j} ∈ |x| so that χ_{k}(x − b_{j}) ≤ χ_{k}(x). Indeed, assume that this is not true for some b_{j}. Then this means (x + l′_{[k]}, b_{j}) > 1 + b_{j}^{2} for every b_{j} ∈ |x|.

Since (l′_{[k]}, b_{j}) ≤ 0, we get (K + x, b_{j}) ≥ 0 for every b_{j} ∈ |x|. This implies K(x) + x^{2} ≥ 0, or χ(x) ≤ 0, which contradicts the fact that Γ is rational.

This, in particular, also shows that χ_{k_{r}}(x) ≥ 0 for any x ≥ 0. From the proof of 6.1(b) we get that χ_{k_{r}}(x) ≥ 0 for any x. In other words, χ(l′_{[k]} + x) ≥ χ(l′_{[k]}) for any x ∈ L. This and 5.3 also imply the last statement about m_{k}.

6.4 Characterization of weakly elliptic singularities via (R_{can}, χ_{can}) or ℍ(R_{can}). A normal surface singularity is called weakly elliptic if any/all of its resolution graphs is/are weakly elliptic. A resolution graph Γ is weakly elliptic if min_{x>0} χ(x) = 0 (cf. [6]). [This definition is valid even if the link is not a rational homology sphere, but in order to be consistent with all the above notations, we will assume this fact as well.]

6.5 Proposition. Let Γ be a connected, negative definite plumbing graph whose plumbed three-manifold is a rational homology sphere. Then the following facts are equivalent:

(a) Γ is weakly elliptic;

(b) R_{can} = R({n_{i}}, {n_{ij}}) for some index set I, #I = l + 1 ≥ 2, and n_{i} = 0 for any i ∈ I, and n_{ij} = 1 for any i ≠ j;

(Hb) ℍ(R_{can}) = T_{0}^{+} ⊕ ( T_{0}(1) )^{⊕l} for some l ≥ 1.

Proof a ⇔ b follows from 6.1, and b ⇔ Hb from 6.1(d) and 3.6.

6.6 Remarks. (a) The results 6.1(d), 6.3 and 6.5 can also be interpreted as follows: The grading χ_{can} of R_{can} satisfies min χ_{can} ≥ 0 (or, equivalently min χ_{can} = 0) if and only if Γ is either rational or weakly elliptic. In this situation, if ℍ_{red}(R_{can}) = 0 then Γ is rational, otherwise it is weakly elliptic.

(b) Let k_{r} = K + 2l′_{[k]} as above. Then min χ_{k_{r}} ≥ min χ. Indeed, by 6.1 (ab), min χ_{k_{r}} = χ_{k_{r}}(x) for some effective cycle x ≥ 0. But for such a cycle χ_{k_{r}}(x) = χ(x) − (l′_{[k]}, x) ≥ χ(x).

(c) In particular, if Γ is rational or weakly elliptic, then for any spin^{c}-structure [k] one has min χ_{k_{r}} = 0. Indeed, 0 = χ_{k_{r}}(0) ≥ min χ_{k_{r}} ≥ min χ = 0.

6.7 Remark. One has the following connections with singularity theory. (Since these facts are not exactly in the spirit of the present paper, we omit the details.) One can verify that l in 6.5 is exactly the length of the elliptic sequence in the sense of S.S.-T. Yau, or, equivalently, it is the length of the Laufer’s sequence. (For their definitions and equivalence, see [23].) If the graph Γ is minimally elliptic (in the sense of Laufer [6]) then l = 1. The opposite statement is not necessarily true: one can find elliptic graphs with l = 1 which are not numerically Gorenstein (i.e. K is not an integral cycle), hence which are not minimally elliptic. E.g., the following graph Γ has these properties:

[figure: −1; −4; −2; −3; −4; Γ :; R_{can} :; ⋮; 0]

[Here the two minimal points of R_{can} correspond to the zero cycle and to Artin’s fundamental cycle.]

7 Generalities about computation sequences

7.1 The computation of the groups ℍ (cf. section 9) is based on the techniques of computation sequences, cf. e.g. with [5, 6, 9, 23]. These objects were successfully used in the study of the (resolution of) normal surface singularities (see e.g. [10] for more details). Some of the next statements and proofs rhyme perfectly with some of those computations, e.g. with the proof of the existence of Artin’s fundamental cycle, with Laufer’s algorithm which provides this fundamental cycle, or with the construction of Yau’s elliptic sequence. Nevertheless, in order to be self-contained, and since we also wish to treat the case of an arbitrary spin^{c}-structure, we will provide all the details.

7.2 Definition. Sequences x_{0}, x_{1}, …, x_{t} ∈ L with x_{l+1} = x_{l} + b_{j(l)} (0 ≤ l < t), where j(l) is determined by some principles fixed in each individual case, are called computation sequences connecting x_{0} and x_{t}.

In this section we will fix a spin^{c}-structure [k] and we will fix its distinguished representative k_{r} = K + 2l′_{[k]}. In order to simplify the notation, in this section we write k for k_{r}. Recall that S_{[k]} = {x ∈ L : (x + l′_{[k]}, b_{j}) ≤ 0 for every j ∈ J }.

We will write m_{j}(z) for the coefficients of a cycle z = ∑ m_{j}(z)b_{j}.

7.3 Lemma. For any x ∈ L, there exists a unique minimal y = y(x) ∈ L with the properties x ≤ y and y ∈ S_{[k]}.

Proof First notice that (since B is negative definite) there exists a cycle y > 0 so that (y, b_{j}) < 0 for every j ∈ J . Then all the coefficients of y are strict positive (cf. also with 6.1(a)). Hence, some integral multiple ay of it will satisfy ay > x and ay ∈ S_{[k]}. For the existence of a unique minimal element with these properties it is enough to verify that if y_{1}, y_{2} ∈ S_{[k]} then y := min{y_{1}, y_{2}} is also in S_{[k]}. Indeed, for any j, at least for one index i ∈ {1, 2} one has b_{j} ∉ |y_{i} − y|. Then (y + l′_{[k]}, b_{j}) = (y_{i} + l′_{[k]}, b_{j}) − (y_{i} − y, b_{j}) ≤ 0.

7.4 Lemma. (Generalized Laufer’s algorithm, cf. [5]) Fix an x ∈ L. Construct a computation sequence x_{0}, …, x_{t} by the following algorithm. Set x_{0} = x. Assume that x_{l} is already constructed. If x_{l} ∉ S_{[k]}, i.e. (x_{l} + l′_{[k]}, b_{j}) > 0 for some index j, then choose one of them for j(l), and write x_{l+1} = x_{l}+b_{j(l)}. If x_{l} ∈ S_{[k]}, then stop and write t = l. Then x_{t} is exactly y(x) considered in 7.3.

Moreover, this computations sequence satisfies χ_{k}(x_{l+1}) ≤ χ_{k}(x_{l}) for every 0 ≤ l < t.

Proof We will show by induction that x_{l} ≤ y(x) for any 0 ≤ l < t. Indeed, assume that x_{l} ≤ y(x), and x_{l+1} = x_{l} + b_{j}. Then we have to verify that m_{j}(x_{l}) < m_{j}(y(x)). Assume that this is not true, i.e. b_{j} ∉ |y(x) − x_{l}|. Then (x_{l} + l′_{[k]}, b_{j}) = (y(x) + l′_{[k]}, b_{j}) − (y(x) − x_{l}, b_{j}) ≤ 0, a contradiction.

On the other hand, if each x_{l} ≤ y(x), then the algorithm should stop with some x_{t} ∈ S_{[k]}, and x_{t} ≤ y(x). Then the minimality of y(x) guarantees that x_{t} = y(x).

The inequality follows from χ_{k}(x_{l+1}) − χ_{k}(x_{l}) = 1 − (x_{l} + l′_{[k]}, b_{j}) ≤ 0.

7.5 Remark. (a) If we start with x = b_{j} (for some/any j), and k = K, then y(x) is exactly Artin’s fundamental cycle x_{min}. It is the minimal strict effective cycle with (x_{min}, b_{j}) ≤ 0 for every j. [Since x_{min} ≥ ∑_{j} b_{j}, hence also x_{min} ≥ b_{j} for every j ∈ J , the proof of 7.4 also shows that starting with x_{0} = b_{j} we get x_{min} independently of the choice of the index j ∈ J . In fact, there is a computation sequence connecting x_{0} = b_{j} and x_{min} such that x_{s−1} = ∑_{j} b_{j} and χ(x_{l}) = 1 for any 0 ≤ j < s, where s = #J .]

It is known that Γ is rational if and only if χ(x_{min}) = 1 [1, 2], and Γ is weakly elliptic if and only if χ(x_{min}) = 0 [6]. (In general, χ(x_{min}) ≤ 1.)

(b) Assume that above Γ is a rational graph, and k = K (hence l′_{[k]} = 0 and χ_{k} = χ). Consider the computation sequence 7.4 connecting x_{0} := b_{j} and x_{min}. It satisfies χ(x_{l+1}) ≤ χ(x_{l}) for any l. One the other hand χ(b_{j}) = 1 and χ(x_{min}) = 1 (by the rationality of Γ). Hence χ(x_{l+1}) = χ(x_{l}) for any l; in other words, (x_{l}, b_{j(l)}) = 1 for any l (a fact first noticed by Laufer [5]).

In the next paragraphs we wish to generalize the above facts 7.3 and 7.4. We prefer to write J = {0} ∪ J^{∗}, hence to distinguish a base element b_{0}.

7.6 Lemma. For any integer i ≥ 0, there exists a unique cycle x(i) ∈ L with the following properties:

(a) m_{0}(x(i)) = i;

(b) (x(i) + l′_{[k]}, b_{j}) ≤ 0 for any j ∈ J^{∗};

(c) x(i) is minimal with properties (a-b).

Moreover, the cycle x(i) satisfies x(i) ≥ 0.

Proof First we verify that there exists at least one x ∈ L satisfying (a-b). By 7.3, there exists x̃ ∈ S_{[k]} with x̃ ≥ b_{0}. Take an integer a ≥ 1 such that (a − 1)l′_{[k]} is an integral cycle and h := m_{0}(ax̃ + (a − 1)l′_{[k]}) − i ≥ 0. Then define x := ax̃ + (a − 1)l′_{[k]} − hb_{0}. By a computation m_{0}(x) = i and (x + l′_{[k]}, b_{j}) = a(x̃ + l′_{[k]}, b_{j}) − h(b_{0}, b_{j}) ≤ 0 for j ≠ 0.

Next, one can verify that there exists a unique minimal element with properties (a-b). The proof is similar to the proof of 7.3, and it is left to the reader.

Finally we verify that x(i) ≥ 0. Write x(i) in the form x_{1} − x_{2} with x_{1} ≥ 0, x_{2} ≥ 0, |x_{1}| ∩ |x_{2}| = ∅. Consider an index j ∈ J^{∗}. If b_{j} ∉ |x_{1}| then one has (l′_{[k]} − x_{2}, b_{j}) ≤ (l′_{[k]} − x_{2} + x_{1}, b_{j}) = (x(i) + l′_{[k]}, b_{j}) ≤ 0. Similarly, for any b_{j} ∈ |x_{1}| one has (l′_{[k]} − x_{2}, b_{j}) ≤ (l′_{[k]}, b_{j}) ≤ 0. Hence l′_{[k]} − x_{2} ∈ (l′_{[k]} + L) ∩ S_{ℚ}, which implies x_{2} = 0 via the minimality of l′_{[k]}, cf. 5.4.

7.7 Lemma. The computation sequence connecting x(i) and x(i+1). For any integer i ≥ 0 consider a computation sequence constructed as follows. Set x_{0} = x(i), x_{1} = x(i)+b_{0}. Assume that x_{l} (l ≥ 1) is already constructed. If x_{l} does not satisfy 7.6(b) then there exists some j ∈ J^{∗} with (x_{l} + l′_{[k]}, b_{j}) > 0. Then choose one of these indices for j(l), and write x_{l+1} = x_{l} + b_{j(l)}. If x_{l} satisfies 7.6(b), then stop and write t = l. Then x_{t} is exactly x(i + 1).

Moreover, this computations sequence satisfies χ_{k}(x_{l+1}) ≤ χ_{k}(x_{l}) for any 0 < l < t.

Corresponding to l = 0 one has χ_{k}(x(i) + b_{0}) − χ_{k}(x(i)) = 1 − (x(i) + l′_{[k]}, b_{0}). In other words, χ_{k}(x(i) + b_{0}) > χ_{k}(x(i)) if and only if x(i) ∈ S_{[k]}.

Proof Repeat the arguments used in the proof of 7.4.

Lemma 7.7 has the following easy generalization (with the same proof):

7.8 Lemma. Assume that x ∈ L satisfies m_{0}(x) = i and x ≤ x(i) for some i ≥ 0. Consider a similar computation sequence as in 7.7. Namely, set x_{0} = x. Assume that x_{l} is already constructed. If for some j ∈ J^{∗} one has (x_{l} +l′_{[k]}, b_{j}) > 0 then take x_{l+1} = x_{l} +b_{j(l)}, where j(l) is such an index j. If x_{l} satisfies 7.6(b), then stop and write t = l. Then x_{t} is exactly x(i). Moreover, this computations sequence satisfies χ_{k}(x_{l+1}) ≤ χ_{k}(x_{l}) for any 0 ≤ l < t.

Notice that, even if it is not explicitly emphasized in its notation, the cycles {x(i)}_{i≥0} depend on the choice of the distinguished vertex and of the spin^{c}- structure [k].

8 Almost-rational graphs. HF^{+}(−M, [k]) = ℍ^{+}(Γ, [k]).

8.1 Definition. Assume that the plumbing graph Γ is a negative definite connected tree. We say that Γ is almost-rational (in short AR) if there exists a vertex j_{0} ∈ J of Γ such that replacing e_{j_{0}} by some e′_{j_{0}} ≤ e_{j_{0}} we get a rational graph Γ′. In general, the choice of j_{0} is not unique. Once the distinguished vertex j_{0} is fixed, we write J = {0}∪J^{∗} such that the index 0 ∈ J corresponds to this vertex.

8.2 Examples. The set of AR graphs is (surprisingly) large.

1) All rational graphs are AR. Indeed, take e′_{j_{0}} = e_{j_{0}} for any vertex j_{0}.

2) Any weakly elliptic graph is AR. This can be proved as follows. Take a computation sequence x_{0}, …, x_{t} for x_{min} as in 7.5. Since Γ is weakly elliptic if and only if χ(x_{min}) = 0 (see e.g. 7.5), we get from a similar discussion like in 7.5 that (x_{l}, b_{j(l)}) = 1 all the time in the computation sequence, excepting exactly one l = l_{0} when this intersection = 2 (and b_{j(l_{0})} ∈ |x_{l_{0}}| as well). Then take j(l_{0}) for the distinguished vertex. If we replace e_{j(l_{0})} by a strict smaller integer, then for some t′ ≤ t, the sequence x_{0}, …, x_{t′} will be a computation sequence for the new fundamental cycle with (x_{l}, b_{j(l)}) = 1 for any l < t′. Hence χ of the new fundamental cycle will be 1, a fact which characterizes the rational singularities.

3) Any star-shaped graph is AR. Indeed, first blow down all the (−1)-vertices different from the central vertex (this transformation preserves the AR graphs); let this new graph be Γ̄. Then take for j_{0} the central vertex of Γ̄, and take for −e′_{j_{0}} an integer larger than the number of adjacent vertices of the central vertex of Γ̄. Then the modified graph will become rational. This follows from the fact that a graph is rational provided that −e_{j} ≥ δ_{j} is satisfied for all its vertices j (cf. 6.2, or [10]). [In other words, if one takes for the distinguished vertex of Γ the central vertex, and one takes −e_{j_{0}} sufficiently large, than one gets a rational graph.]

The same argument shows that the graphs considered by Ozsváth and Szabó in [21] (and in Theorem 2.4.6) are AR.

4) The class of AR graphs is closed while taking subgraphs and decreasing the Euler numbers e_{j} (since the class of rational graphs is so).

5) Not any graph is AR. For example, if Γ has two (or more) vertices, both with decoration −e_{j} ≤ δ_{j} − 2, then Γ is not AR. E.g.:

[figure: −2; −1; −13; −1; −2; −3; −3; (a); −4; −2; −2; −4; −4; −4; −4; −4; (b)]

Example (a) was already considered in [21], section 4 (cf. also with 8.4(b)). Example (b) has the following property: if we delete one of the two (−2)- vertices, then all the components of the remaining graph are rational. Still the graph itself is not AR.

8.3 Theorem. Assume that the plumbing graph Γ is AR, and let M = M(Γ) be the oriented plumbed 3-manifold associated with it. Then for any [k] ∈ Spin^{c}(M), HF_{odd}^{+}(−M, [k]) = 0, and one has an isomorphism:

HF_{even}^{+}(−M, [k]) = ℍ^{+}(Γ, [k]).

In particular,

d(M, [k]) = max_{k′∈[k]} \frac{(k′)^{2} + s}{4} = \frac{k_{r}^{2} + s}{4} − 2 min χ_{k_{r}} = \frac{K^{2} + s}{4} − 2 min_{x∈L} χ(l′_{[k]} + x);

and

χ(HF^{+}(−M, [k])) = rank_{ℤ} ℍ_{red}^{+}(Γ, [k]) = rank_{ℤ} ℍ_{red}(R_{[k]}, χ_{[k]}).

If Γ is rational or weakly elliptic then

d(M, [k]) = \frac{k_{r}^{2} + s}{4} = \frac{K^{2} + s}{4} − 2χ(l′_{[k]}).

Proof The proof is similar to the proof of theorem 2.1 of [21] (cf. 2.4.6 here), but the starting point of the inductive procedure is the characterization 6.3 of rational singularities.

For any graph Γ and vertex j ∈ J , we consider the following two modified graphs:

• Γ \ j is obtained from Γ by deleting the vertex j and all the edges adjacent to j.

• Γ_{j}^{−} is obtained from Γ by modifying the decoration e_{j} into e_{j} − 1.

Notice that, in general, Γ\j is not connected, write {(Γ\j)_{c}}_{c} for its connected components. In this case, M(Γ \ j) = #_{c}M((Γ \ j)_{c}), and all the invariants HF^{+} and ℍ^{+} are defined in this case as well. Moreover, one can verify that if Γ is negative definite, then Γ_{j}^{−} and all the components of Γ\j are also negative definite.

The proof is based on the following commutative diagram (see [21], section 2). We will write HF_{o/e}^{+} for HF_{odd/even}^{+}.

[figure: HF_{o}^{+}(−M(Γ \ j)) → HF_{e}^{+}(−M(Γ)) →^{A^{+}} HF_{e}^{+}(−M(Γ_{j}^{−})) →^{B^{+}} HF_{e}^{+}(−M(Γ \ j)) →^{C^{+}} HF_{o}^{+}(−M(Γ)); T_{Γ}^{+}; T_{Γ_{j}^{−}}^{+}; T_{Γ\j}^{+}; ℍ^{+}(Γ)  ⟶^{𝔸^{+}} ℍ^{+}(Γ_{j}^{−})  ⟶^{𝔹^{+}} ℍ^{+}(Γ \ j)]

One has the following properties:

(i) the first line is exact (cf. [21] section 2 and 10.12 [18]);

(ii) 𝔹^{+} ∘ 𝔸^{+} = 0, and 𝔸^{+} is injective (cf. 2.10 [21]).

Step 1. First, let us prove that for any rational graph Γ one has: (a) the morphism T_{Γ}^{+} : HF_{e}^{+}(−M(Γ)) → ℍ^{+}(Γ) is an isomorphism, (b) HF_{o}^{+}(−M) = 0, in particular, by 4.8 and 6.3 HF_{red}^{+}(−M) = 0.

If s denotes the number of vertices of Γ, then this property is true for s ≤ 3. Indeed, these cases are covered by [21] (cf. 2.4.6). By induction, assume that (a-b) is true for any rational graph with number of vertices < s. Then, by the general theory (see e.g. 7.2 [17]), it is true even for those non-connected graphs whose connected components are rational with total number of vertices < s.

Consider now an arbitrary rational graph Γ with s vertices. It is well-known that if G is a rational graph and j is one of its vertices, , then G_{j}^{−} is also rational. Applying this fact several times starting from Γ, we get a graph which has the property −e_{j} ≥ δ_{j} for all of its vertices. For such a graph we already know that (a-b) are true (cf. 2.4.6). Hence, going back along this sequence of graphs, we are in the following situation at each inductive step:

(iii) T_{Γ_{j}^{−}}^{+} and T_{Γ\j}^{+} are isomorphisms;

(iv) HF_{o}^{+}(−M(Γ_{j}^{−})) = HF_{o}^{+}(−M(Γ \ j)) = 0.

Then the following fact follows easily from the above diagram and (i)-(ii)-(iii):

(v) ker 𝔹^{+} = Im 𝔸^{+}.

Moreover,

(vi) B^{+} is onto.

This follows by a similar argument as the proof of 2.8 in [21]: Since both Γ and Γ_{j}^{−} are negative definite, the two cobordisms inducing A^{+} and B^{+} are both negative definite, hence the third cobordism (inducing C^{+}) is not. In particular, it induces the trivial map on HF^{∞}. But HF_{red}^{+}(−M(Γ \ j)) = 0 (by induction, since Γ \ j has < s vertices), it follows that C^{+} = 0.

This induction ends the proof of step 1.

Step 2. Assume that Γ is an AR graph with distinguished vertex j_{0}. Then all the components of Γ\j_{0} are rational, hence T_{Γ\j_{0}}^{+} is an isomorphism. Moreover, by the same argument as in (vi), we get that C^{+} = 0 again. Then one can use a similar inductive procedure as above connecting the graph Γ to a rational graph by a sequence of AR graphs, at each time decreasing e_{j_{0}} by one.

For the last formulas, see 4.8, 4.6.1(4), 5.3 and 6.6(c).

8.4 Remarks. The above proof can be pushed further to obtain the following:

(a) By the same inductive proof, one can obtain the same result as in 8.3 for a slightly larger class than AR graphs. Namely, assume that Γ has the following property: there exists a vertex j_{0} such that (i) all the components of Γ \ j_{0} are rational; and (ii) replacing e_{j_{0}} by some e′_{j_{0}} ≤ e_{j_{0}} we get an AR graph. Then the statements of 8.3 are valid. Notice that the graph 8.2(5,b) satisfies the above requirement (but it is not AR). (Nevertheless, see also 9.4(c).)

(b) Assume that Γ has the following property: there exists a vertex j_{0} such that replacing e_{j_{0}} by some e′_{j_{0}} ≤ e_{j_{0}} we get an AR graph. Then Γ\j_{0} is an AR graph, and the proof of 8.3 is still valid excepting (in step 2) the triviality of C^{+}. Hence, for such graphs we get an isomorphism T^{+} : HF_{even}^{+}(−M, [k]) → ℍ^{+}(Γ, [k]), but the vanishing of HF_{odd}^{+}(−M, [k]) does not follow. (This result and argument is the perfect analog of Theorem 2.2 of [21].)

Notice that the graph 8.2(5,a) satisfies the above requirement. Its homology HF^{+} is computed in (4.3) [21], and HF_{odd}^{+} is non-zero indeed.

9 AR graphs. The computation of ℍ(R_{k_{r}}, χ_{k_{r}}).

In this section we assume that Γ is an AR graph with a distinguished vertex 0 ∈ J . We will use the notation J^{∗} = J \ {0}. [k] will denote a fixed spin^{c}- structure with its distinguished characteristic element k_{r} = K + 2l′_{[k]}. Similarly as in section 8, in order to simplify the notations, sometimes we write k for k_{r}. For each i ≥ 0, we consider the cycles x(i), defined in 7.6, associated with the distinguished vertex 0 ∈ J and spin^{c}-structure [k].

9.1 Lemma. Assume that Γ is a AR graph and fix i ≥ 0. Then the following facts hold.

(a) Consider an arbitrary x ∈ L with m_{0}(x) = i. Then χ_{k}(x) ≥ χ_{k}(x(i)).

(b) Consider any y > x(i) with m_{0}(y) = m_{0}(x(i)) = i. Then there exists at least one b_{j} ∈ |y − x(i)| (j ∈ J^{∗}) such that χ_{k}(y − b_{j}) ≤ χ_{k}(y).

(c) Consider a computation sequence which connects x_{0} = x(i) and x_{t} = x(i + 1), cf. 7.7. Then (x_{l} + l′_{[k]}, b_{j(l)}) = 1, or equivalently, χ_{k}(x_{l+1}) = χ(x_{l}) for any 0 < l < t. In particular, χ_{k}(x(i) + b_{0}) = χ_{k}(x(i + 1)).

Proof (a) Write x = x(i) − y_{1} + y_{2} with y_{r} ≥ 0, b_{0} ∉ |y_{r}| for r = 1, 2, and (∗) |y_{1}| ∩ |y_{2}| = ∅. Then χ_{k}(x) = χ_{k}(x(i) − y_{1}) + χ(y_{2}) + (y_{1}, y_{2}) − (x(i) + l′_{[k]}, y_{2}) ≥ χ_{k}(x(i) − y_{1}). Indeed, (y_{1}, y_{2}) ≥ 0 because of (∗), and −(x(i) + l′_{[k]}, y_{2}) ≥ 0 from the definition of x(i). Finally, χ(y_{2}) ≥ 0, since the subgraph supported by |y_{2}| can be consider as the subgraph of the modified graph Γ′, which is rational, hence the subgraph supported by |y_{2}| itself is rational. On the other hand, by 7.8, χ(x(i) − y_{1}) ≥ χ(x(i)).

(b) Assume that for any b_{j} ∈ |y − x(i)| one has χ_{k}(y − b_{j}) > χ_{k}(y). This is equivalent with (b_{j}, y + l′_{[k]}) ≥ 2 + b_{j}^{2}. Since (b_{j}, x(i) + l′_{[k]}) ≤ 0, we get that (b_{j}, y − x(i)) ≥ 2 + b_{j}^{2}, or (b_{j}, y − x(i) + K) ≥ 0. Hence (∗) χ(y − x(i)) ≤ 0. By similar argument as in (a), y − x(i) is supported by a rational graph, which contradicts (∗).

(c) For any 0 ≤ l ≤ t define z_{l} := x_{l}−x(i). Then clearly z_{0} = 0, z_{1} = b_{j(0)} = b_{0}, and z_{t} = x(i + 1) − x(i). Consider (z_{l}, b_{j(l)}) = (x_{l} + l′_{[k]}, b_{j(l)}) − (x(i) + l′_{[k]}, b_{j(l)}). Since (x_{l} + l′_{[k]}, b_{j(l)}) > 0 from the construction of {x_{l}}, −(x(i) + l′_{[k]}, b_{j(l)}) ≥ 0 from the definition of x(i), we get (z_{l}, b_{j(l)}) > 0. Notice that j(l) ∈ J^{∗} (for l > 0) and b_{0} ∉ |z_{l}|, hence the values (z_{l}, b_{j(l)}) will stay unmodified even if we replace our graph Γ with the rational graph Γ′ (i.e. if we decrease e_{v_{0}} ). Since (z_{l}, b_{j(l)}) > 0, we get that (in Γ′) the sequence {x_{l}}_{0<l≤t} can be completed to a computation sequence connecting b_{j(0)} and x_{min}, associated with the canonical characteristic element K. But then 7.5 shows that (z_{l}, b_{j(l)}) = 1 for any 0 < l < t. Clearly, this is true in Γ as well. Using again the above identity we get that (x_{l} + l′_{[k]}, b_{j(l)}) = 1 for any 0 < l < t, or equivalently χ_{k}(x_{l+1}) = χ_{k}(x_{l}) (cf. the proof of 7.4).

Our goal is the identification of the graded tree (R_{k}, χ_{k}) (k = k_{r}). First we identify the local minimum points V_{1} = {v ∈ V(R_{k}) : δ_{v} = 1} of χ_{k}, cf. 3.3. Fix one v ∈ V_{1}, and set χ_{k}(v) = n_{v}. Let C_{v} be the connected component of L̄_{k,≤n_{v}} corresponding to v.

9.2 Lemma. (a) C_{v} has a unique maximal element x_{v} (with respect to ≤);

(b) x_{v} ∈ S_{[k]}; (c) Set i_{v} := m_{0}(x_{v}). Then x(i_{v}) ∈ C_{v}.

Proof Since v is a local minimum point, (∗) χ_{k}(x) = n_{v} for any x ∈ C_{v}. Assume that x, x + b_{i} and x + b_{j} are in C_{v} (with i ≠ j) . By (∗) one gets that (x + l′_{[k]}, b_{i}) = (x + l′_{[k]}, b_{j}) = 1. Then by a computation, χ_{k}(x + b_{i} + b_{j}) = χ_{k}(x) − (b_{i}, b_{j}) ≤ χ_{k}(x). By the fact that v is a local minimum, one gets χ_{k}(x + b_{i} + b_{j}) = n_{v}. In particular, x + b_{i} + b_{j} ∈ C_{v} as well. This is enough for (a). Indeed, in the presence of two maximal elements, one can connect them by a path in such a way that χ_{k} is constant along the path. Then any subsequence of type x+b_{i}, x, x+b_{j} can be replaced by x+b_{i}, x+b_{i}+b_{j}, x+b_{j}. The repeated application of this leads to a contradiction. For (b) write χ_{k}(x_{v} + b_{j}) > χ_{k}(x_{v}) for any j, which gives (x_{v} + l′_{[k]}, b_{j}) ≤ 0. (b) and the minimal property of x(i_{v}) gives x(i_{v}) ≤ x_{v}. Notice also that m_{0}(x(i_{v})) = m_{0}(x_{v}). Using inductively 9.1(b), we can construct a computation sequence connecting x(i_{v}) with x_{v} along which χ_{k} is non-decreasing. But since v is a local minimum, χ_{k} along this computation sequence should be constant. Hence all the cycles of this sequence are in C_{v}; in particular, x(i_{v}) ∈ C_{v}.

In the next theorem we recover the graded tree (R_{k}, χ_{k}) from the χ_{k} -values {χ_{k}(x(i))}_{i≥0} of the cycles {x(i)}_{i≥0}. (Recall that k = k_{r}.)

9.3 Theorem. (a) There exists an integer l such that χ_{k}(x(i+1)) ≥ χ_{k}(x(i)) for any i ≥ l. In particular, by taking τ(i) := χ_{k}(x(i)) for any i ≥ 0, the graded tree (R_{τ} , χ_{τ}) is well-defined (cf. 3.4(3)).

(b) (R_{k}, χ_{k}) = (R_{τ} , χ_{τ}).

Proof First we define a map ψ : V_{1}(R_{k}, χ_{k}) → {x(i)}_{i≥0} by v ↦ x(i_{v}), where i_{v} is determined in 9.2. We claim that ψ is injective. Indeed, assume that for v_{1} ≠ v_{2} one has i_{v_{1}} = i_{v_{2}} . 9.2(c) guarantees that χ_{k}(v_{1}) = χ_{k}(x(i_{v_{1}})) = χ_{k}(x(i_{v_{2}})) = χ_{k}(v_{2}). Let this integer be n. Consider in L̄_{k,≤n} the components C_{v_{r}} corresponding to v_{r}, r = 1, 2. By 9.2(c), x(i_{v_{r}}) ∈ C_{v_{r}}, a fact which contradicts with C_{v_{1}} ∩ C_{v_{2}} = ∅.

Assume now that there exists a sequence of integers i_{1}, i_{1} + 1, …, i_{2} such that χ_{k}(x(i_{1})) > χ_{k}(x(i_{1} + 1)) = ⋯ = χ_{k}(x(i_{2} − 1)) < χ_{k}(x(i_{2})). Set n = χ_{k}(x(i_{1} + 1)). Then in L̄_{k,≤n} there is a connected component C_{v} which contains x(i_{1}+1); let v be the corresponding vertex of R_{k}. By 9.1(a), v is a local minimum point of R_{k}. Since R_{k} has only a finitely many local minimum points, there exists l such that for i ≥ l such a sequence i_{1}, …, i_{2} cannot occur. This, together with lim_{i→∞} χ_{k}(x(i)) = ∞ proves (a).

Notice also that if v ∈ V_{1}(R_{k}, χ_{k}), then χ(v) = x(i_{v}) defines a local minimum vertex of (R_{τ} , χ_{τ}) This follows from the above discussion and from the properties of the computation sequences connecting the different x(i)’s.

Let ψ_{∗} : V_{1}(R_{k}) → V_{1}(R_{τ}) be induced by ψ. In fact, the previous discussion also show that ψ_{∗} is a bijection compatible with the gradings. In order to finish the proof, consider two different local minimum points u, v ∈ V_{1} of R_{k}. We have to verify that χ_{k}(sup(u, v)) = χ_{τ} (sup(ψ_{∗}(u), ψ_{∗}(v))). Using the above notation, consider i_{u} and i_{v} and assume i_{u} < i_{v}. Then consider one index i with i_{u} < i < i_{v} such that χ_{k}(x(i)) ≥ χ_{k}(x(j)) for any i_{u} ≤ j ≤ i_{v}. Then, in R_{τ}, n_{uv} = χ_{k}(sup(ψ_{∗}(u), ψ_{∗}(v))) is exactly χ_{k}(x(i)). We show that the same is true for χ_{k}(sup(u, v)) in R_{k}. Indeed, any path connecting x(i_{u}) and x(i_{v}) will have at least one element whose b_{0}-coefficient is exactly i (since m_{0}(u) = i_{u} < i < i_{v} = m_{0}(v)). Hence by 9.1(a), χ_{k}(sup(u, v)) ≥ χ_{k}(x(i)). But the iterated application of 7.7, and 9.1(c), provides a computation sequence connecting x(i_{u}) and x(i_{v}) along which χ_{k} is ≤ χ_{k}(x(i)); hence the equality follows.

9.4 Remarks. (a) 9.3 shows that for almost rational graphs, any graded tree (R_{k}, χ_{k}) is completely determined by the values of χ_{k} along a very natural infinite computation sequence (depending on k) which contains the elements {x(i)}_{i≥0}. This sequence can be constructed by iterating 7.7, i.e. by gluing the corresponding sequences which connect x(i) with x(i + 1) for all i ≥ 0.

In particular, all the important vertices of R_{k} can be represented by some special elements in L which can be arranged in an increasing linear order (with respect to ≤). [This property is extremely useful in the study of the weakly elliptic singularities, [9], but here we realize that it is a more general phenomenon.]

(b) The set {x(i)}_{i} sometimes is not very economical: only some of the i’s carry substantial information which will survive in (R_{τ} , χ_{τ}). E.g., for rational singularities, χ(x(i + 1)) ≥ χ(x(i)), hence only the information χ(x(0)) = 0 is preserved in R_{τ}.

(c) Part (c) of Lemma 9.1 cannot be extended for the class of graphs considered in 8.4(a). In the case of that class, one can still identify (R_{k}, χ_{k}) with some (R_{τ} , χ_{τ}), but in the definition of τ it is not enough to take only the collection of cycles x(i) along the infinite computation sequence, but one has to add some other special cycles as well.

(d) As we already mentioned, one of our interests is the numerical invariant

−sw_{M,[k]}^{OSz} = χ(HF^{+}(−M, [k]) + \frac{d(M, [k])}{2}.

In the case of AR graphs, this equals

rank_{ℤ} ℍ_{red}(R_{k_{r}}, χ_{k_{r}}) − min χ_{k_{r}} + \frac{k_{r}^{2} + s}{8}.

The term k_{r}^{2} + s (from K^{2} + s and l′_{[k]}) is clear. The next fact provides the other term.

9.5 Corollary. For any spin^{c}-structure [k] consider the representative k_{r} = K + 2l′_{[k]}. Then

rank_{ℤ} ℍ_{red}(R_{k_{r}}, χ_{k_{r}}) − min χ_{k_{r}} = ∑_{i≥0} max{0, −1 + (b_{0}, x(i) + l′_{[k]})}.

[In fact, rank_{ℤ} ℍ_{red}(R_{k′}, χ_{k′}) is independent of the choice of k′ ∈ [k].] Proof We apply 3.8. Indeed, using 9.1(c), χ_{k}(x(i))−χ_{k}(x(i+1)) = χ_{k}(x(i))− χ_{k}(x(i) + b_{0}) = −1 + (b_{0}, x(i) + l′_{[k]}).

9.6 Corollary. Consider a normal surface singularity (X, 0), and X̃ be a fixed resolution of it. Conider any holomorphic line bundle L ∈ P ic(X̃) on X̃ with c_{1}(L) = −l′_{[k]} (here L′ ≈ H^{2}(X̃, ℤ) by Poincaré dulaity). Assume that the resolution graph is AR. Then

(a) h^{1}(X̃, L) ≤ rank_{ℤ} ℍ_{red}(R_{k_{r}}, χ_{k_{r}}) − min χ_{k_{r}}.

(b) If (X, 0) is rational then both sides of this inequality (a) are zero.

(c) If (X, 0) is weakly elliptic, and [k] = [K], then rank_{ℤ} ℍ_{red}(R_{can}) = l and min χ_{can} = 0 (cf. 6.5); in particular: p_{g} ≤ l.

This answers positively (the inequality part of the main conjecture [12], modified as in 2.5(IV)) for AR resolution graphs. For Gorenstein weakly elliptic singularities, the equality p_{g} = l follows from [9]. For a generalization of the conjecture [12] and the construction of those line bundles L for which one expects equality above, see [11].

Proof We need only to prove (a). For this use the above infinite computational sequence {x_{l}}_{l} (cf. 9.4(a)). First notice that (using the theorem on formal functions, and the identification of the elements x_{l} = ∑_{j} n_{j}b_{j} ∈ L with cycles ∑_{j} n_{j}E_{j}, cf. 5.1.1), one has

H^{1}(X̃, L) = lim_{←} H^{1}(x_{l}, L|_{x_{l}}).

Next, consider the cohomology long exact sequences associated with the short exact sequences obtained from 0 → O_{b_{i(l)}}(−x_{l}) → O_{x_{l+1}} → O_{x_{l}} → 0 by tensoring with L. This provides the exact sequence

→ H^{1}(ℙ^{1}, O_{ℙ^{1}}(−(b_{i(l)}, x_{l} + l′_{[k]}))) → H^{1}(L|_{x_{l+1}}) → H^{1}(L|_{x_{l}}) → 0.

Then use 9.1(c) and 9.5.

10 Example. Lens spaces.

10.1 Notations. We invite the reader to refresh the notations of 2.1. In particular, we recall that {g_{j}}_{j∈J} denotes the dual base in L′, cf. the last paragraph of 2.1.

10.2 The plumbing graph of lens spaces. Consider the continued fraction

\frac{p}{q} = [k_{1}, k_{2}, …, k_{s}] := k_{1} − \frac{1}{k_{2} − \frac{1}{... − \frac{1}{k_{s}}}},

where k_{j} ≥ 2 for any j. Then

[figure: ⋯; −k_{1}; −k_{2}; −k_{s}; −k_{s−1}]

is the (minimal) plumbing graph Γ of the lens space M = L(p, q). M (and Γ) can be considered as the link (resp. the minimal resolution graph) of the cyclic quotient singularity ℂ^{2}/ℤ_{p}, where the action is ξ ∗ (x, y) = (ξx, ξ^{q}y). M can also be obtained as a −p/q surgery on the unknot in S^{3}.

For any 1 ≤ i ≤ j ≤ s we write the continued fraction [k_{i}, …, k_{j}] as a rational number n_{ij}/d_{ij} with n_{ij} > 0 and gcd(n_{ij}, d_{ij}) = 1. We also set n_{i,i−1} := 1 and n_{ij} := 0 for j < i − 1. Since n_{ij}/d_{ij} = k_{i} − d_{i+1,j}/n_{i+1,j}, one gets d_{ij} = n_{i+1,j} and n_{ij} = k_{i}n_{i+1,j} − n_{i+2,j} for any i ≤ j. In fact, any such identity has its symmetric version. For this it is helpful to notice that n_{ij} is the determinant of −B_{(i,j)}, where B_{(i,j)} is the bilinear form associated with the

graph [figure: ⋯; −k_{i}; −k_{j}].

This symmetry in the present case reads as: n_{ij} = k_{j}n_{i,j−1} − n_{i,j−2} for any i ≤ j. Using these, for any 1 ≤ j ≤ s one obtains by induction:

n_{1j} · n_{2s} = p · n_{2j} + n_{j+2,s} and n_{js}n_{1,s−1} = p · n_{j,s−1} + n_{1,j−2}.  (1)

Clearly n_{1s} = p and n_{2s} = q. We write q′ := n_{1,s−1}. Then by (1) qq′ ≡ 1 modulo p (and 0 < q′ < p).

10.3 The group H, the spin^{c} structures, and the elements l′_{[k]}. Clearly L′/L = H = ℤ_{p}, and [g_{s}] = g_{s} + L is one of its generators (with this choice we can use the present formulas in the next section as well). Indeed, [g_{j}] = [n_{j+1,s}g_{s}] in H (1 ≤ j ≤ s).

Similarly, the set of spin^{c}-structures on M is the set of orbits {[−ag_{s}]}_{0≤a<p} = {−ag_{s} + L}_{0≤a<p} (we prefer to use this sign, since −g_{s} is effective). More precisely, this correspondence is [k] = K + 2(−ag_{s} + L), where a runs from 0 to p − 1. In order to emphasize the role of a, we also use the notation l′_{[−ag_{s}]} for l′_{[k]}.

For any 0 ≤ a < p write

l′_{[−ag_{s}]} = −(a_{1}g_{1} + a_{2}g_{2} + ⋯ + a_{s}g_{s}).

Since (l′_{[−ag_{s}]}, b_{j}) ≤ 0, we get a_{j} ≥ 0 for any j. In the next discussion we will clarify the relationship between the integer 0 ≤ a < p (which codifies Spin^{c}(L(p, q))) and the system E(a) := (a_{1}, …, a_{s}) (the coefficients of the corresponding minimal vectors l′_{[−ag_{s}]}).

10.3.1 Lemma. For i ≤ j one has ∑_{t=i}^{j} n_{i,t−1}a_{t} < n_{ij} and ∑_{t=i}^{j} n_{t+1,j}a_{t} < n_{ij}.

Proof We verify the second set of inequalities by induction over j − i. If i = j, then consider v := l′_{[−ag_{s}]} − b_{i} ∈ l′_{[−ag_{s}]} + L. Clearly (v, b_{t}) ≤ 0 for any t ≠ i. Since l′_{[−ag_{s}]} is minimal in (l′_{[−ag_{s}]} + L) ∩ S_{ℚ}, we get that (v, b_{i}) should be positive. Hence −a_{i} + k_{i} > 0.

Next, for the case j = i+1, notice that by the inductive step, l := (k_{i}−a_{i})b_{i+1}+ b_{i} > 0. Then consider v := l′_{[−ag_{s}]} − l ∈ l′_{[−ag_{s}]} + L. Then for any t ≠ j, the value (v, b_{t}) ≤ 0 automatically. Again by the minimality of l′_{[−ag_{s}]}, we get that (v, b_{j}) > 0, or n_{ij} − n_{jj}a_{i} − a_{j} > 0. The induction follows similarly.

In particular, the entries of (a_{1}, …, a_{s}) satisfy the system of inequalities:

(SI) \{ \begin{matrix}a_{1} ≥ 0, …, a_{s} ≥ 0 \\ n_{i+1,s}a_{i} + n_{i+2,s}a_{i+1} + ⋯ + n_{ss}a_{s−1} + a_{s} < n_{is} for any 1 ≤ i ≤ s. \end{matrix}

By this system one can identify the possible spin^{c}-structures with the possible combinations (a_{1}, …, a_{s}). For this, first notice that the integer a can be recovered from (a_{1}, …, a_{s}) by

a = n_{2s}a_{1} + n_{3s}a_{2} + ⋯ + n_{ss}a_{s−1} + a_{s}.  (1)

Indeed, modulo L, one has the following identities:

ag_{s} ≡ −l′_{[−ag_{s}]} ≡ ∑_{j=1}^{s} a_{j}g_{j} ≡ (∑_{j=1}^{s} a_{j}n_{j+1,s})g_{s}.

Since ∑ a_{j}n_{j+1,s} < p (from (SI), i = 1), the above identity follows.

Next, any 0 ≤ a < p determines inductively the entries a_{1}, ⋯, a_{s} by the formula

a_{i} = [\frac{a − ∑_{t=1}^{i−1} n_{t+1,s}a_{t}}{n_{i+1,s}} ] (1 ≤ i ≤ s).

Geometry & Topology, Volume X (20XX)

In fact, this also shows that the set of integral solutions of the system (SI) is exactly the set of all possible combinations (a_{1}, …, a_{s}) associated with the integers 0 ≤ a < p by the above procedure.

10.3.2 As a curiosity, we mention that the above system (SI) can also be interpreted in language of “generalized” continued fractions. For any 1 ≤ i ≤ s write r_{i} := n_{is}/n_{i+1,s}. Then

\frac{a}{p} = \frac{a_{1} + \frac{a_{2} + \frac{⋯ + \frac{a_{s−1} + \frac{a_{s}}{r_{s}}}{⋯}}{r_{3}}}{r_{2}}}{r_{1}}.

The inequalities (SI) imply that all the possible fractions in the above expression are < 1; and this property guarantees the uniqness of the entries (a_{1}, …, a_{s}) in this continued fraction (for any given 0 ≤ a < p).

10.3.3 In fact,

l′_{[−(p−1)g_{s}]} = g_{s} + ∑_{i=1}^{s} b_{i} = −( (k_{1} − 1)g_{1} + ∑_{i=2}^{s}(k_{i} − 2)g_{i}).  (1)

This follows from the fact that −(p − 1)g_{s} ≡ g_{s} modulo L, and g_{s} + ∑ b_{j} is in the unit cube ∑ r_{j}b_{j}, 0 < r_{j} < 1 for any 1 ≤ j ≤ s. [Notice that in general it is not true that l′_{[−ag_{s}]} is in the unit cube.] (1) can be generalized by the following algorithm and identity (the verification is left to the reader).

The set of all possible combinations E(a) = (a_{1}, …, a_{s}), 0 ≤ a < p, in the order E(p − 1), …, E(0), can be generated inductively as follows. Start with E(p − 1) = (k_{1} − 1, k_{2} − 2, …, k_{s} − 2). Assume that E(a) = (a_{1}, …, a_{s}) is already determined. Then, if a_{s} > 0, then E(a − 1) = (a_{1}, …, a_{s−1}, a_{s} − 1). If a_{i} = ⋯ = a_{s} = 0, but a_{i−1} ≠ 0, then

E(a − 1) = (a_{1}, …, a_{i−2}, a_{i−1} − 1, k_{i} − 1, k_{i+1} − 2, …, k_{s} − 2).

In particular

l′_{[−(a−1)g_{s}]} − l′_{[−ag_{s}]} = g_{s} + ∑_{t=i}^{s} b_{t}.

Before we start the list of invariants, we prove the following identities for E(a) = (a_{1}, …, a_{s}). In the sequel {x} := x − [x] denotes the fractional part of the real number x.

10.4 Lemma. [aq′/p] = ∑_{t=1}^{s} a_{t}n_{t+1,s−1} and {aq′/p} = (∑_{t=1}^{s} a_{t}n_{1,t−1})/p.

Proof Via 10.3.1(1), we have to show that

∑_{t=1}^{s} a_{t}n_{t+1,s−1} ≤ \frac{n_{1,s−1}}{p} ∑_{t=1}^{s} a_{t}n_{t+1,s} < 1 + ∑_{t=1}^{s} a_{t}n_{t+1,s−1}.

This follows from the second part of 10.2(1) and the first inequality of 10.3.1 (applied for (i, j) = (1, s)). The second part follows from the expression of [aq′/p] and the second part of 10.2(1).

10.5 The Ozsváth-Szabó invariant HF^{+}(±M). Since Γ is rational, by 6.3 and 8.3, HF_{red}^{+}(±M) = 0 and HF^{+}(±M, [k]) = T_{±d}^{+}, where

d := d(M, [k]) = \frac{k_{r}^{2} + s}{4} = \frac{K^{2} + s}{4} − 2χ(l′_{[k]}).

Notice that (see e.g. (7.1) of [12]):

\frac{K^{2} + s}{4} = \frac{p − 1}{2p} − 3 · s(q, p),

where s(q, p) denotes the Dedekind sum

s(q, p) = ∑_{l=0}^{p−1} ((\frac{l}{p}))((\frac{ql}{p})), where ((x)) = \{ \begin{matrix}\{x\} − 1/2 & if & x ∈ ℝ \ ℤ \\ 0 & if & x ∈ ℤ. \end{matrix}

(In fact, this formula for K^{2} + s for cyclic quotients goes back to the work of Hirzebruch.)

Next, we wish to determine χ(l′_{[−ag_{s}]}).

10.5.1 Proposition. For any 0 ≤ a < p one has:

χ(l′_{[−ag_{s}]}) = \frac{a(1 − p)}{2p} + ∑_{j=1}^{a} \{\frac{jq′}{p} \}.

Proof Using the notations of 10.3.3 (in particular a_{t} = 0 for t ≥ i), we have

χ(l′_{[−(a−1)g_{s}]}) = χ(l′_{[−ag_{s}]} + g_{s} + ∑_{t=i}^{s} b_{t}) =

χ(l′_{[−ag_{s}]}) + χ(g_{s} + ∑_{t=i}^{s} b_{t}) − (l′_{[−ag_{s}]}, g_{s} + ∑_{t=i}^{s} b_{t}).

Since l′_{[−ag_{s}]} = − ∑_{t=1}^{i−1} a_{t}g_{t} and −pg_{s} = ∑_{j} n_{1,j−1}b_{j}, and from 10.4, one gets

−(l′_{[−ag_{s}]}, g_{s} + ∑_{t=i}^{s} b_{t}) = ∑_{t} a_{t}(g_{t}, g_{s}) = − ∑_{t} a_{t}n_{1,t−1}/p = −{aq′/p}.

Moreover, χ(g_{s}+∑_{t=i}^{s} b_{t}) = χ(g_{s})+χ(∑_{t=i}^{s} b_{t})−1 = χ(g_{s}) by an easy induction. Notice that the b_{s}-coefficient of K (i.e. (K, g_{s})) is −1 + (q′ + 1)/p (cf. 5.2 [12]), and g_{s}^{2} = −q′/p, hence χ(g_{s}) = (p − 1)/(2p). Hence the formula follows by induction over a (since χ(l′_{[0]}) = 0).

10.5.2 Clearly, with the choice g_{1} as a generator, by a similar proof one has

χ(l′_{[−ag_{1}]}) = \frac{a(1 − p)}{2p} + ∑_{j=1}^{a} \{\frac{jq}{p} \}.

10.6 The Casson-Walker invariant is λ(L(p, q)) = p · s(q, p)/2 (see e.g. [12](5.3)).

10.7 The Reidemeister-Turaev torsion. One can use for the computation of the sign-refined Reidemeister-Turaev torsion the formula [12] (5.7) (cf. also with 3.5 [loc.cit.]). In that article, the canonical spin^{c}-structure σ_{can} was defined via c_{1}(σ_{can}) = −K, and an arbitrary spin^{c} structure via the action σ = h_{σ} · σ_{can} for some h_{σ} ∈ H. This reads as −c_{1}(σ) = −c_{1}(σ_{can}) − 2h_{σ}, which in our case corresponds to k_{r} = K + 2l′_{[k]}. In particular, for [−ag_{s}], h_{σ} is the class of ag_{s} (see also 5.1).

Then we apply [12] (5.7). Here we identify any character χ of H by the root of unity ξ := χ([g_{s}]). Then we get:

𝒯_{M,[−ag_{s}]}(1) = \frac{1}{p} ∑ \frac{ξ^{−a}}{(ξ − 1)(ξ^{q} − 1)},

where the sum is over all p-roots of unity ξ with ξ ≠ 1. If a = 0 then

𝒯_{M,[0]}(1) = \frac{1}{p} ∑ \frac{1}{(ξ^{q} − 1)(ξ − 1)} = \frac{p − 1}{4p} − s(q, p).

For a > 0, we use the same inductive step as above, namely

𝒯_{M,[−(a−1)g_{s}]}(1) − 𝒯_{M,[−ag_{s}]}(1) = \frac{1}{p} ∑ \frac{ξ^{−a}}{ξ^{q} − 1} = \frac{1}{p} ∑ \frac{ξ^{−aq′}}{ξ − 1} = \{\frac{aq′}{p} \} + \frac{1 − p}{2p}.

Adding all these identities, and using 10.5.1, one gets

𝒯_{M,[−ag_{s}]}(1) = \frac{p − 1}{4p} − s(q, p) − χ(l′_{[−ag_{s}]}).

10.8 The identity sw_{M,[k]}^{OSz} = sw_{M,[k]}^{T CW}. Using the above formulas, one gets

𝒯_{M,[k]}(1) − \frac{λ(M)}{|H|} = \frac{d(M, [k])}{2},

which proves the identity sw_{M,[k]}^{OSz} = sw_{M,[k]}^{T CW} mentioned in 2.5.

Since ∑_{[k]} 𝒯_{M,[k]}(1) = 0, one also gets

−λ(M) = ∑_{[k]∈Spin^{c}(M)} \frac{k_{r}^{2} + s}{8},

or, equivalently:

∑_{[k]} χ(l′_{[k]}) = \frac{p − 1}{4} − p · s(q, p).

11 Example. Seifert 3-manifolds

11.1 In this section we will use mainly the notations of [13] (which also provides a list of classical references for Seifert 3–manifolds).

Assume that M is a Seifert rational homology sphere with negative orbifold Euler number e (equivalently, the corresponding plumbing graph is negative definite). The minimal plumbing graph Γ is star-shaped with ν arms; we assume ν ≥ 3. [The case ν < 3 is completely covered by the previous section.] Γ can be determined from the (normalized) Seifert invariants {(α_{l}, ω_{l})}_{l=1}^{ν} and e as follows. Here 0 ≤ ω_{l} < α_{l} for any 1 ≤ l ≤ ν, and e = e_{0} + ∑_{l} ω_{l}/α_{l}, where e_{0} is the decoration of the central vertex. The other decorations can be determined from the continued fraction α_{l}/ω_{l} = [k_{1}^{l}, k_{2}^{l}, ⋯, k_{s_{l}}^{l}]; the vertices {v_{j}^{l}}_{j=1}^{s_{l}} situated on the l^{th}–arm are decorated by e_{j}^{l} = −k_{j}^{l} in such a way that the vertex v_{1}^{l} is connected with the central vertex.

We denote by b_{0} and b_{j}^{l} (1 ≤ j ≤ s_{l}; 1 ≤ l ≤ ν) the base elements of L, respectively by g_{0} and {g_{j}^{l}}_{jl} the dual base elements of L′ (cf. 2.1). As we already mentioned (cf. 8.2(3)), such a graph Γ is AR with its central vertex distinguished.

We will also use the following notations: ε := (2 − ν + ∑_{l} 1/α_{l})/e denotes the “exponent of M ” (in some articles −ε is the “log discrepancy of the central vertex”). For any l we consider 0 < ω′_{l} < α_{l} with ω_{l}ω′_{l} ≡ 1 modulo α_{l}.

The following identities are well-known (they can be deduced e.g. from [12], section 5).

(g_{0}, g_{0}) = \frac{1}{e}; (g_{0}, g_{s_{l}}^{l}) = \frac{1}{eα_{l}}; (g_{s_{t}}^{t}, g_{s_{l}}^{l}) = \frac{1}{eα_{t}α_{l}} (t ≠ l); (g_{s_{l}}^{l}, g_{s_{l}}^{l}) = \frac{1}{eα_{l}^{2}} − \frac{ω′_{l}}{α_{l}} .

−(K, g_{0}) = 1 + ε; −(K, g_{s_{l}}^{l}) = 1 + \frac{ε − ω′_{l}}{α_{l}} .

χ(g_{0}) = \frac{1}{2} + \frac{ε}{2} − \frac{1}{2e}; χ(g_{s_{l}}^{l}) = \frac{1}{2} + \frac{ε}{2α_{l}} − \frac{1}{2eα_{l}^{2}} .

11.2 Notations. For any fixed 1 ≤ l ≤ ν, we consider similar notations as in the previous section. For any 1 ≤ i ≤ j ≤ s_{l}, we write n_{ij}^{l}/d_{ij}^{l} := [k_{i}^{l}, …, k_{j}^{l}] (with n_{ij}^{l} > 0 and gcd(n_{ij}^{l}, d_{ij}^{l}) = 1); and we set n_{i,i−1}^{l} := 1 and n_{ij}^{l} := 0 for j < i − 1. Clearly, they satisfy the identities considered in 10.2. Moreover, α_{l} = n_{1,s_{l}}^{l}, ω_{l} = n_{2,s_{l}}^{l}, and ω′_{l} = n_{1,s_{l}−1}^{l}.

11.3 The group H. In the abelian group H = L′/L one has [g_{j}^{l}] = [n_{j+1,s_{l}}^{l}g_{s_{l}}^{l}]. In fact, one has the following (Abelian) presentation (cf. [15]):

H = ⟨g_{0}, g_{s_{1}}^{1}, g_{s_{2}}^{2}, … g_{s_{ν}}^{ν} | e_{0}g_{0} + ∑_{l=1}^{ν} ω_{l}g_{s_{l}}^{l} = 0, g_{0} = α_{l}g_{s_{l}}^{l} for all l⟩.

The order of H is −eα_{1} ⋯ α_{ν}, and the order o of the subgroup ⟨g_{0}⟩ is o = −eα, where α := lcm(α_{1}, …, α_{ν}) (cf. [15]).

11.4 Spin^{c}-structures and representatives l′_{[k]}. We fix a spin^{c}-structure [k] = K + 2(l′_{[k]} + L), or equivalently, the orbit l′_{[k]} + L in L′/L. We assume that the distinguished representative l′_{[k]} ∈ L′ has the following form

l′_{[k]} = −a_{0}g_{0} − ∑_{j,l} a_{j}^{l}g_{j}^{l}.  (1)

For any fixed 1 ≤ l ≤ ν, we associate with the collection (a_{1}^{l}, …, a_{s_{l}}^{l}) the integer (cf. 10.3):

a_{l} := ∑_{t≥1} n_{t+1,s_{l}}^{l}a_{t}^{l}.  (2)

The integers {a_{0}; a_{j}^{l}}_{lj}, respectively {a_{0}; a_{l}}_{l} satisfy the following inequalities:

11.5 Proposition. Consider l′_{[k]} and the notations (1) and (2) above. Then

(SI)  \{ \begin{matrix}a_{0} ≥ 0; , a_{j}^{l} ≥ 0 & (1 ≤ l ≤ ν, 1 ≤ j ≤ s_{l}); \\ ã_{j}^{l} := ∑_{t≥j} n_{t+1,s_{l}}^{l}a_{t}^{l} < n_{js_{l}}^{l} & (1 ≤ l ≤ ν, 1 ≤ j ≤ s_{l}); \\ 1 + a_{0} + ie_{0} + ∑_{l} [\frac{iω_{l}+a_{l}}{α_{l}} ] ≤ 0 & for any i > 0. \end{matrix}

In particular, the set of integers (a_{0}, a_{1}, …, a_{ν}) satisfy

(SI_{red}) \{ \begin{matrix}a_{0} ≥ 0; , α_{l} > a_{l} ≥ 0 & (1 ≤ l ≤ ν); \\ 1 + a_{0} + ie_{0} + ∑_{l} [\frac{iω_{l}+a_{l}}{α_{l}} ] ≤ 0 & for any i > 0. \end{matrix}

Moreover, there is a one-to-one correspondence between the integer solutions of (SI), respectively of (SI_{red}), provided by the correspondence (2) above (cf. also with 10.3).

Proof The first two set of inequalities follow from (SI) 10.3. For the last one, we proceed as follows. For any i > 0 we define y(i) := ib_{0} + ∑_{l,j} u_{j}^{l}b_{j}^{l} ∈ L, where

u_{1}^{l} := [\frac{in_{2s_{l}}^{l} + ã_{1}^{l}}{n_{1s_{l}}^{l}} ], u_{2}^{l} := [\frac{u_{1}^{l}n_{3s_{l}}^{l} + ã_{2}^{l}}{n_{2s_{l}}^{l}} ], ⋯, u_{s_{l}}^{l} := [\frac{u_{s_{l}−1}^{l} + ã_{s_{l}}^{l}}{n_{s_{l}s_{l}}^{l}} ].

Consider v := l′_{[k]} − y(i). The point is that (v, b_{j}^{l}) ≤ 0 for any 1 ≤ l ≤ ν and 1 ≤ j ≤ s_{l}. Indeed, if j < s_{l}, from the definition of u_{j}^{l} one has u_{j}^{l}n_{j,s_{l}}^{l} ≤ u_{j−1}^{l}n_{j+1,s_{l}}^{l} + ã_{j}^{l}, or, using 10.2 (namely n_{j,s_{l}}^{l} = k_{j}n_{j+1,s_{l}}^{l} − n_{j+2,s_{l}}^{l}) and ã_{j}^{l} = ã_{j+1}^{l} + n_{j+1,s_{l}}^{l}a_{j}^{l}:

−a_{j}^{l} − u_{j−1}^{l} + k_{j}u_{j}^{l} ≤ \frac{u_{j}^{l}n_{j+2,s_{l}}^{l} + ã_{j+1}^{l}}{n_{j+1,s_{l}}^{l}} ,

hence −a_{j}^{l} −u_{j−1}^{l} +k_{j}u_{j}^{l} ≤ the integral part of the right hand side, which equals u_{j+1}^{l}. In other words, (v, b_{j}^{l}) = −a_{j}^{l} − u_{j−1}^{l} + k_{j}u_{j}^{l} − u_{j+1}^{l} ≤ 0. (The case j = s_{l} is easy.)

Since v = l′_{[k]} − y(i) ∈ l′_{[k]} + L and y(i) > 0, by the minimality of l′_{[k]} in (l′_{[k]} + L) ∩ S_{ℚ} we deduce that v ∉ S_{ℚ}. Since (v, b_{j}^{l}) ≤ 0 for any l and j, we get that (v, b_{0}) > 0. This is equivalent to −a_{0} − ie_{0} − ∑_{l} u_{1}^{l} > 0, hence the third inequality of (SI) follows.

For (SI_{red}), notice that a_{l} = ã_{1}^{l} < n_{1s_{l}}^{l} = α_{l} from (SI). The correspondence between the integral solutions of the two system of inequalities is clarified in 10.3; namely, is given by E(a_{l}) = (a_{1}^{l}, …, a_{s_{l}}^{l}).

The point is that the set of solutions of (SI) (or, equivalently, of (SI_{red})) corresponds exactly to the set of all possible spin^{c}-structures. In other words, if {a_{0}; a_{j}^{l}}_{jl} satisfies (SI_{red}), then it is the set of coefficients of some l′_{[k]}. This follows from the next fact.

11.6 Proposition. For any c_{0} ≥ 0 and α_{l} > c_{l} ≥ 0 consider E(c_{l}) = (c_{1}^{l}, …, c_{s_{l}}^{l}) as in 10.3. This defines l′ := −c_{0}g_{0} − ∑_{l,j} c_{j}^{l}g_{j}^{l} ∈ S_{ℚ}. If l′ is not minimal in (l′ + L) ∩ S_{ℚ} then for some i > 0 one has

1 + c_{0} + ie_{0} + ∑_{l} [\frac{iω_{l} + c_{l}}{α_{l}} ] > 0.

Proof Define the integers c̃_{j}^{l}, u_{j}^{l} and the cycle y(i) (for any i > 0) in a similar way as in (SI) and in previous proof, by replacing a_{j}^{l} by c_{j}^{l}. Then the argument of the previous proof repeated gives that y(i) satisfies (a) m_{0}(y(i)) = i, and (b) (l′ − y(i), b_{j}^{l}) ≤ 0 for any l and j. The point is that for any fixed i > 0, y(i) is the unique maximal effective cycle which satisfy both (a) and (b). This follows from the very definition of the integers u_{j}^{l}, its proof is similar to the proof of 11.11 and it is left to the reader.

Now, assume that l′ = l′_{[k]} + x for some effective non-zero cycle x ∈ L. Set i := m_{0}(x). Since x satisfy both (a) and (b) for this i, from the maximality of y(i) one gets y(i) ≥ x. Consider now l′ − y(i) = l′_{[k]} + x − y(i). Since (l′ − y(i), b_{j}^{l}) ≤ 0 one gets that (l′_{[k]} + x − y(i), b_{j}^{l}) ≤ 0 as well. Moreover, (l′_{[k]} + x − y(i), b_{0}) ≤ (l′_{[k]}, b_{0}) ≤ 0 (the first inequality from b_{0} ∉ |y(i) − x| and x − y(i) ≤ 0, the second from l′_{[k]} ∈ S_{ℚ}). Hence l′_{[k]} + x − y(i) ∈ S_{ℚ} with x − y(i) ≤ 0. From the minimality of l′_{[k]} one gets x = y(i). Hence (b_{0}, l′ − y(i)) = (b_{0}, l′_{[k]}) ≤ 0, or −c_{0} − ie_{0} − ∑_{l}[iω_{l} + c_{l}/α_{l}] ≤ 0.

11.7 Corollary. The following sets are in one-to-one correspondence:

(1) Spin^{c}(M), or equivalently, the set of distinguished representatives k_{r} = K + 2l′_{[k]};

(2) the set of integral solutions {a_{0}; a_{j}^{l}}_{lj} of (SI);

(3) the set of integral solutions {a_{0}; a_{1}, …, a_{ν}} of (SI_{red}).

The set {a_{0}; a_{j}^{l}}_{lj} is the coefficient set of l′_{[k]}, {a_{0}, …, a_{ν}} is obtained from {a_{0}; a_{j}^{l}}_{lj} by (2) 11.4. Notice also that in L′ one has

−l′_{[k]} ≡ a_{0}g_{0} + ∑_{l,j} a_{j}^{l}g_{j}^{l} ≡ a_{0}g_{0} + ∑_{l} a_{l}g_{s_{l}}^{l} (modulo L).

In particular, the cardinality of all these sets is the same |H| = |e|α_{1} ⋯ α_{ν}. The author knows no direct proof of this fact (for (2) or (3)).

11.8 The values χ(l′_{[k]}). Assume that [k] corresponds (by the correspondence 11.7) to the set {a_{0}; a_{1}, …, a_{ν}}. Assume that a_{l} > 0 for some fixed 1 ≤ l ≤ ν. Let [k_{l}^{−}] be the spin^{c}-structure corresponding to {a_{0}; a_{1}, …, a_{l−1}, a_{l} − 1, a_{l+1}, …, a_{ν}} (clearly, this set of integers also satisfies (SI_{red})). Then, by a similar method as in (the proof of) 10.5.1, we get

χ(l′_{[k_{l}^{−}]}) = χ(l′_{[k]}) + χ(g_{s_{l}}^{l}) + \frac{a_{0}}{eα_{l}} + ∑_{t≠l} \frac{a_{t}}{eα_{t}α_{l}} + ∑_{t≥1} a_{t}^{l}(g_{t}^{l}, g_{s_{l}}^{l}).

By a calculation (using 10.4 and 10.5.1):

∑_{t≥1} a_{t}^{l}(g_{t}^{l}, g_{s_{l}}^{l}) = ∑_{t≥1} a_{t}^{l}(\frac{n_{t+1,s_{l}}^{l}}{eα_{l}^{2}} − \frac{n_{1,t−1}^{l}}{α_{l}} ) = \frac{a_{l}}{eα_{l}^{2}} − \{\frac{a_{l}ω′_{l}}{α_{l}} \}.

Therefore,

χ(l′_{[k_{l}^{−}]}) − χ(l′_{[k]}) = χ(g_{s_{l}}^{l}) + \frac{1}{eα_{l}} (a_{0} + ∑_{t=1}^{ν} \frac{a_{t}}{α_{t}} ) − \{\frac{a_{l}ω′_{l}}{α_{l}} \}.

Hence, by induction (by decreasing the coefficients (a_{1}, …, a_{ν}) to (0, …, 0)), χ(l′_{[k]}) can be computed in terms of χ(l′_{[−a_{0}g_{0}]}). But, using the identities 11.1, one has:

−χ(l′_{[−a_{0}g_{0}]}) = \frac{1}{2}( −a_{0}(K, g_{0}) + a_{0}^{2}g_{0}^{2} ) = \frac{a_{0}^{2}}{2e} + \frac{a_{0}}{2} (1 + ε).

Summing up these identities, and using ã := a_{0} + ∑_{l} a_{l}/α_{l}, one gets

−χ(l′_{[k]}) = ∑_{l=0}^{ν} \frac{a_{l}}{2} + \frac{εã}{2} + \frac{ã^{2}}{2e} − ∑_{l=1}^{ν} ∑_{i=1}^{a_{l}} \{\frac{iω′_{l}}{α_{l}} \}.

For another expression see 11.15 (r∗).

11.9 The invariant k_{r}^{2} + s. Recall that k_{r}^{2} + s = K^{2} + s − 8χ(l′_{[k]}). χ(l′_{[k]}) is computed above in 11.8, while K^{2} + s in [12] (5.5):

K^{2} + s = ε^{2}e + e + 5 − 12 ∑_{l=1}^{ν} s(ω_{l}, α_{l}).

11.10 The cycles x(i) (i ≥ 0). We fix a spin^{c}-structure [k] which corresponds to the set {a_{0}; a_{1}, …, a_{ν}} by the correspondence 11.7. Recall that the graded root (R_{k_{r}}, χ_{k_{r}}), in particular the ℤ[U]-module ℍ(R_{k_{r}}, χ_{k_{r}}), can be completely recovered from the cycles x(i) (i ≥ 0) (cf. sections 7 and 9). The next result determines these cycles in terms of the Seifert invariants of M and the integers {a_{0}, …, a_{ν}}.

11.11 Proposition. For any l consider E(a_{l}) = (a_{1}^{l}, …, a_{s_{l}}^{l}) (cf. 11.5 and 10.3), and define ã_{j}^{l} := ∑_{t≥j} n_{t+1,s_{l}}^{l}a_{t}^{l} (1 ≤ j ≤ s_{l}) (cf. (SI) in 11.5).

(1) Define the integers {v_{j}^{l}} (1 ≤ l ≤ ν, 1 ≤ j ≤ s_{l}) inductively by

v_{1}^{l} := ⌈\frac{iω_{l} − a_{l}}{α_{l}} ⌉ = ⌈\frac{in_{2,s_{l}}^{l} − ã_{1}^{l}}{n_{1,s_{l}}^{l}} ⌉; v_{j}^{l} := ⌈\frac{v_{j−1}^{l}n_{j+1,s_{l}}^{l} − ã_{j}^{l}}{n_{j,s_{l}}^{l}} ⌉ (1 < j ≤ s_{l}).

Set z(i) := ib_{0} + ∑_{l,j} v_{j}^{l}b_{j}^{l}. Then (l′_{[k]} + z(i), b_{j}^{l}) ≤ 0 for any l and j.

(2) Assume that an effective cycle z(i) := ib_{0} + ∑_{l,j} v_{j}^{l}b_{j}^{l} (where {v_{j}^{l}}_{lj} are some integers) satisfies (l′_{[k]} + z(i), b_{j}^{l}) ≤ 0 for any l and j. Then

v_{1}^{l} ≥ ⌈\frac{iω_{l} − a_{l}}{α_{l}} ⌉ and v_{j}^{l} ≥ ⌈\frac{v_{j−1}^{l}n_{j+1,s_{l}}^{l} − ã_{j}^{l}}{n_{j,s_{l}}^{l}} ⌉ (1 < j ≤ s_{l}).

(3) In particular, z(i) defined in (1) equals x(i), the minimal effective cycle which satisfies (l′_{[k]} + x(i), b_{j}^{l}) ≤ 0 for any l and j.

Proof The proof of (1) has the same spirit as the proof of 11.5. From the definition of v_{j}^{l} one gets n_{j,s_{l}}^{l}v_{j}^{l} ≥ v_{j−1}^{l}n_{j+1,s_{l}}^{l} − ã_{j}^{l}, which is equivalent with

k_{j}^{l}v_{j}^{l} − v_{j−1}^{l} + a_{j}^{l} ≥ \frac{v_{j}^{l}n_{j+2,s_{l}}^{l} − ã_{j+1}^{l}}{n_{j+1,s_{l}}^{l}} .

Now, using the definition of v_{j+1}^{l} for the right hand side, one gets −a_{j}^{l} + v_{j−1}^{l} − k_{j}^{l}v_{j}^{l} + v_{j+1}^{l} ≤ 0. (Here, if j = 1 then v_{j−1}^{l} := i.) For (2), notice that (l′_{[k]} + x(i), b_{s_{l}}^{l}) ≤ 0 is equivalent with the wanted inequality for j = s_{l}. In general, for arbitrary j, the inequality follows from (l′_{[k]} + x(i), v_{j}) ≤ 0, where v_{j} = ∑_{t≥j} n_{t+1,s_{l}}^{l}b_{t}^{l}.

11.12 The values τ(i) (i ≥ 0). Recall that τ(i) := χ_{k_{r}}(x(i)) (i ≥ 0). If i = 0 then x(0) = 0, hence τ(0) = 0. From 9.1(c) and 11.11, one gets that for i ≥ 0

τ(i + 1) − τ(i) = 1 − (l′_{[k]} + x(i), b_{0}) = 1 + a_{0} − ie_{0} + ∑_{l} [\frac{−iω_{l} + a_{l}}{α_{l}} ].

In particular, for i = 0 one has τ(1) − τ(0) = 1 + a_{0} ≥ 1. In general,

τ(i) = ∑_{t=0}^{i−1} (1 + a_{0} − te_{0} + ∑_{l} [\frac{−tω_{l} + a_{l}}{α_{l}} ] ).

11.13 The Ozsváth-Szabó invariant. Using the general results 4.8, 8.3 and 9.3 and

min χ_{k_{r}} = min_{i≥0} τ(i),

one gets

HF_{even}^{+}(−M, [k]) = ℍ^{+}(Γ, [k]) = ℍ(R_{τ} , χ_{τ})[−\frac{k_{r}^{2} + s}{4}].

Clearly, from its very definition, the graded root (R_{τ} , χ_{τ}) is completely determined from the values {τ(i)}_{i≥0}. Moreover, the numerical invariants can be determined as follows:

±d(±M, [k]) = \frac{k_{r}^{2} + s}{4} − 2 min_{i≥0} τ(i),

and (using 3.8 as well):

χ(HF^{+}(−M, [k])) = min_{i≥0} τ(i) + ∑_{i≥0} max \{0, −1 − a_{0} + ie_{0} − ∑_{l=1}^{ν} [\frac{−iω_{l} + a_{l}}{α_{l}} ] \}.

(Since e = e_{0} + ∑ ω_{l}/α_{l} < 0, this sum is finite.)

11.14 Remark. Assume that [k] = [K], i.e. l′_{[k]} = 0 (or, equivalently, a_{0} = ⋯ = a_{ν} = 0). Then the above sum becomes (the Dolgachev-Pinkham) invariant

DP := ∑_{i≥0} max \{0, −1 + ie_{0} − ∑_{l} [\frac{−iω_{l}}{α_{l}} ]\},

and the above identity

χ(HF^{+}(−M, can)) − min χ_{can} = DP.

This is interesting for two reasons.

(1) If M is the link of a normal surface singularity with geometric genus p_{g}, then 9.6 reads as p_{g} ≤ DP. In fact, if M is a rational homology sphere, and it is the link of a singularity which admits a good ℂ^{∗}-action (a fact which automatically implies that M is a Seifert 3–manifold), then by results of Dolgachev and Pinkham, p_{g} = DP (see e.g. [22]). It is remarkable, that in that context, the expression DP is deduced by a rather different argument (sitting in algebraic geometry).

This inequality (and identity for singularities with good ℂ^{∗}-action) is compatible with the conjecture [12], see also [13] for this special situation.

(2) The identity sw_{M,can}^{OSz} = sw_{M,can}^{T CW} in this case [k] = [K] is equivalent to

T_{M,can}(1) − \frac{λ(M)}{|H|} = \frac{K^{2} + s}{8} + DP.

This identity was verified in [13]. In the sequel we verify the corresponding identity valid for an arbitrary spin^{c}-structure.

11.15 The identity sw_{M,[k]}^{OSz} = sw_{M,[k]}^{T CW}. Using the above facts, this identity is equivalent to

𝒯_{M,[k]}(1) − \frac{λ(M)}{|H|} = χ(HF^{+}(−M, [k])) − min χ_{k_{r}} + \frac{k_{r}^{2} + s}{8}.  (∗)

We fix a spin^{c}-structure [k] which corresponds to {a_{0}; a_{1}, …, a_{ν}} as above (cf. 11.7). For the completness of the description, we also need (cf. [13]):

P̂_{1}(t) := \frac{(t^{α} − 1)^{ν−2}}{∏_{l}(t^{α/α_{l}} − 1)}.

Then by [13] (see also 5.1 and 10.7), and with the notation ã = a_{0} + ∑_{l} a_{l}/α_{l} one has

𝒯_{M,[k]}(1) = lim_{t→1} ( ∑_{i≥ã/e} max \{0, 1+a_{0}−ie_{0}+∑_{l} [\frac{−iω_{l} + a_{l}}{α_{l}} ]\}t^{oi+αã}− \frac{1}{|H|}P̂_{1}(t) ).

Using the inequalities (SI) (cf. 11.5) one deduce that the contribution corresponding to the indices i < 0 is zero. Then using max{0, x}−max{0, −x} = x, we get that

𝒯_{M,[k]}(1) − χ(HF^{+}(−M, [k])) + min χ_{k_{r}}

equals the limit

L := lim_{t→1} (P_{[k]}(t) − \frac{1}{|H|}P̂_{1}(t) ),

where

P_{[k]}(t) := ∑_{i≥0} (1 + a_{0} − ie_{0} + ∑_{l} [\frac{−iω_{l} + a_{l}}{α_{l}} ])t^{oi+αã}.

In particular, the identity (∗) is equivalent to λ(M)/|H|+(k_{r}^{2} +s)/8 = L. This identity was verified in [13] for the canonical spin^{c}-structure (cf. comment 5.1).

Therefore, it is enough to check a relative identity (the difference between the two identities corresponding to an arbitrary [k] and the canonical [K]). Since k_{r}^{2} + s = K^{2} + s − 8χ(l′_{[k]}), this relative identity is

−χ(l′_{[k]}) = lim_{t→1} (P_{[k]}(t) − P_{[K]}(t)).  (r∗)

Notice that once this identity is proved, it also provides a new formula for χ(l′_{[k]}). In order to verify (r∗) we consider the Laurent expansions of P_{[k]}(t) with respect to t^{o} − 1. They have a pole of order two, and the above limit shows that the part with negative exponents is independent of [k]. For [K] it was determined in [13], hence one has:

P_{[k]}(t) = \frac{−e}{(t^{o} − 1)^{2}} + \frac{−e − eε/2}{t^{o} − 1} + higher order terms.

In the sequel we write P (t) ∼ Q(t) if lim_{t→1}(P (t) − Q(t)) = 0.

In the verification of (r∗), we consider the same inductive steps as in 11.8. First, we verify (r∗) for l′_{[k]} = −a_{0}g_{0}. Then (using o = −eα as well) one gets

P_{[k]}(t) − P_{[K]}(t) = P_{[K]}(t)(t^{αa_{0}} − 1) + ∑_{i≥0} a_{0}t^{oi+αa_{0}}

∼ −(t^{αa_{0}} − 1)(\frac{e}{(t^{o} − 1)^{2}} + \frac{e + eε/2}{t^{o} − 1}) + \frac{a_{0}t^{αa_{0}}}{1 − t^{o}} ∼ \frac{a_{0}^{2}}{2e} + \frac{a_{0}}{2} (1 + ε).

This is compatible with χ(l′_{[−a_{0}g_{0}]}) given in 11.8.

Next, consider [k] and [k_{l}^{−}] as in 11.8. Notice that

[\frac{−iω_{l} + a_{l}}{α_{l}} ] − [\frac{−iω_{l} + a_{l} − 1}{α_{l}}]

is 1 if −iω_{l} + a_{l} is divisible by α_{l}, otherwise is zero. The divisibility is satisfied if i = {ω′_{l}a_{l}/α_{l}}α_{l} + jα_{l} for some j ≥ 0. Therefore,

P_{[k]}(t) − P_{[k_{l}^{−}]}(t) = P_{[k]}(t)(1 − t^{−α/α_{l}}) + ∑_{iω_{l}≡a_{l}} t^{oi+αã−α/α_{l}}

∼ \frac{t^{oα_{l}\{ \frac{ω′_{l}a_{l}}{α_{l}} \}+αã− \frac{α}{α_{l}}}}{1 − t^{oα_{l}}} + (t^{− \frac{α}{α_{l}}} − 1)(\frac{e}{(t^{o} − 1)^{2}} + \frac{e + eε/2}{t^{o} − 1})

∼ \frac{1}{2} + \frac{ε}{2α_{l}} − \frac{1}{2eα_{l}^{2}} + \frac{ã}{eα_{l}} − \{\frac{ω′_{l}a_{l}}{α_{l}} \}.

This agrees with χ(l′_{[k_{l}^{−}]}) − χ(l′_{[k]}) from 11.8, hence (r∗) and (∗) follow by induction.

Geometry & Topology, Volume X (20XX)

11.16 Corollary. Again, since ∑_{[k]} 𝒯_{M,[k]}(1) = 0, one gets

λ(M) = ∑_{[k]∈Spin^{c}(M)} χ(HF^{+}(M, [k])) − \frac{d(M, [k])}{2}.

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