ON THE HEEGAARD FLOER HOMOLOGY OF S_{−d}^{3}(K) AND UNICUSPIDAL RATIONAL PLANE CURVES

ANDRÁS NÉMETHI

Abstract. We present a formula for the Heegaard Floer homology of −S_{−d}^{3}(K) (where d is a positive integer and K is an arbitrary algebraic knot in S^{3}) in terms of the Alexander polynomial of K, and relate the result with the theory of normal surface singularities and the geometry of unicuspidal rational projective plane curves.

1. Introduction

The ultimate goal of the present article is to describe a connection between the Heegaard Floer theory and some open problems in algebraic geometry (singularity theory). The connecting bridge is the 3-manifold M = S_{−d}^{3}(K), obtained by (−d)-surgery along an arbitrary algebraic knot K ⊂ S^{3} (where d is an arbitrary positive integer). From the point of view of the Heegaard Floer theory (introduced and intensively studied by Ozsváth and Szabó, see e.g. [26, 28]), we present a complete description of the Heegaard Floer homology HF^{+}(−M). The formula is proved in [21]. The proof is based on the results of [20], and also on the fact that M is a plumbed manifold associated with an almost rational negative definite connected graph. The formula involves the integer d, and the complete information which codifies the isotopy type of K ⊂ S^{3}. This can be expresses either in terms of the Alexander polynomial of K, or by the semigroup associated with the corresponding local plane curve singularity. These invariants are reviewed in section 2.

In fact, in the spirit of [20], we determine the graded roots associated with M (which contain all the needed information about HF^{+}). Their definitions, properties are reviewed in section 3, together with the recipe how one recovers from them the Heegaard Floer homologies.

Section 4 contains the description of the graded roots associated with M in terms of the Alexander polynomial of K. Recently, in [29], Ozsváth and Szabó computed HF^{+}(S_{d}^{3}(K)) for any knot K and integer surgery coefficient d in terms of the filtered chain homotopy type of the Heegaard Floer complex associated with K. The advantage of our result is that the input of the theorem (namely, the Alexander polynomial) is more elementary, it is the simplest homological object associated with K ⊂ S^{3}.

In section 5 we verify that the renormalized Euler characteristic of the Heegaard Floer homology coincides with the Reidemeister-Turaev torsion normalized by the Casson-Walker invariant. Although this follows from a very recent general statement [30], we find its direct verification instructive: some of the involved formulas will be exploited in the last section as well.

The last two sections contains the connections with algebraic geometry. Section 5 is in the spirit of [22], and tries to exploit the new information of the previous sections in order to answer the question: is it possible to find an optimal topological upper bound for the geometric genus (and similar analytic invariants) of some normal surface singularities?

1991 Mathematics Subject Classification. Primary. 57M27, 57R57, 58Kxx, 14E15, 14B15; Secondary.

14J17, 32S25, 32S45.

Key words and phrases. 3-manifolds, ℚ-homology spheres, Ozsváth-Szabó homology, Heegaard Floer homology, Seiberg-Witten invariants, Reidemeister-Turaev torsion, Casson-Walker invariant, knot surgeries, normal surface singularities, plane curve singularities, Kulikov singularities.

The author is partially supported by NSF grant DMS-0304759.

2 András Némethi

But, definitely, the main goal of the article is section 6: it contains the main motivation of the author in the 3-manifolds S_{−d}^{3}(K). Most of material of this section is based on [6]. It formulates a conjecture about the existence of unicuspidal rational projective plane curves. The main message is that the conjectured topological criterion has an intimate relationship with Heegaard Floer theory.

2. The manifold M = S_{−d}^{3}(K)

2.1. Review of algebraic knots. A general reference for this subsection is the book of Brieskorn and Knörrer [4], see also [5]. Let K_{f} ⊂ S^{3} be the link of an irreducible complex plane curve singularity f : (ℂ^{2}, 0) → (ℂ, 0). More precisely, for ϵ > 0 sufficiently small, write S^{3} = {z ∈ ℂ^{2} : ||z|| = ϵ} and take the transversal intersection K_{f} := {f = 0} ∩ S^{3} ↪ S^{3}. The natural orientations of S^{3} and of the regular part of {f = 0} induces a natural orientation on K_{f}. Since f is irreducible, K_{f} ≈ S^{1}. We will assume that {f = 0} is not smooth at the origin, i.e. K_{f} ⊂ S^{3} is not the unknot.

It is well-known that any embedded algebraic knot K_{f} ⊂ S^{3} as above is an iterated torus knot. Its isotopy type is completely characterized by any of the following invariants listed below.

2.1.1. The Newton pairs of f consist of s ≥ 1 pairs of integers {(p_{i}, q_{i})}_{i=1}^{s}, where p_{i} ≥ 2, q_{i} ≥ 1, q_{1} > p_{1} and gcd(p_{i}, q_{i}) = 1.

2.1.2. In some algebraic construction one prefers to replace the Newton pairs by the Puiseux pairs; they will be not used here. On the other hand, in topological constructions, it is preferable to replace them by the ‘linking pairs’ (or, the decorations of the splice diagram, cf. [5]) (p_{i}, a_{i})_{i=1}^{s}, where

a_{1} = q_{1} and a_{i+1} = q_{i+1} + p_{i+1}p_{i}a_{i} for i ≥ 1.

2.1.3. Let Δ(t) be the Alexander polynomial of K_{f} ⊂ S^{3}, or equivalently, the characteristic polynomial of the algebraic monodromy acting on the first homology of the Milnor fiber of f. It is normalized by Δ(1) = 1. In terms of the (p_{i}, a_{i})_{i} pairs it is

Δ(t) = \frac{(t^{a_{1}p_{1}p_{2}···p_{s}} − 1)(t^{a_{2}p_{2}···p_{s}} − 1) ⋯ (t^{a_{s}p_{s}} − 1)(t − 1)}{(t^{a_{1}p_{2}···p_{s}} − 1)(t^{a_{2}p_{3}···p_{s}} − 1) ⋯ (t^{a_{s}} − 1)(t^{p_{1}···p_{s}} − 1)}.

The degree of Δ, or equivalently, the first Betti number of the Milnor fiber of f, is the Milnor number μ of f.

2.1.4. The semigroup Γ of the germ f is a sub-semigroup of ℕ defined by Γ := {i_{0}(f, g) : g ∈ O_{(ℂ^{2},0)}}, where i_{0}(f, g) denotes the local intersection multiplicity of f and g (which equals the codimension of the ideal (f, g) in O_{(ℂ^{2},0)}). For g invertible i_{0}(f, g) is zero, hence 0 ∈ Γ.

It is known that Γ is generated by the integers β̄_{0} = p_{1}p_{2} ⋯ p_{s}, β̄_{k} = a_{k}p_{k+1} ⋯ p_{s} for 1 ≤ k ≤ s−1, and β̄_{s} = a_{s}. Moreover, #(ℕ \ Γ) is finite. Its cardinality is the delta-invariant δ of f, which in this case also equals μ/2, cf. [15]. The largest element of ℕ \ Γ is μ − 1. In fact, for 0 ≤ k ≤ μ − 1 one has: k ∈ Γ if and only if μ − 1 − k ∉ Γ.

It is also known that δ = μ/2 also equals the minimal Seifert genus of K_{f} ⊂ S^{3}.

2.1.5. The embedded minimal good dual resolution graph of the germ f has the following shape

[figure: K_{f}; ⋯; −1; v_{0}; G(f) :]

On the Heegaard Floer homology of S_{−d}^{3}(K) 3

Above we emphasize only the vertices of degree one and three. The dash-line between two such vertices replaces a string [figure: ⋯] . The number of vertices of degree three is exactly s. In general, any vertex v is decorated by the self-intersection of the corresponding irreducible exceptional divisor E_{v}. In the above diagram we put only the decoration of the vertex v_{0}, which corresponds to the unique (−1)-curve, and which also supports the strict transform of {f = 0}. The other decorations are not really essential in our next discussions; for the description of the complete graph see e.g. [4] or [16], section 4.I.

The above diagram can also be identified with the plumbing graph of (S^{3}, K_{f}). In this case the self-intersections are the corresponding Euler numbers of the S^{1}-bundles, all the surfaces used in the plumbing construction are S^{2}’s, and K_{f} is a generic fiber of the unique (−1)-bundle.

The above resolution graph G(f) will be denoted by the following schematic diagram:

[figure: K_{f}; −1; v_{0}; G^{∗}]

The polynomial Δ(t) can also be deduced from G(f) by A’Campo’s formula [1]. Notice that (if we disregard the arrowhead) the graph G^{∗} can be blown down completely (since (ℂ^{2}, 0) is smooth).

We emphasize again, the information codified in the following objects – isotopy class of K_{f} ⊂ S^{3}, set of Newton pairs, set of linking pairs, Alexander polynomial, semigroup or the embedded resolution graph – are completely equivalent.

2.2. Definition of the polynomial Q. One has the following identity connecting the Alexander polynomial Δ and the semigroup Γ – the readers interested only in the topological aspects, can take it as the ‘definition’ of Γ – cf. [9]:

\frac{Δ(t)}{1 − t} = ∑_{k∈Γ} t^{k}.

Since Δ(1) = 1 and Δ′(1) = δ (use e.g. the formula of 2.1.3), one gets

Δ(t) = 1 + δ(t − 1) + (t − 1)^{2} · Q(t)

for some polynomial Q(t) = ∑_{i=0}^{μ−2} α_{i}t^{i} of degree μ − 2 with integral coefficients. We claim that all the coefficients {α_{i}}_{i=0}^{μ−2} are strict positive, and, in fact:

δ = α_{0} ≥ α_{1} ≥ ⋯ ≥ α_{μ−2} = 1.

Indeed, by the above identity, one has δ + (t − 1)Q(t) = ∑_{k∉Γ} t^{k} or

Q(t) = ∑_{k∉Γ} \frac{t^{k} − 1}{t − 1} = ∑_{k∉Γ}(t^{k−1} + ⋯ + t + 1).

This shows that

α_{i} = #{k ∉ Γ : k > i};

hence, the coefficients α_{i} measure the distribution of the semigroup Γ in ℕ.

2.3. The surgery 3-manifold M = S_{−d}^{3}(K). In the sequel we fix an oriented algebraic knot K_{f} ⊂ S^{3} as above, sometimes denoted simply by K. The main object of the present article is the oriented manifold M := S_{−d}^{3}(K_{f}) obtained by −d-surgery along K_{f} ⊂ S^{3}, where d > 0 is a positive integer.

It is easy to see (e.g. by Kirby calculus, see e.g. [7]) that M can be represented by the following plumbing diagram:

[figure: v_{+}; −d − m_{f}; −1; v_{0}; G^{∗}; G(M) :]

Above:

m_{f} = a_{s}p_{s},

or equivalently, in the language of the embedded resolution graph of f (cf. 2.1.5), m_{f} is the multiplicity of the pull back of the germ f along the (−1)-curve. Topologically, e.g. if one uses Kirby calculus and blows down all the vertices of G^{∗}, m_{f} is a sum ∑_{j} m_{j}^{2} of squares of a sequence of linking numbers m_{j}. It is interesting to notice that, in the world of plane curve singularities, this sequence of m_{j}’s is exactly the ‘multiplicity sequence’ of the singularity f (which is also a complete topological invariant of the isotopy type of K_{f} ⊂ S^{3}.)

Take a point ∗ ∈ S^{3} \ K_{f} and identify S^{3} \ ∗ with ℝ^{3}. Then a Kirby diagram of M is given by the knot K = K_{f} ⊂ ℝ^{3} with decoration (surgery coefficient) −d. In particular, the Kirby diagram of the manifold −M (M with reversed orientation) – whose Heegaard Floer homology will be computed in the sequel – is given by the mirror image K̄ of K (with respect to any plane in ℝ^{3}) with surgery coefficient d; i.e. −M = S_{d}^{3}(K̄).

2.4. Peculiar properties of the graph G(M).

2.4.1. The plumbing graph G(M) is a connected tree, and – as far as d > 0 – is negative definite. In particular, G(M) is the minimal good dual resolution graph of a normal surface singularity. Notice that the minimal resolution of such a singularity contains only one irreducible exceptional divisor, which is rational, it has self-intersection −d and a singular point of local topological type (S^{3}, K_{f}).

2.4.2. | det(G(M))| = d. In fact, H := H_{1}(M, ℤ) = ℤ_{d}. There is natural identification of the spin^{c}- structures of M with the classes [k] ∈ ℤ/dℤ (or, with the integers k = 0, 1, …, d − 1), [k] = [0] corresponding to the ‘canonical’ spin^{c}-structure (cf. [22]). The spin^{c}-structures of M and −M are naturally identified.

2.4.3. Our goal is to compute the Ozsváth-Szabó Heegaard Floer homology of M. For some particular plumbed 3-manifolds associated with negative definite plumbing graphs this is determined in [28] and [20]. Notice that, in general, G(M) may have many ‘bad vertices’ (a crucial obstruction for [28], for the precise definition see [loc. cit.]), hence [28] cannot be applied. [In fact, the number of ‘bad vertices’ in the present case is #{1 ≤ i ≤ s : p_{i} < q_{i}}.] Nevertheless, [20] works by the following reason. First recall (cf. [20]) that a graph is called almost rational, or AR, if it has at least one vertex with the following property: if one replaces the decoration (Euler-number) of the corresponding vertex by a smaller (more negative) integer, then one gets a rational graph (in the sense of Artin [3]). The point is that the Heegaard Floer homology of a plumbed manifolds associated with AR-graphs can be computed by an algorithm of [20].

2.4.4. Lemma. G(M) is an AR-graph.

Proof. There exists a particular family of rational graphs – the so called sandwiched graphs (associated with sandwiched singularities) – which are characterized as follows: they are subgraphs of (not necessarily minimal) connected negative definite plumbing graphs which can be completely blown down (cf. e.g. [31]). We will show that if we modify the −1 decoration of v_{0} in G(M) into −2 (let us call this new graph by G(M)_{−2}), we get a sandwiched graph. Indeed, consider the graph G^{∗}. Blow up the −1 vertex v_{0}. The new decoration of v_{0} will be −2, while a new −1 vertex was born. Then blow up this new vertex (m_{f} + d − 1) times. Then one gets a graph which contains G(M)_{−2} as a subgraph. □

2.4.5. Using the results of [20], the (proof of the) above lemma has also the following consequence: the plumbed 3-manifold associated with G(M)_{−2} is an L-space (in the sense of Ozsváth-Szabó, i.e. its reduces Heegaard Floer homology is trivial).

2.4.6. As we will see in section 6, G(M) is a ‘Kulikov graph’.

3. Graded roots

3.1. Preliminary remark. The Heegaard Floer homology of M (in fact, of −M) will be computed in a combinatorial way from the graph G(M). Following [20], we do this via some intermediate objects, the d graded roots associated with G(M), each corresponding to a spin^{c}-structure of M. As we will see, the graded roots contain more information than the Heegaard Floer homology, a fact which hopefully will be more exploited in the future. On the other hand, compared with the homology, they have an enormous advantage: they can much-much easier be identified and described (because, probably, they preserve the geometric information of M at more canonical level). This phenomenon will be perfectly exemplified by the present M = S_{−d}^{3}(K_{f}) case as well.

In this section we provide a short review of abstract graded roots; for more details, see [20] where they were introduced.

3.2. Definition of a graded root (R, χ).

(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;

(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]. Their set is denoted by V_{lm}. In fact, V_{lm} coincides with the set of vertices with adjacent degree one.

(3) 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 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.

3.3. 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 R := ∐_{i} R_{i}/_{∼} is a graded root with χ(v̄_{i}^{k}) = k.

Clearly V_{lm} is a subset of {v̄_{i}^{n_{i}}}_{i∈I}. They are equal if in (ii) all the inequalities are strict. (Otherwise some indices are superfluous, and the corresponding R_{i}’s produce no contribution.)

(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}).

To any graded root, one can associate a natural ℤ[U]-module. This will be described in the next paragraphs. First, we start with some notations.

3.4. ℤ[U]-module notations. Consider the graded ℤ[U]-module ℤ[U, U^{−1}], and (following [28]) denote by T_{0}^{+} its quotient by the submodule U · ℤ[U]. This has a grading in such a way that deg(U^{−h}) = 2h (h ≥ 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 ℤmodule, 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, fix an arbitrary ℚ-graded ℤ[U]-module P with h-homogeneous elements P_{h}. Then for any r ∈ ℚ we denote by P [r] the same module graded in such a way that P [r]_{h+r} = P_{h}. Then set T_{r}^{+} := T_{0}^{+}[r] and T_{r}(n) := T_{0}(n)[r].

3.5. Definition. The ℤ[U]-module associated with a graded root. For any graded root (R, χ), let ℍ(R, χ) 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).

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 h ∈ ℤ if for each v ∈ V with ϕ(v) ≠ 0, ϕ(v) ∈ T_{0}^{+} is homogeneous of degree h − 2χ(v).

One can verify easily, that ℍ(R, χ) can also be computed as follows (cf. proposition (3.6) in [20]):

3.6. Proposition. Let (R, χ) be a graded root. We order V_{lm} 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_{k} := {v_{1}, …, v_{k}} ⊊ V_{lm}, then let v_{k+1} be an arbitrary vertex in V_{lm} \J_{k} with χ(v_{k+1}) = min_{v∈V_{lm}\J_{k}} 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_{k}. 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 direct sum decomposition of graded ℤ[U]-modules:

ℍ(R, χ) = T_{2m}^{+} ⊕ ℍ_{red}(R, χ).

Proposition 3.6 for (R_{τ} , χ_{τ}) gives the following:

3.7. Corollary. (See (3.8) in [20].) Let (R_{τ} , χ_{τ}) be a graded root associated with some function τ : {0, …, L} → ℤ, cf. 3.3(3), which satisfies τ(1) > τ(0) = 0. Then the ℤ-rank of ℍ_{red}(R_{τ} , χ_{τ}) is:

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

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

4. The Heegaard Floer homology HF^{+}(−M)

4.1. Preliminaries. For any oriented rational homology 3-sphere M the Heegaard Floer homology HF^{+}(M) was introduced by Ozsváth and Szabó in [26] (cf. also with their long list of articles). In fact, HF^{+}(M) is a ℤ[U]-module with a ℚ-grading compatible with the ℤ[U]-action, where deg(U) = −2. Additionally, HF^{+}(M) also has an (absolute) ℤ_{2}-grading; HF_{even}^{+}(M), respectively HF_{odd}^{+}(M), denote the part of HF^{+}(M) with the corresponding parity.

Moreover, HF^{+}(M) has a natural direct sum decomposition of ℤ[U]-modules (compatible with all the gradings) corresponding to the spin^{c}-structures of M:

HF^{+}(M) = ⊕_{[k]∈Spin^{c}(M)} HF^{+}(M, [k]).

For any spin^{c}-structure [k], one has a graded ℤ[U]-module isomorphism

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

where HF_{red}^{+}(M, [k]) 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])).

From the Ozsváth-Szabó theory one can recover the (modified) Seiberg-Witten topological invariant of (M, [k]) (see [30]) via

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

Assume now that M = S_{−d}^{3}(K) as above. Next theorem provides a purely combinatorial description of HF^{+}(−M, [k]) from G(M). More precisely, HF^{+}(−M, [k]) will be expressed in terms of the delta-invariant δ, the coefficients {α_{i}}_{i} of the polynomial Q (cf. 2.2), and the integer d.

Notice also that one can recover HF^{+}(M, [k]) from HF^{+}(−M, [k]) via (7.3) [26] and (1.1) [27].

4.2. Theorem. For each fixed k = 0, 1, …, d−1, – corresponding to the d different spin^{c}-structures of M – one defines the following objects :

• l_{k} := [\frac{2δ−2−k}{d} ]; r_{k} := \frac{(d+2δ−2−2k)^{2}−d}{4d};

• a function τ_{k} : {0, 1, …, 2l_{k} + 2} → ℤ by

\{ \begin{matrix}τ_{k}(2l) = \frac{l(l−1)}{2}d − l(δ − 1 − k); \\ τ_{k}(2l + 1) = τ_{k}(2l + 2) + α_{k+ld}. \end{matrix}

• and the graded root (R_{τ_{k}}, χ_{τ_{k}}) associated with τ_{k}. Then the following facts hold:

HF_{odd}^{+}(−M, [k]) = 0;

HF_{even}^{+}(−M, [k]) = ℍ(R_{τ_{k}}, χ_{τ_{k}})[r_{k}]; d(−M, [k]) = 2 · min τ_{k} + r_{k}.

In fact, the author proved in [21] a similar result for any M = S_{−r}^{3}(K), where r is an arbitrary positive rational number (but the result definitely is more technical). Since the present article aims the connection of the integral surgery manifold with some constructions in algebraic geometry, we decided to present above only the M = S_{−d}^{3}(K) case. The proof, including the general case, will be published elsewhere.

The value τ_{k}(2l + 1) (for 0 ≤ l ≤ l_{k}) can also be expressed as

τ_{k}(2l + 1) = τ_{k}(2l) + {γ ∈ Γ : γ ≤ k + ld}.

Since 0 ∈ Γ, τ_{k}(2l + 1) is strict larger than both τ_{k}(2l) and τ_{k}(2l + 2).

This fact together with 4.2 and 3.7 imply:

4.3. Corollary.

sw^{OSz}(M, [k]) = −sw^{OSz}(−M, [k]) = \frac{r_{k}}{2} − ∑_{l≥0} α_{k+ld}.

4.4. Remark. In the next section we will discuss some properties – in different geometric situations – of the coefficients {α_{i}}_{i}. Using the above formulae, they can be reinterpreted in terms of the shape of the corresponding graded roots, or in terms of the structure of the ℤ[U]-module HF^{+}(−M).

5. Connection with the Reidemeister-Turaev torsion

5.1. Recall that the Ozsváth-Szabó Heegaard Floer homology substitutes the (modified) Seiberg-Witten topological invariant of (M, [k]) via its (renormalized) Euler characteristic sw^{OSz}(M, [k]). In fact, the same is true for the sign refined Reidemeister-Turaev torsion normalized with the Casson-Walker invariant. Although recently it was proved in [30] that these two objects are the same (using a surgery formula for sw^{OSz}), we will verify this fact directly. This is not only instructive, but it is also a good test for all the above formulas. Moreover, these new formulas may provide some insight in the forthcoming relations (cf. e.g. section 7).

Recall that the normalized Reidemeister-Turaev torsion is

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

where λ(M) is the Casson-Walker invariant of M (normalized as in [12] (4.7)), |H| is the order of H = H_{1}(M, ℤ), and 𝒯_{M,[k]}(1) is the Reidemeister-Turaev torsion associated with M and the spin^{c}-structure [k]. In the sequel we will follow the notations (and formulas) of [22].

5.2. The torsion. The Reidemeister-Turaev torsion is computed in Theorem (5.7) of [22] from the plumbing graph via its Fourier transform. In our case H is the cyclic group ℤ_{d}. After the identification of the spin^{c}-structure (similarly as in (10.6) in [20], and taking for v_{∗} in (5.7)[22] the vertex v_{+}), one gets the Fourier-Dedekind sum:

𝒯_{M,[k]}(1) = \frac{1}{d} ^{∑′} \frac{ξ^{−k} · Δ(ξ)}{(1 − ξ)^{2}},

where k = 0, 1, … d − 1 (corresponding to the different spin^{c}-structures), and ∑′ denotes the sum over all the d-roots of unity ξ not equal to 1. Recall that Δ(t) = 1 + (t − 1)δ + (t − 1)^{2}Q(t). Using [22], Appendix (B7) one has

\frac{1}{d} ^{∑′} \frac{ξ^{−k}}{(ξ − 1)^{2}} = −\frac{d^{2} − 1 + 6(1 + k)^{2} − 6d(1 + k)}{12d}.

By an elementary computation

\frac{1}{d} ^{∑′} \frac{ξ^{−k}}{ξ − 1} = \frac{2k − d + 1}{2d}, and \frac{1}{d} ^{∑′} ξ^{−k}Q(ξ) = −\frac{1}{d}Q(1) + ∑_{l} α_{k+ld}.

5.3. The Casson-Walker invariant. From the Walker-Lescop surgery formula (see [12], page 13) applied for M = S_{−d}^{3}(K), one gets

\frac{λ(M)}{d} = \frac{(d − 1)(d − 2)}{24d} − \frac{1}{2d}(Δ^{#})′′(1),

where Δ^{#}(t) is the symmetrized Alexander ‘polynomial’ t^{−δ}Δ(t). Clearly (Δ^{#})′′(1) = δ−δ^{2}+2Q(1). All the formulas together show sw^{OSz}(M, [k]) = sw^{T CW} (M, [k]).

6. Connection with local normal surface singularities

6.1. As we already mentioned in 2.4.1, G(M) is a connected negative definite plumbing graph. In particular, it can be the (minimal good) resolution graph of a normal surface singularity (X, 0). This is equivalent with the fact that the link of (X, 0) is M (i.e. M = ρ^{−1}(ϵ), for 0 < ϵ ≪ 1 and some real analytic function ρ : X → [0, ∞) with ρ^{−1}(0) = 0).

The next discussion is in the spirit of [22], where we proposed to find a topological upper bound (in terms of the Seiberg-Witten invariant of the link) for the geometric genus of a normal surface singularity. The interested reader is invited to consult [22, 23, 24, 14], or the review article [18].

Fix a normal surface singularity (X, 0) whose link is M. Fix one of its sufficiently small (Stein) representatives X and consider its minimal good resolution π : X̃ → X. The irreducible components {E_{j}}_{j∈J} of the exceptional set π^{−1}(0) will be indexed by J, E_{+} corresponds to the index + (or, to the vertex v_{+}). Corollary (9.6)(a) of [20], via the present 4.2 and 4.3 reads as follows.

6.2. Theorem. Consider any holomorphic line bundle L ∈ P ic(X̃) on X̃ whose first Chern class c_{1}(L) ∈ H^{2}(X̃, ℤ) satisfies the following numerical conditions: c_{1}(L)(E_{j}) = 0 for any j ≠ + while c_{1}(L)(E_{+}) = k for some fixed k ∈ {0, 1, …, d − 1}. Then

h^{1}(X̃, L) ≤ ∑_{l≥0} α_{k+ld}.

In fact, we expect (cf. [22] and [19]) that the above upper bound is an optimal topological upper bound. In order to be more cautious (cf. e.g. with [14]), we formulate it as a question:

6.3. Question. Is it possible to find a special analytic singularity (X, 0) (with M as its link), and convenient line bundles L ∈ P ic(X̃) (with topological properties as in 6.2) satisfying

h^{1}(X̃, L) = ∑_{l≥0} α_{k+ld} ?

Recall that h^{1}(X̃, OX̃) is the geometric genus p_{g}(X, 0) of the singularity (X, 0). The above inequality 6.2 in the case k = 0 and L = OX̃ gives a topological upper bound for the geometric genus of any analytic singularity whose link is M:

p_{g}(X, 0) ≤ ∑_{l≥0} α_{ld}.  (∗)

The above Question 6.3 can be made even more precise. To do this, first we mention that G(M) is a ‘Kulikov graph’ (Karras called them ‘Kodaira graphs’). These particular graphs were studied intensively by Kulikov [11], Karras [10], Stevens [32] and others. The point is that there is a ‘nice’ family of normal surface singularities with link M. Its construction is the following. Consider the plumbing graph

[figure: v_{+}; −m_{f}; −1; v_{0}; G^{∗}; G_{0}(M) :]

Clearly, the corresponding plumbed 3-manifold is S_{0}^{3}(K). The point is that this graph is negative semi-definite, hence by a theorem of Winters [34], there exists a holomorphic proper map h : Y → D (where D is a small disc, and Y is a smooth surface) such that h^{−1}(0) is a normal crossing divisor whose dual graph is G_{0}(M), and the generic fiber of h is a smooth curve. Then one blows up d generic points of the irreducible component of h^{−1}(0) corresponding to the vertex v_{+}. If one disregard the new exceptional curves, then one gets a curve combination whose dual graph is exactly G(M). In particular, these curves can be contracted providing a singularity (X, 0) with dual resolution graph G(M). The analytic structure of (X, 0) has some nice properties (e.g. h induces a nice function on it, for details see e.g. [32]). A singularity constructed in this way is called Kulikov singularity.

6.4. Question. Is it true that p_{g}(X, 0) = ∑_{l≥0} α_{ld} for a Kulikov singularity (X, 0) with minimal good resolution graph G(M) ?

7. Connection with global projective plane geometry

This section is a report about some results of [6]. In fact, this section contains one of the main motivations of the author in the 3-manifold S_{−d}^{3}(K).

7.1. There is a special value of d which is particularly important from the point of view of unicuspidal rational projective plane curves.

Assume that C ⊂ ℙ^{2} is a complex projective plane curve of degree d. If C is smooth then its genus is (d − 1)(d − 2)/2. If C has some singular points, then each singular point kills some cycles from the first homology. E.g., we will discuss here the case when C has exactly one singular point p ∈ C, C at p is locally irreducible (i.e. C is unicuspidal), and C is rational (i.e. of genus zero). In this case one must have μ(C, p) = (d − 1)(d − 2).

Conversely, we can fix the topological type of an algebraic knot K ⊂ S^{3} as in 2.1, and we assume that its Milnor number μ equals (d − 1)(d − 2) for some integer d. The main question (probably as old as the classical algebraic geometry itself) is their algebraic realization:

7.2. Open problem. Classify all the topological types K ⊂ S^{3} and integers d which can be realized by a unicuspidal rational plane curve as above.

7.2.1. Notice that the restriction μ = (d − 1)(d − 2) is very far to have a classifying power. E.g., let us consider only the torus knots K_{a,b} of type (a, b) (with a < b) and set d = 5. Then μ = (a−1)(b−1) for any plane curve singularity with link K_{a,b} ⊂ S^{3}, hence (a − 1)(b − 1) = 4 · 3 = 12. The possible candidates for (a, b) are (2, 13), (3, 7) and (4, 5). The point what we want to make is that (2, 13) and (4, 5) can be realized algebraically, while (3, 7) not.

7.2.2. The open problem 7.2 has a complete answer (known by the author) only for d ≤ 6 – or for some families of curves defined by some other restrictions (e.g. for curves C with logarithmic Kodaira dimension κ̄(ℙ^{2} \ C) < 2).

7.3. In the next paragraphs we explain the connection of 7.2 with the 3-manifold S_{−d}^{3}(K).

Assume that the pair (K ⊂ S^{3}; d) can be realized by a rational unicuspidal projective curve C, whose equation is {g_{d}(x, y, z) = 0} – here g_{d} is a homogeneous polynomial of degree d in the projective coordinates (x, y, z). Then one can take the affine surface singularity g_{d} : ℂ^{3} → ℂ. Obviously, the singular locus of g^{−1}(0) is one-dimensional, hence its link is not even smooth. Nevertheless, if one takes a perturbation given by a generic homogeneous polynomial g_{d+1} of degree d + 1, then g := g_{d} + g_{d+1} : (ℂ^{3}, 0) → (ℂ, 0) defines an isolated singularity whose link M(g) := {g = 0} ∩ S_{ϵ}^{5} (for 0 < ϵ ≪ 1) – and basically all the numerical invariants of g – are independent of the choice of g_{d+1}. [These singularities are called superisolated , for more information, see e.g. [13, 2, 14].]

The point is that, in the above situation, M(g) is exactly S_{−d}^{3}(K) (a fact proved by Luengo in [13]). In particular, we can run a topological machinery for S_{−d}^{3}(K) (e.g. all the results of section 4), or involve analytic results about g (cf. section 5) in order to collect some information about C, or to transfer the classification problem 7.2 into some other field.

E.g., it is not difficult to prove that the geometric genus of {g = 0} depends only on d, in fact it is d(d − 1)(d − 2)/6 (cf. [14]). Then (∗) in section 6 reads as follows:

7.4. Corollary. If the local topological type K ⊂ S^{3} can be realized by the singularity of a rational unicuspidal curve of degree d then

∑_{l≥0} α_{ld} ≥ d(d − 1)(d − 2)/6.

Notice that this statement is sitting completely in the classical algebraic geometry of projective plane curves (although we were lead to it via the Euler characteristic of the Heegaard Floer homology).

Nevertheless, for this special d, one can reprove – even improve – 7.4 using arguments of classical algebraic geometry. Since μ − 2 = d(d − 3), the {α_{ld}}_{l≥0} coefficients are α_{0}, α_{d}, …, α_{d(d−3)}. In fact, we know that α_{0} = δ = μ/2 = (d − 1)(d − 2)/2 and α_{d(d−3)} = 1.

7.5. Proposition. [6] For all l = 0, 1, …, d − 3 one has

α_{(d−3−l)d} ≥ (l + 1)(l + 2)/2.

Proof. Recall that γ ∈ Γ if and only if μ − 1 − γ ∉ Γ. Using 2.2 and this property one has

α_{(d−3−l)d} = #{γ ∉ Γ : γ > (d − 3 − l)d} = #{γ ∈ Γ : γ ≤ ld}.

Consider now the projective space ℙ of dimension (l + 1)(l + 2)/2 − 1 of plane curves of degree l. Let t ↦ ϕ(t) be a local parametrization (normalization) of the singularity (C, p). If h is the equation of a plane curve H of degree l, then the ‘contact’ of H and C can be measured by the vanishing of the coefficients of h(ϕ(t)) = ∑_{k≥0} a_{k}t^{k}. Assume that a_{γ} = 0 for γ ∈ Γ and γ ≤ ld. Them by the very definition of the semigroup Γ, one has that a_{k} = 0 for all k ≤ ld. Assume that α_{(d−3−l)d} < (l + 1)(l + 2)/2. Then the equations a_{γ} (γ ∈ Γ, γ ≤ ld) on ℙ determine a non-empty set. In particular, there exists at least one curve H of degree l whose intersection multiplicity at p with C is > ld. This, by Bezout theorem, can happen only if C and H have a common component. But this is not the case because C is irreducible and l < d. □

Now we arrived to the main statement of the present section:

7.6. Conjecture. [6] If the local topological type K ⊂ S^{3} can be realized by the singularity of a rational unicuspidal curve of degree d then for any l = 0, 1, …, d − 3 one has

α_{(d−3−l)d} = (l + 1)(l + 2)/2.  (C_{l})

[For the more general cuspidal case, see [6].]

7.7. For the convenience of the reader we will provide some additional re-interpretations of 7.6. In the spirit of the Reidemeister-Turaev torsion, let us consider the expression

R(t) := \frac{1}{d} ∑_{ξ^{d}=1} \frac{Δ(ξt)}{(1 − ξt)^{2}} − \frac{1 − t^{d^{2}}}{(1 − t^{d})^{3}} .  (R)

Then by a similar computation as in section 5, one gets that R(t) = N(t^{d}), where

N(t) = ∑_{l=0}^{d−3} (α_{(d−3−l)d} − \frac{(l + 1)(l + 2)}{2})t^{d−3−l}.

In particular, Conjecture 7.6 is equivalent with the vanishing R(t) ≡ 0.

Notice that since Δ is symmetric, using (R), R(t), or N(T) itself is a symmetric polynomial. In particular, (C_{l}) is true if and only if (C_{d−3−l}) is true.

In fact, the results of section 5 shows that

R(1) = −sw^{T CW} (M(g), [0]) + \frac{r_{0}}{2} − p_{g}(\{g = 0\}, 0).

The vanishing of the left hand side is predicted by [22] (but consult also [14]).

In terms of the semigroup Γ, the above identities can be rewritten as

∑_{γ∈Γ} t^{⌈γ/d⌉} = \frac{1 − t^{d}}{(1 − t)^{2}} = 1 + 2t + 3t^{2} + ⋯ (d − 1)t^{d−2} + d(t^{d−1} + t^{d} + ⋯),  (#)

which for t = 1 gives

−sw(M, [0]) + \frac{r_{0}}{2} = ∑_{γ∉Γ} ⌈γ/d⌉.

The meaning of (#) is the following:

In any interval I_{l} := ( (l − 1)d, ld] there are exactly min{l + 1, d} elements of Γ.

This is a very special distribution property of the semigroup Γ.

But the most amazing re-interpretation is in terms of Theorem 4.2, k = 0 – in fact, we strongly believe that this should be the starting point of the proof of 7.6. Notice that if one rewrites the coefficients α_{ld} according to the conjecture 7.6 (and also uses 2δ = (d − 1)(d − 2)), the graded roots (R_{τ_{0}}, χ_{τ_{0}}) depend only on the integer d. Let us denote this graded root by (R, χ)(d) and let U(d) be its ℤ[U]-module ℍ(R, χ)(d))[r_{0}] (provided by 3.5). [We invite the reader to write explicitly down (ℝ, χ)(d) and U(d).] Then 7.6 reads as follows:

If the local topological type K ⊂ S^{3} can be realized by the singularity of a rational unicuspidal curve of degree d then the graded root (associated with the ‘canonical spin^{c}-structure’) of G(S_{−d}^{3}(K)) should coincide with the ‘universal’ graded root (R, χ)(d). In particular, the Heegaard Floer homology of −S_{−d}^{3}(K) should be exactly the ‘universal’ ℤ[U]-module U(d).

7.8. Example. Consider again d = 5 and the torus knots K_{a,b} where (a, b) = (2, 13), (3, 7), (4, 5) (cf. 7.2.1). The semigroup Γ is generated by the integers a and b. It is easy to see that (2, 13) and (4, 5) satisfy the semigroup distribution property, while (3, 7) not. It is even more interesting that, although the corresponding semigroups in the cases (2, 13) and (4, 5) are definitely different, the corresponding graded roots and associated ℤ[U]-modules are the same (and coincide with the corresponding universal objects), while for (3, 7) this coincidence fails.

The above conjecture is also supported by the following positive result:

7.9. Theorem. [6] Conjecture 7.6 is true whenever the logarithmic Kodaira dimension κ̄ := κ̄(ℙ^{2} \ C) is < 2.

In fact, we verified even for κ̄ = 2 many curves (see [6]). We wish to emphasize that, in fact, even after an intense search, we were not able to find any pair (K ⊂ S^{3}, d) which satisfies the conjecture but it is not realizable geometrically: In our very optimistic dreams we hope that the above conjecture is an ‘if and only if’ conjecture providing a complete classification criterion for the existence of rational unicuspidal curves: (K ⊂ S^{3}, d) is realizable if and only if α_{d(d−3−l)} = (l + 1)(l + 2)/2 for any l = 0, …, d − 3.

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Department of Mathematics, Ohio State University, Columbus, OH 43210, USA; and

Rényi Institute of Mathematics, Budapest, Hungary

   E-mail address: nemethi@math.ohio-state.edu; and nemethi@renyi.hu
   URL: http://www.math.ohio-state.edu/~nemethi