SEIBERG-WITTEN INVARIANTS AND SURFACE SINGULARITIES III (SPLICINGS AND CYCLIC COVERS)

ANDRÁS NÉMETHI AND LIVIU I. NICOLAESCU

Abstract. We verify the conjecture formulated in [31] for suspension singularities of type g(x, y, z) = f(x, y) + z^{n}, where f is an irreducible plane curve singularity. More precisely, we prove that the modified Seiberg-Witten invariant of the link M of g, associated with the canonical spin^{c} structure, equals −σ(F)/8, where σ(F) is the signature of the Milnor fiber of g. In order to do this, we prove general splicing formulae for the Casson-Walker invariant and for the sign refined Reidemeister-Turaev torsion (in particular, for the modified Seiberg-Witten invariant too). These provide results for some cyclic covers as well. As a by-product, we compute all the relevant invariants of M in terms of the Newton pairs of f and the integer n.

1. Introduction

The present article is a natural continuation of [31] and [32], and it is closely related to [28, 29, 30]. In [31] we formulated a very general conjecture which connects the topological and the analytical invariants of a complex normal surface singularity whose link is a rational homology sphere.

Even if we restrict ourselves to the case of hypersurface singularities, the conjecture is still highly non-trivial. The “simplified” version for this case reads as follows.

1.1. Conjecture. [31] Let g : (ℂ^{3}, 0) → (ℂ, 0) be a complex analytic germ which defines an isolated hypersurface singularity. Assume that its link M is a rational homology sphere. Denote by sw_{M}^{0}(σ_{can}) the modified Seiberg-Witten invariant of M associated with the canonical spin^{c} structure σ_{can} (cf. 2.7). Moreover, let σ(F) be the signature of the Milnor fiber F of g. Then

−sw_{M}^{0}(σ_{can}) = σ(F)/8.  (1)

The goal of the present paper is to verify this conjecture for suspension hypersurface singularities. More precisely, in 6.15 we prove the following.

1.2. Theorem. Let f : (ℂ^{2}, 0) → (ℂ, 0) be an irreducible plane curve singularity. Fix an arbitrary positive integer n such that the link M of the suspension singularity g(x, y, z) := f(x, y) + z^{n} is a rational homology sphere (cf. 6.2(3)). Then 1.1(1) holds.

The numerical identity 1.1(1) covers a very deep qualitative analytic-rigidity phenomenon.

2000 Mathematics Subject Classification. Primary. 14B05, 14J17, 32S25, 57M27, 57R57. Secondary.

14E15, 32S45, 57M25.

Key words and phrases. (links of) normal surface singularities, suspension singularities, plane curve singularities, Newton pairs, branched cyclic covers of 3-manifolds, geometric genus, Seiberg-Witten invariants of ℚ-homology spheres, Reidemeister-Turaev torsion, Casson-Walker invariant, splicing, plumbing.

The first author is partially supported by NSF grant DMS-0088950; the second author is partially supported by NSF grant DMS-0071820.

From a topological point of view, any normal two-dimensional analytic singularity (X, 0) is completely characterized by its link M, which is an oriented 3-manifold. Moreover, by a result of Neumann [34], any decorated plumbing (or resolution) graph of (X, 0) carries the same information as M. A property of (X, 0) will be called topological if it can be determined from any of these graphs.

A very intriguing issue, which has generated intense research efforts, is the possibility of expressing the analytic invariants of (X, 0) (like the geometric genus p_{g}, multiplicity, etc.) or the smoothing invariants (if they exist, like the signature σ(F) or the topological Euler-characteristic χ(F) of the Milnor fiber F) in terms of the topology of M.

1.3. A short historical survey. M. Artin proved in [3, 4] that the rational singularities (i.e. the vanishing of p_{g}) can be characterized completely from the resolution (or plumbing) graph. In [18], H. Laufer extended Artin’s results to minimally elliptic singularities, showing that Gorenstein singularities with p_{g} = 1 can be characterized topologically. Additionally, he noticed that the program breaks for more complicated singularities (see also the comments in [31] and [27]). On the other hand, the first author noticed in [27] that Laufer’s counterexamples do not signal the end of the program. He conjectured that if we restrict ourselves to the case of those Gorenstein singularities whose links are rational homology spheres then p_{g} is topological. This was carried out explicitly for elliptic singularities in [27] (partially based on some results of S. S.-T. Yau, cf. e.g. with [44]).

For Gorenstein singularities which have smoothing (with Milnor fiber F), the topological invariance of p_{g} can be reformulated in terms of σ(F) and/or χ(F). Indeed, via some results of Laufer, Durfee, Seade, Wahl and Steenbrink, any of p_{g}, σ(F) and χ(F) determines the remaining two modulo K^{2} + #V (for the precise identities see e.g. [24] or [31]). Here, K^{2} + #V is defined as follows. For a given resolution, one takes the canonical divisor K, and the number #V of irreducible components of the exceptional divisor of the resolution. Then K^{2} + #V can be deduced from the resolution graph, and is independent of the choice of the graph. In particular, it is an invariant of the link M (for details, see [31] (2.12)).

For example, the identity which connects p_{g} and σ(F) is

8p_{g} + σ(F) + K^{2} + #V = 0.  (2)

This connects the above facts about p_{g} with the following list of results about σ(F).

Fintushel and Stern proved in [11] that for a hypersurface Brieskorn singularity whose link is an integral homology sphere, the Casson invariant λ(M) of the link M equals −#R(M)/2, where #R(M) is the number of conjugacy classes of irreducible SU(2)-representations of π_{1}(M). It turns out that this number is σ(F)/8 (a fact conjectured by Neumann and Wahl). This result was generalized by Neumann and Wahl in [36]. They proved the same statement for all Brieskorn-Hamm complete intersections and suspensions of irreducible plane curve singularities (with the same assumption about the link). Moreover, they conjectured the validity of the formula for any isolated complete intersection singularity whose link is an integral homology sphere.

In [31] the authors extended the above conjecture for smoothing of Gorenstein singularities with rational homology sphere link. Here the Casson invariant λ(M) is replaced by a certain Seiberg-Witten invariant sw_{M}^{0}(σ_{can}) of the link associated with the canonical spin^{c} structure of M. sw_{M}^{0}(σ_{can}) is the difference of a certain Reidemeister-Turaev sign-refined torsion invariant and the Casson-Walker invariant, for details see 2.7. If M is an integral homology sphere, then the torsion invariant vanishes.

In fact, the conjecture in [31] is more general. A part of it says that for any ℚ-Gorenstein singularity whose link is a rational homology sphere, one has

8p_{g} − 8sw_{M}^{0}(σ_{can}) + K^{2} + #V = 0.  (3)

Notice that (in the presence of a smoothing and of the Gorenstein property, e.g. for any hypersurface singularity) (3) via (2) is exactly (1). The identity (3) was verified in [31] for cyclic quotient singularities, Brieskorn-Hamm complete intersections and some rational and minimally elliptic singularities. [32] contains the case when (X, 0) has a good ℂ^{∗}-action.

Finally we mention, that recently Neumann and Wahl initiated in [37] an all-embracing program about those ℚ-Gorenstein singularities whose link is rational homology sphere. They conjecture that the universal abelian cover of such a singularity is an isolated complete intersection whose equations, together with the action of H_{1}(M), can be recovered from M (modulo some equisingular and equivariant deformation). In particular, this implies the topological invariance of p_{g} as well. We hope that our efforts in the direction of the conjecture formulated in [31] will contribute to the accomplishment of this program as well.

On the other hand, the theory of suspension hypersurface singularities also has its own long history. This class (together with the weighted-homogeneous singularities) serve as an important “testing and exemplifying” family for various properties and conjectures. For more information, the reader is invited to check [29, 30] and the survey paper [28], and the references listed in these articles. Se also [26] for topological characterizations of some other analytic invariants (cf. 6.8(1)).

1.4. A few words about the proof. If M is the link of g = f + z^{n} (as in 1.2), then M has a natural splice decomposition into Seifert varieties of type Σ(p, a, m). Moreover, in [29](3.2) the first author established an additivity formula for σ(F) compatible with the geometry of this decomposition. On the other hand, for any Brieskorn singularity (x, y, z) ↦ x^{p} + y^{a} + z^{m} (whose link is Σ(p, a, m)) the conjectured identity (1) is valid by [31, 32]. Hence it was natural to carry out the proof of 1.2 by proving an additivity result for sw_{M}^{0}(σ_{can}) with respect to the splice decomposition of M into Seifert varieties.

This additivity result is proved in 6.14 (as an outcome of all the preparatory results of the previous sections) but its proof contains some surprising steps.

Our original plan was the following. First, we would identify the splicing data of M. Then, for such splicing data, we would establish separate splicing formulæ, for the Casson-Walker invariant and for the Reidemeister-Turaev sign-refined torsion with the hope that we can do this in the most general topological context, not relying on any (additional) special features the links of singularities might have.

This program worked without a hitch for the Casson-Walker invariant, thanks to the results of Fujita [12] and Lescop [21] (see section 3). On the other hand the splicing formulæ for the torsion turned out to be extremely involved in the generality we originally sought. Like magic, the combinatorial complexity of these formulæ reduces dramatically precisely in the context relevant to our investigation.

The torsion computations require the explicit description of the supports of all the relevant characters, and then the computation of some sophisticated Fourier-Dedekind sums. The computation is feasible because these sums are not random. They have two very subtle special features which follow from various properties of irreducible plane curve singularities. The first one is a numerical inequality (5.1(6)) (measuring some special algebraic property). The second (new) property is the alternating property of their Alexander polynomial (5.2).

In section 4 we establish different splicing formulas for sw_{M}^{0}(σ_{can}), and we show the limits of a possible additivity. We introduce even a new invariant D which measures the non-additivity property of sw_{M}^{0}(σ_{can}) with respect to (some) splicing data (see e.g. 4.9) or (some) cyclic covers (see 4.11). This invariant vanishes in the presence of the alternating property of the involved Alexander polynomial.

This shows clearly (and rather surprisingly) that the behavior of sw_{M}^{0}(σ_{can}) with respect to splicing and cyclic covers (constructions topological in nature) definitely prefers some special algebraic situations. For more comments, see 4.12, 5.4 and 6.8.

Section 5 contains the needed results for irreducible plane curve singularities and the Algebraic Lemma used in the summation of the Fourier-Dedekind sums mentioned above.

In section 6 we provide a list of properties of the link of f + z^{n}. Here basically we use almost all the partial results proved in the previous sections. Most of the formulae are formulated as inductive identities with respect to the number of Newton pairs of f.

1.5. Notations. All the homology groups with unspecified coefficients are defined over the integers. In section 3, s(·, ·) denotes Dedekind sums, defined by the same convention as in [12, 21] or [31]. For definitions and detailed discussions about the involved invariants, see [31]. For different properties of hypersurface singularities, the reader may consult [2].

2. Preliminaries and notations

2.1. Oriented knots in rational homology spheres. Let M be an oriented 3-manifold which is a rational homology sphere. Fix an oriented knot K ⊂ M, denote by T (K) a small tubular neighborhood of K in M, and let ∂T (K) be its oriented boundary with its natural orientation. The natural oriented meridian of K, situated in ∂T (K), is denoted be m. We fix an oriented parallel ℓ in ∂T (K) (i.e. ℓ ∼ K in H_{1}(T (K))). If ⟨, ⟩ denotes the intersection form in H_{1}(∂T (K)), then ⟨m, ℓ⟩ = 1 (cf. e.g. Lescop’s book [21], page 104; we will use the same notations m and l for some geometric realizations of the meridian and parallel as primitive simple curves, respectively for their homology classes in H_{1}(∂T (K))).

Obviously, the choice of ℓ is not unique. In all our applications, ℓ will be characterized by some precise additional geometric construction.

Assume that the order of the homology class of K in H_{1}(M) is o > 0. Consider an oriented surface F_{oK} with boundary oK, and take the intersection λ := F_{oK} ∩ ∂T (K). λ is called the longitude of K. The homology class of λ in H_{1}(∂T (K)) can be represented as

λ = oℓ + km, for some integer k.

Set gcd(o, |k|) = δ > 0. Then λ can be represented in ∂T (K) as δ primitive torus curves of type (o/δ, k/δ) with respect to ℓ and m.

2.2. Dehn fillings. Let T (K)^{∘} be the interior of T (K). For any homology class a ∈ H_{1}(∂T (K)), which can be represented by a primitive simple closed curve in ∂T (K), one defines the Dehn filling of M \ T (K)^{∘} along a by

(M \ T (K)^{∘})(a) = M \ T (K)^{∘} ∐_{f} S^{1} × D^{2},

where f : ∂(S^{1} × D^{2}) → ∂T (K) is a diffeomorphism which sends {∗} × ∂D^{2} to a curve representing a.

2.3. Linking numbers. Consider two oriented knots K, L ⊂ M with K ∩ L = ∅. Fix a Seifert surface F_{oK} of oK (cf. 2.1) and define the linking number Lk_{M}(K, L) ∈ ℚ by the “rational” intersection (F_{oK} ·L)/o. In fact, Lk_{M}(K, ·) : H_{1}(M \K, ℚ) → ℚ is a well-defined homeomorphism and Lk_{M}(K, L) = Lk_{M}(L, K). For any oriented knot L on ∂T (K) one has (see e.g. [21, 6.2.B]):

Lk_{M}(L, K) = ⟨L, λ⟩/o.  (1)

For any oriented knot K ⊂ M one has the obvious exact sequence

0 → ℤ ⟶^{α} H_{1}(M \ T (K)) ⟶^{j} H_{1}(M) → 0,  (2)

where α(1_{ℤ}) = m. If K ⊂ M is homologically trivial then this sequence splits. Indeed, let ϕ be the restriction of Lk_{M}(K, ·) to H_{1}(M \ K) = H_{1}(M \ T (K)). Then ϕ has integer values and ϕ ∘ α = 1_{ℤ}. This provides automatically a morphism s : H_{1}(M) → H_{1}(M \ T (K)) such that j ∘ s = 1 and α ∘ ϕ + s ∘ j = 1; in particular with ϕ ∘ s = 0 too. In fact, s(H_{1}(M)) = T ors H_{1}(M \ T (K)). Moreover, under the same assumption o = 1, one has the isomorphisms

H_{2}(M, K) →^{∂} H_{1}(K) = ℤ and H_{1}(M) → H_{1}(M, K).  (3)

Sometimes, in order to simplify the notations, we write H for the group H_{1}(M).

The finite group H carries a natural symmetric bilinear form b_{M} : H^{⊗2} → ℚ/ℤ defined by b_{M}([K], [L]) = Lk_{M}(K, L) (mod ℤ), where L and K are two representatives with K ∩ L = ∅. If Ĥ denotes the Pontryagin dual Hom(H, S^{1}) of H, then [K] ↦ b_{M}([K], ·) is an isomorphism H → Ĥ.

2.4. (M, K) represented by plumbing. The main application of the present article involves algebraic links (M, K) which can be represented by plumbing. We recall the notations briefly (for more details, see e.g. [31] or [35]).

We will denote by Γ(M, K) the plumbing graph of a link K ⊂ M. The vertices v ∈ V are decorated by the Euler numbers e_{v} (of the S^{1}-bundles over E_{v} ≈ S^{2} used in the plumbing construction). The components of the link K are represented by arrows in Γ(M, K): if an arrow is attached to the vertex v then the corresponding component of K is a fixed fiber of the S^{1}-bundle over E_{v}. (We think about an arrow as an arrowhead connected to v by an edge.) If we delete the arrows then we obtain a plumbing graph Γ(M) of M. Let δ_{v} (resp. δ̄_{v}) be the degree (i.e. the number of incident edges) of the vertex v in Γ(M) (resp. in Γ(M, K)). Evidently δ̄_{v} − δ_{v} is exactly the number of arrows supported by the vertex v.

Since M is a rational homology sphere, Γ is a tree.

Let {I_{uv}}_{u,v∈V} be the intersection matrix associated with Γ; i.e. I_{uu} = e_{v}, and for u ≠ v the entry I_{uv} = 0 or 1 depending that u and v are connected or not in Γ. Since M is rational homology sphere, I is non-degenerate. In fact

| det(I)| = |H|.  (4)

The generic fiber of the S^{1}-bundle over E_{v} is denoted by g_{v}, and we use the same notation for its homology class in H_{1}(M) as well. By the above discussion, if u ≠ v then Lk_{M}(g_{u}, g_{v}) is well-defined. If u = v then we write Lk_{M}(g_{u}, g_{u}) for Lk_{M}(g_{u}, g′_{u}) where g_{u} and g′_{u} are two different fibers of the S^{1}-bundle over E_{v}.

For any fixed vertex u ∈ V, we denote by b⃗(u) the column vector with entries =1 on the place u and zero otherwise. We define the column vector w⃗(u) (associated with the knot g_{u} ⊂ M and its order o(u)) as the solution of the (non-degenerate) linear system I · w⃗(u) = −o(u)b⃗(u). The entries {w_{v}(u)}_{v∈V} are called the weights associated with g_{u}.

Then the inverse matrix I^{−1} of I, the set of weights {w_{v}(u)}_{v∈V}, and the linking pairing Lk_{M} satisfy:

−I_{uv}^{−1} = \frac{w_{v}(u)}{o(u)} = Lk_{M}(g_{u}, g_{v}) for any u and v ∈ V.  (5)

In particular, w_{v}(u) ∈ ℤ for any u and v. In fact, if I is negative definite, then the integers w_{v}(u) are all positive.

2.5. (M, K) represented by splice diagram. If M is an integral homology sphere, and (M, K) has a plumbing representation, then there is an equivalent graph-codification of (M, K) in terms of the splice (or Eisenbud-Neumann) diagram, for details see [10].

The splice diagram preserves the “shape” of the plumbing graph (e.g. there is a oneto-one correspondence between those vertices v with δ_{v} ≠ 2 of the splice, respectively of the plumbing graphs), but in the splice diagram one collapses into an edge each string of the plumbing graph. Moreover, the decorations are also different. In the splice diagram, each vertex has a sign ϵ = ±1, which in all our cases will be ϵ = +1, hence we omit them. Moreover, if an end of an edge is attached to a vertex v with δ_{v} ≥ 3, then it has a positive integer as its decoration. The arrows have the same significance.

One of the big advantages of the of the splice diagram is that it codifies in an ideal way the splicing decomposition of M into Seifert pieces. (In fact, the numerical decorations are exactly the Seifert invariant of the corresponding Seifert splice-components.)

Therefore, in some cases it is much easier and more suggestive to use them. (Nevertheless, we will use them only in those cases when we really want to emphasize this principle, e.g. in the proof of 4.11, or when it is incomparably easier to describe a construction with them, e.g. in 4.13.) The reader is invited to consult the book of Eisenbud and Neumann [10] for the needed properties. The criterion which guarantees that (M, K) is algebraic is given in 9.4; the equivalence between the splice and plumbing graphs is described in sections 20-22; the splicing construction appears in section 8.

2.6. The Alexander polynomial. Assume that K is a homologically trivial oriented knot in M. Let V be the Alexander matrix of K ⊂ M, and V^{∗} its transposed (cf. [21], page 26).

In the literature one can find different normalizations of the Alexander “polynomial”. The most convenient for us, which makes our formulae the simplest possible, is

Δ_{M}^{♮}(K)(t) := det(t^{1/2} V − t^{−1/2} V^{∗}).

In the surgery formula 3.4 we will need Lescop’s normalization [21], in this article denoted by Δ_{M}^{L}(K)(t). They are related by the identity (cf. [21, 2.3.13]):

Δ_{M}^{♮}(K)(t) = Δ_{M}^{L}(K)(t)/|H|.

Then (see e.g. [21, 2.3.1]) one has:

Δ_{M}^{♮}(K)(1) = Δ_{M}^{L}(K)(1)/|H| = 1.  (6)

We also prefer to think about the Alexander polynomial as a characteristic polynomial. For this notice that V is invertible over ℚ, hence one can define the “monodromy operator” ℳ := V^{−1}V^{∗}. Then set:

Δ_{M}(K)(t) := det(I − tℳ) = det(V^{−1}) · t^{r} · Δ_{M}^{♮}(K)(t).  (7)

If (M, K) can be represented by a negative definite plumbing graph, then by a theorem of Grauert [13] (M, K) is algebraic, hence by Milnor fibration theorem, it is fibrable. In this case, ℳ is exactly the monodromy operator acting on the first homology of the (Milnor) fiber. Moreover, Δ_{M}(K)(t) can be computed from the plumbing graph by A’Campo’s theorem [1] as follows (see also [10]). Assume that K = g_{u} for some u ∈ V, then

\frac{Δ_{M}(g_{u})(t)}{t − 1} = ∏_{v∈V} (t^{w_{v}(u)} − 1)^{δ̄_{v}−2}.  (8)

Notice that (6) guarantees that Δ_{M}(K)(1) = det(V^{−1}), from (8) we get Δ_{M}(K)(1) > 0, and from the Wang exact sequence of the fibration |Δ_{M}(K)(1)| = |H|. Therefore:

Δ_{M}(K)(1) = det(V)^{−1} = |H|.  (9)

More generally, if (M, K) has a negative definite plumbing representation, and K = g_{u} for some u, then for any character χ ∈ Ĥ we define Δ_{M,χ}(g_{u})(t) via the identity

\frac{Δ_{M,χ}(g_{u})(t)}{t − 1} := ∏_{v∈V} (t^{w_{v}(u)}χ(g_{v}) − 1)^{δ̄_{v}−2},  (10)

and we write

Δ_{M}^{H}(g_{u})(t) := \frac{1}{|H|} · ∑_{χ∈Ĥ} Δ_{M,χ}(g_{u})(t).  (11)

In section 6 we will need the following analog of (9) in the case when K = g_{u} ⊂ M is not homologically trivial (but (M, K) has a negative definite plumbing representation):

lim_{t→1} (t − 1) ∏_{v∈V} (t^{w_{v}(u)} − 1)^{δ̄_{v}−2} = |H|/o(u).  (12)

This follows e.g. from [31] A10(b). (In fact, |H|/o(u) has the geometric meaning of |T ors H_{1}(M \ K)|, and (12) can also be deduced from the Wang exact sequence of the monodromy, similarly as above.)

2.7. The Seiberg-Witten invariant. If M is a rational homology 3-sphere, then the set Spin^{c}(M) of the spin^{c} structures of M is a H-torsor. If M is the link of a normal surface singularity (or, equivalently, if M has a plumbing representation with a negative definite intersection matrix), then Spin^{c}(M) has a distinguished element σ_{can}, called the canonical spin^{c} structure (cf. [31]).

To describe the Seiberg-Witten invariants one has to consider an additional geometric data belonging to the space of parameters

P = {u = (g, η); g = Riemann metric, η = closed two-form}.

Then for each spin^{c} structure σ on M one defines the (σ, g, η)-Seiberg-Witten monopoles. For a generic parameter u, the Seiberg-Witten invariant sw_{M}(σ, u) is the signed monopole count. This integer depends on the choice of the parameter u and thus it is not a topological invariant. To obtain an invariant of M, one needs to alter this monopole count by the Kreck-Stolz invariant KS_{M}(σ, u) (associated with the data (σ, u)), cf. [23] (or see [17] for the original “spin version”). Then, by [8, 23, 25], the rational number

\frac{1}{8}KS_{M}(σ, u) + sw_{M}(σ, u)

is independent of u and thus it is a topological invariant of the pair (M, σ). We denote this modified Seiberg-Witten invariant by sw_{M}^{0}(σ). In general, it is very difficult to compute sw_{M}^{0}(σ) using this definition. Fortunately, it has another realization as well. For any spin^{c} structure σ on M, we denote by

𝒯_{M,σ} = ∑_{h∈H} 𝒯_{M,σ}(h) h ∈ ℚ[H]

the sign refined Reidemeister-Turaev torsion associated with σ (for its detailed description, see [40]). We think of 𝒯_{M,σ} as a function H → ℚ given by h ↦ 𝒯_{M,σ}(h). The augmentation map 𝔞𝔲𝔤 : ℚ[H] → ℚ is defined by ∑ a_{h} h ↦ ∑ a_{h}. It is known that 𝔞𝔲𝔤(𝒯_{M,σ}) = 0.

Denote the Casson-Walker invariant of M by λ_{W} (M) [42]. It is related with Lescop’s normalization λ(M) [21, §4.7] by λ_{W} (M) = 2λ(M)/|H|. Then by a result of the second author [38], one has:

sw_{M}^{0}(σ) = 𝒯_{M,σ}(1) − λ_{W} (M)/2.  (13)

Below we will present a formula for 𝒯_{M,σ} in terms of Fourier transform. Recall that a function f : H → ℂ and its Fourier transform f̂: Ĥ → ℂ satisfy:

f̂(χ) = ∑_{h∈H} f(h)χ̄(h); f(h) = \frac{1}{|H|} ∑_{χ∈Ĥ} f̂(χ)χ(h).

Here Ĥ denotes the Pontryagin dual of H as above. Notice that f̂(1) = 𝔞𝔲𝔤(f), in particular 𝒯̂_{M,σ}(1) = 𝔞𝔲𝔤(𝒯_{M,σ}) = 0. Therefore,

𝒯_{M,σ}(1) = \frac{1}{|H|} ∑_{χ∈Ĥ\\{1\}} 𝒯̂_{M,σ}(χ).  (14)

Now, assume that M is represented by a negative definite plumbing graph. Fix a non-trivial character χ ∈ Ĥ \ {1} and an arbitrary vertex u ∈ V with χ(g_{u}) ≠ 1. Then set

P̂_{M,χ,u}(t) := ∏_{v∈V} (t^{w_{v}(u)}χ(g_{v}) − 1)^{δ_{v}−2},  (15)

where t ∈ ℂ is a free variable. Then, by [31] (5.8), the Fourier transform 𝒯̂_{M,σ_{can}} of 𝒯_{M,σ_{can}} is given by

𝒯̂_{M,σ_{can}}(χ̄) = lim_{t→1} P̂_{M,χ,u}(t).  (16)

This limit is independent of the choice of u, as long as χ(g_{u}) ≠ 1. In fact, by [31], even if χ(g_{u}) = 1, but u is adjacent to some vertex v with χ(g_{v}) ≠ 1, u does the same job.

3. Some general splicing formulae

3.1. The splicing data. We will consider the following geometric situation. We start with two oriented 3-manifolds M_{1} and M_{2}, both rational homology spheres. For i = 1, 2, we fix an oriented knot K_{i} in M_{i}, and we use the notations of 2.1 with the corresponding indices i = 1, 2. In this article we will consider a particular splicing, which is motivated by the geometry of the suspension singularities. The more general case will be treated in a forthcoming paper.

On the pair (M_{2}, K_{2}) we impose no additional restrictions. But, for i = 1, we will consider the following working assumption:

WA1: Assume that o_{1} = 1, i.e. K_{1} is homologically trivial in M_{1}. Moreover, we fix the parallel ℓ_{1} exactly as the longitude λ_{1}. Evidently, k_{1} = 0.

Finally, by splicing, we define a 3-manifold M (for details, see e.g. [12]):

M = M_{1} \ T (K_{1})^{∘} ∐_{A} M_{2} \ T (K_{2})^{∘},

where A is an identification of ∂T (K_{2}) with −∂T (K_{1}) determined by

A(m_{2}) = λ_{1}, and A(ℓ_{2}) = m_{1}.  (1)

3.2. The closures \overline{M}_{i}. Once the splicing data is fixed, one can consider the closures \overline{M}_{i} of M_{i} \ T (K_{i})^{∘} (i = 1, 2) with respect to A (cf. [5] or [12]) by the following Dehn fillings:

\overline{M}_{2} = (M_{2} \ T (K_{2})^{∘})(A^{−1}(y_{1})), \overline{M}_{1} = (M_{1} \ T (K_{1})^{∘})(A(y_{2})),

where δ_{i}y_{i} := λ_{i} (i = 1, 2). Using (1) one has A^{−1}(y_{1}) = m_{2}, hence

\overline{M}_{2} = M_{2}.

Moreover, A(y_{2}) = A((o_{2}ℓ_{2} + k_{2}m_{2})/δ_{2}) = (o_{2}m_{1} + k_{2}λ_{1})/δ_{2}, hence:

\overline{M}_{1} = (M_{1} \ T (K_{1})^{∘})(μ), where μ := (o_{2}m_{1} + k_{2}λ_{1})/δ_{2}.

In fact, \overline{M}_{1} can be represented as a (p, q)-surgery of M_{1} along K_{1}. The integers (p, q) can be determined as in [21], page 8: μ is homologous to qK_{1} in T (K_{1}), hence q = k_{2}/δ_{2}. Moreover, p = Lk_{M_{1}}(μ, K_{1}), which via 2.3(1) equals ⟨(o_{2}m_{1} + k_{2}λ_{1})/δ_{2}, λ_{1}⟩ = o_{2}/δ_{2}. Therefore,

\overline{M}_{1} = M_{1}(K_{1}, p/q) = M_{1}(K_{1}, o_{2}/k_{2}).  (2)

3.3. Fujita’s splicing formula for the Casson-Walker invariant. Using the above expressions for the closures, (1.1) of [12], in the case of the above splicing (with A = f^{−1}), reads as

λ_{W} (M) = λ_{W} (M_{2}) + λ_{W} (M_{1}(K_{1}, o_{2}/k_{2})) + s(k_{2}, o_{2}).  (3)

Additionally, if we assume that K_{2} is homologically trivial in M_{2} (i.e. o_{2} = 1), and we fix ℓ_{2} as λ_{2} (i.e. k_{2} = 0), then (3) transforms into:

λ_{W} (M) = λ_{W} (M_{1}) + λ_{W} (M_{2}).  (4)

3.4. Walker-Lescop surgery formula. Now, we will analyze the manifold M_{1}(K_{1}, p/q) obtained by p/q-surgery, where p = o_{2}/δ_{2} > 0 and q = k_{2}/δ_{2} (not necessarily positive). First notice (cf. [21, 1.3.4]) that |H_{1}(M_{1}(K_{1}, p/q))| = p · |H_{1}(M_{1})|. Using this, the surgery formula (T2) from [21], page 13, and the identification λ_{W} (·) = 2λ(·)/|H_{1}(·, ℤ)|, one gets:

λ_{W} (M_{1}(K_{1}, p/q)) = λ_{W} (M_{1}) + Cor,  (5)

where the correction term Cor is

Cor := \frac{q}{p} · \frac{Δ_{M_{1}}^{L}(K_{1})′′(1)}{|H_{1}(M_{1})|} − \frac{p^{2} + 1 + q^{2}}{12pq} + sign(q)(\frac{1}{4} + s(p, q)).

Using (3), (5) and the reciprocity law of the Dedekind sums (for p > 0):

s(q, p) + sign(q)s(p, q) = −\frac{sign(q)}{4} + \frac{p^{2} + 1 + q^{2}}{12pq} ,

one gets the following formula:

3.5. Theorem. The splicing formula for Casson-Walker invariant. Consider a splicing manifold M characterized by the data described in 3.1. Then:

λ_{W} (M) = λ_{W} (M_{1}) + λ_{W} (M_{2}) + \frac{k_{2}}{o_{2}} · Δ_{M_{1}}^{♮}(K_{1})′′(1).

3.6. The splicing property of the group H_{1}(M, ℤ). In the next paragraphs we analyze the behavior of H_{1}(·, ℤ) under the splicing construction 3.1.

First notice that by excision, for any q, one has:

H_{q}(M, M_{2} \ T (K_{2})^{∘}) = H_{q}(M_{1} \ T (K_{1})^{∘}, ∂T (K_{1})) = H_{q}(M_{1}, K_{1}).

Therefore, the long exact sequence of the pair (M, M_{2} \ T (K_{2})^{∘}) reads as:

0 → H_{2}(M_{1}, K_{1}) ⟶^{∂_{1}} H_{1}(M_{2} \ T (K_{2})^{∘}) → H_{1}(M) → H_{1}(M_{1}, K_{1}) → 0.

Using the isomorphisms 2.3(3), ∂_{1} can be identified with ∂_{1}(1_{ℤ}) = m_{2}, hence coker∂_{1} = H_{1}(M_{2}), cf. 2.3(2). Therefore, 2.3(3) gives the exact sequence

0 → H_{1}(M_{2}) ⟶^{i} H_{1}(M) ⟶^{p} H_{1}(M_{1}) → 0.  (6)

This exact sequence splits. Indeed, let s̄ be the composition of s_{1} : H_{1}(M_{1}) → H_{1}(M_{1} \ T (K_{1})) (cf. 2.3) and H_{1}(M_{1} \ T (K_{1})) → H_{1}(M) (induced by the inclusion). Then p ∘ s̄ = 1. In particular,

H_{1}(M) = im(i) ⊕ im(s̄) ≈ H_{1}(M_{2}) × H_{1}(M_{1}).  (7)

Notice that any [K] ∈ H_{1}(M_{1}) can be represented (via s_{1}) by a representative K in M_{1} \ T (K_{1}) providing a class in T ors H_{1}(M_{1} \ T (K_{1})). Write o for its order, and take a Seifert surface F, sitting in M_{1} \ T (K_{1}) with ∂F = oK. If L ⊂ M_{1} \ T (K_{1}) with L ∩ K = ∅ then obviously Lk_{M}(K, L) = Lk_{M_{1}}(K, L). Moreover, since F has no intersection points with any curve L ∈ M_{2} \ T (K_{2}), for such an L one gets:

Lk_{M}(K, L) = 0, hence b_{M}(im(s̄), im(i)) = 0.  (8)

By a similar argument,

b_{M}(s̄(x), y) = b_{M_{1}}(x, p(y)) for any x ∈ H_{1}(M_{1}) and y ∈ H_{1}(M).  (9)

The point is that in the next sections we need Lk_{M}(K, ·) for general K ⊂ M_{1} \ K_{1} which is not a torsion element in H_{1}(M_{1} \ K_{1}).

To compute this linking number consider another oriented knot L ⊂ M_{1} \ K_{1} with K ∩ L = ∅. Our goal is to compare Lk_{M_{1}}(K, L) and Lk_{M}(K, L). Assume that the order of the class K in H_{1}(M_{1}) is o. Let F_{oK} be a Seifert surface in M_{1} with ∂F_{oK} = oK which intersects L and K_{1} transversally. It is clear that F_{oK} intersects L exactly in o · Lk_{M_{1}}(K, L) points (counted with sign). On the other hand, it intersects K_{1} in o · Lk_{M_{1}}(K, K_{1}) points. For each intersection point with sign ϵ = ±1, we cut out from F_{oK} the disc F_{oK} ∩ T (K_{1}), whose orientation depends on ϵ. Its boundary is ϵm_{1} which, by the splicing identification, corresponds to ϵℓ_{2} in M_{2} \ T (K_{2}). Using rational coefficients, ℓ_{2} = (1/o_{2})λ_{2} − (k_{2}/o_{2})m_{2}. Some multiple of λ_{2} can be extended to a surface in M_{2}\K_{2} which clearly has no intersection with L. m_{2} by splicing is identified with λ_{1} which has a Seifert surface in M_{1} \ K_{1} which intersect L in Lk_{M_{1}}(K_{1}, L) points. This shows that for K, L ⊂ M_{1} \ T (K_{1}):

Lk_{M}(K, L) = Lk_{M_{1}}(K, L) − Lk_{M_{1}}(K, K_{1}) · Lk_{M_{1}}(L, K_{1}) · k_{2}/o_{2}.  (10)

Assume now that K ∈ M_{2} \ T (K_{2}), and let o be the order of its homology class in H_{1}(M_{2}). Assume that the Seifert surface F of oK intersects K_{2} transversally in t_{2} points. Then F ∩ ∂T (K_{2}) = t_{2}m_{2}, and after the splicing identification this becomes t_{2}λ_{1}. Let F_{1} be the Seifert surface of λ_{1} in M_{1} \ T (K_{1})^{∘}. Then (after some natural identifications) F \ T (K_{2})^{∘} ∐ t_{2}F_{1} is a Seifert surface of oK in M. Therefore, for any L ⊂ M_{1} \ T (K_{1}) one has:

Lk_{M}(K, L) = Lk_{M_{2}}(K, K_{2})·Lk_{M_{1}}(K_{1}, L) for K ⊂ M_{2} \ T (K_{2}) and L ⊂ M_{1} \ T (K_{1}). (11)

By a similar argument:

Lk_{M}(K, L) = Lk_{M_{2}}(K, L) for K, L ⊂ M_{2} \ T (K_{2}).  (12)

3.7. Splicing plumbing manifolds. Some of the results of this article about Reidemeister-Turaev torsion can be formulated and proved in the context of general (rational homology sphere) 3-manifolds, and arbitrary spin^{c} structures. Nevertheless, in this article we are mainly interested in algebraic links, therefore we restrict ourselves to plumbed manifolds.

This can be formulated in the second working assumption:

WA2: (M_{i}, K_{i}) (i = 1, 2) can be represented by a negative definite plumbing graph. Moreover, if M is the result of the splicing (satisfying WA1, cf. 3.1), then M can also be represented by a negative definite plumbing graph.

Assume that the plumbing graphs Γ(M_{1}, K_{1}) and Γ(M_{2}, K_{2}) have the following schematic form (with g_{v_{1}} = K_{1} and g_{v_{2}} = K_{2}):

[figure: ⋮; v_{1}; ⋮; v_{2}]

Then it is not difficult to see that (a possible) plumbing graph Γ(M) for M has the following form (where v_{1} and v_{2} are connected by a string):

[figure: ⋮; …; v_{1}; ⋮; v_{2}]

If V_{i} (i = 1, 2), respectively V, represent the set of vertices of Γ(M_{i}), resp. of Γ(M), then V = V_{1} ∪ V_{2} modulo some vertices with δ = 2. (In particular, in any formula like 2.7(15), V behaves as the union V_{1} ∪ V_{2}.)

3.8. Remark. If one wants to compute 𝒯_{M,σ_{can}} for such plumbed manifolds, then one can apply 2.7(15) and (16). For this, one has to analyze the supports of the characters, and the corresponding weights w_{v}(u). These weights are closely related with the corresponding linking numbers Lk_{M}(g_{u}, g_{v}) (cf. 2.4 (5)), hence the relations 3.6(10)-(11)-(12) are crucial. For characters of type χ ∈ p̂(H_{1}(M_{1})ˆ ) (cf. 3.6(6) or the next proof) u ∈ V_{1}, hence 3.6(10) should be applied for any v ∈ V_{1}. But this is rather unpleasant due to the term Lk_{M_{1}}(g_{u}, K_{1}) · Lk_{M_{1}}(g_{v}, K_{1}) · k_{2}/o_{2}. The description is more transparent if either H_{1}(M_{1}) = 0 or k_{2} = 0.

Therefore, we will consider first these particular cases only. They, as guiding examples, already contain all the illuminating information and principles we need to proceed. For the link of {f(x, y)+z^{n} = 0}, the splicing formula will be made very explicit in 6.13 (based on a rather involved detailed and complete classification of the characters and the regularization terms P̂).

3.9. Theorem. Some splicing formulae for the Reidemeister-Turaev torsion. Assume that M satisfy WA1 (3.1) and WA2 (3.7). Then the following hold: (A) Assume that K_{2} ⊂ M_{2} is also homologically trivial (i.e. o_{2} = 1), and ℓ_{2} = λ_{2} (i.e. k_{2} = 0). Then

𝒯_{M,σ_{can}}(1) = 𝒯_{M_{1},σ_{can}}(1) + 𝒯_{M_{2},σ_{can}}(1).

(B) Assume that M_{1} is an integral homology sphere (i.e. H_{1}(M_{1}) = 0). Then

𝒯_{M,σ_{can}}(1) = ∑_{χ_{2}∈H_{1}(M_{2})ˆ \\{1\}} \frac{𝒯̂_{M_{2},σ_{can}}(χ̄_{2})}{|H_{1}(M_{2})|} · Δ_{M_{1}}(K_{1})(χ_{2}(K_{2})).

In particular, if K_{2} ⊂ M_{2} is homologically trivial, then χ_{2}(K_{2}) = 1 for any χ_{2}, hence by 2.6(9) one gets

𝒯_{M,σ_{can}}(1) = 𝒯_{M_{2},σ_{can}}(1) (and evidently 𝒯_{M_{1},σ_{can}}(1) = 0).

This is true for any choice of ℓ_{2}, i.e. even if k_{2} is non-zero.

Proof. The theorem is a consequence of 2.7(14)-(15)-(16) and 3.6. For this, we have to analyze the characters χ of H_{1}(M). The dual of the exact sequence 3.6(6) is

0 → H_{1}(M_{1})ˆ ⟶^{p̂} H_{1}(M)ˆ ⟶^{î} H_{1}(M_{2})ˆ → 0.

First, consider a character χ of H_{1}(M) of the form χ = p̂(χ_{1}) for some χ_{1} ∈ H_{1}(M_{1})ˆ . Since any χ_{1} ∈ H_{1}(M_{1})ˆ can be represented as b_{M_{1}}(x, ·) for some x ∈ H_{1}(M_{1}) (cf. 2.3), and p̂(b_{M_{1}}(x, ·)) = b_{M}(s̄(x), ·) (cf. 3.6(9)), property 3.6(8) guarantees that χ(g_{v}) = 1 for any v ∈ V_{2}. In particular, for χ = p̂(χ_{1}) with χ_{1} ∈ H_{1}(M_{1})ˆ \ {1}, and for some u ∈ V_{1} with χ_{1}(g_{u}) ≠ 1 (which works for M as well), one gets:

P̂_{M,χ,u}(t) = ∏_{v∈V_{1}} (t^{w_{v}(u)}χ_{1}(g_{v}) − 1)^{δ̄_{v}−2} · ∏_{v∈V_{2}} (t^{w_{v}(u)} − 1)^{δ̄_{v}−2}.

Here, for v ∈ V_{i}, δ̄_{v} means the number of adjacent edges of v in Γ(M_{i}, K_{i}) (i = 1, 2), cf. 2.4. By 3.6(11), for any v ∈ V_{2} one has

w_{v}(u) = o(u)Lk_{M}(g_{u}, g_{v}) = o(u)Lk_{M_{1}}(g_{u}, g_{v_{1}}) · Lk_{M_{2}}(g_{v_{2}}, g_{v}),

hence by 2.6(8)

P̂_{M,χ,u}(t) = P̂_{M_{1},χ_{1},u}(t) · Δ_{M_{2}}(g_{v_{2}})(t^{o(u)Lk_{M_{1}}(g_{u},g_{v_{1}})}).

Taking the limit t → 1, and using 2.6(9), one gets:

𝒯̂_{M,σ_{can}}(χ̄) = 𝒯̂_{M_{1},σ_{can}}(χ̄_{1}) · |H_{1}(M_{2})|.  (13)

Now, we prove (A). In this case M_{1} and M_{2} are symmetric, hence there is a similar term as in (13) for characters χ_{2} ∈ H_{1}(M_{2})ˆ .

On the other hand, if χ = χ_{1}χ_{2} for two non-trivial characters χ_{i} ∈ H_{1}(M_{i})ˆ (i = 1, 2), then one can show that P̂_{M,χ,u}(t) has a root (of multiplicity at least two) at t = 1, hence 𝒯̂_{M,σ_{can}}(χ̄) = 0. Now, use 2.7(14) and |H_{1}(M)| = |H_{1}(M_{1})| · |H_{1}(M_{2})| (cf. 3.6(7)).

For (B), fix a non-trivial character χ_{2} ∈ H_{1}(M_{2})ˆ . The relations in 3.6 guarantee that if we take u ∈ V_{2} with χ_{2}(g_{u}) ≠ 1 (considered as a property of M_{2}) then χ(g_{u}) ≠ 1 as well for χ = î^{−1}(χ_{2}). Moreover, for any v ∈ V_{1} one has: χ(g_{v}) = χ_{2}(K_{2})^{Lk_{M_{1}}(K_{1},g_{v})} (cf. 3.6(11)). Therefore:

P̂_{M,χ,u}(t) = ∏_{v∈V_{1}} (t^{w_{v}(u)}χ(g_{v}) − 1)^{δ̄_{v}−2} · ∏_{v∈V_{2}} (t^{w_{v}(u)}χ(g_{v}) − 1)^{δ̄_{v}−2}

= P̂_{M_{2},χ_{2},u}(t) · Δ_{M_{1}}(g_{v_{1}})(t^{o(u)Lk_{M_{2}}(g_{u},g_{v_{2}})}χ_{2}(K_{2})). □

3.10. Remarks. (1) A similar proof provides the following formula as well (which will be not used later). Assume that M satisfies WA1 and WA2, and k_{2} = 0. Then

𝒯_{M,σ_{can}}(1) = 𝒯_{M_{1},σ_{can}}(1) + ∑_{χ_{2}∈H_{1}(M_{2})ˆ \\{1\}} \frac{𝒯̂_{M_{2},σ_{can}}(χ̄_{2})}{|H_{1}(M_{2})|} · Δ_{M_{1}}^{H}(K_{1})(χ_{2}(K_{2})).

(2) The obstruction term (see also 4.6(1)) which measures the non-additivity of the Casson-Walker invariant (under the splicing assumption WA1) is given by \frac{k_{2}}{o_{2}} ·Δ_{M_{1}}^{♮}(K_{1})′′(1) (cf. 3.5). On the other hand, if H_{1}(M_{1}) = 0, then the obstruction term for the nonadditivity of the Reidemeister-Turaev torsion (associated with σ_{can}) is

∑_{χ_{2}∈H_{1}(M_{2})ˆ \\{1\}} \frac{𝒯̂_{M_{2},σ_{can}}(χ̄_{2})}{|H_{1}(M_{2})|} · ( Δ_{M_{1}}(K_{1})(χ_{2}(K_{2})) − 1).

Notice that they “look rather different” (even under this extra-assumption H_{1}(M_{1}) = 0). In fact, even their nature are different: the first depends essentially on the choice of the parallel ℓ_{2} (see the coefficient k_{2} in its expression), while the second not. In particular, one cannot really hope (in general) for the additivity of the modified Seiberg-Witten invariant.

Therefore, it is really remarkable and surprising, that in some of the geometric situations discussed in the next sections, the modified Seiberg-Witten invariant is additive (though the invariants λ_{W} and 𝒯_{σ_{can}}(1) are not additive, their obstruction terms cancel each other).

4. The basic topological example

4.1. Recall that in our main applications (for algebraic singularities) the involved 3manifolds are plumbed manifolds. In particular, they can be constructed inductively from Seifert manifolds by splicing (cf. also with 2.5). The present section has a double role. First, we work out explicitly the splicing results obtained in the previous section for the case when M_{2} is a Seifert manifold (with special Seifert invariants). On the other hand, the detailed study of this splicing formulae provides us a better understanding of the subtlety of the behavior of the (modified) Seiberg-Witten invariant with respect to splicing and cyclic covers. They will be formulated in some “almost–additivity” properties, where the non-additivity will be characterized by a new invariant D constructed from the Alexander polynomial of (M_{1}, K_{1}).

4.2. The splicing component M_{2}. Assume that M_{2} is the link

Σ = Σ(p, a, n) := {(x, y, z) ∈ ℂ^{3} : x^{p} + y^{a} + z^{n} = 0, |x|^{2} + |y|^{2} + |z|^{2} = 1}

of the Brieskorn hypersurface singularity

X_{2} := {(x, y, z) ∈ ℂ^{3} : x^{p} + y^{a} + z^{n} = 0},

where gcd(n, a) = 1 and gcd(p, a) = 1. Set d :=gcd(n, p). Then M_{2} is a rational homology sphere with the following data (for more details, and for a more complete list of the relevant invariants, see e.g. [31], section 6; cf also with [15, 35]).

(a) The Seifert invariants are:

n/d, p/d, \underbrace{a, a, ⋯, a}_{d},

hence all together there are d + 2 special fibers. These numbers also give (up to a sign) the determinants of the corresponding arms of the plumbing graph of Σ;

(b) The orbifold Euler characteristic is e = −d^{2}/(npa).

(c) H_{1}(M_{2}) = ℤ_{a}^{d−1} (cf. also with 4.3(1) below).

(e) 𝒯_{M_{2},σ_{can}}(1) = \frac{np}{24da}(d − 1)(a^{2} − 1) (see also (∗) in the proof of 4.8).

4.3. The link K_{2}. In this section we will modify slightly the construction of 3.1: we will fix a link with d connected components (instead of a knot), and we will perform splicing along each connected component.

This link K_{2} ⊂ M_{2} is given by the equation {y = 0} in M_{2}. In other words, K_{2} is the union of the d special (Seifert) fibers corresponding to the Seifert invariant a. The components of K_{2} will be denoted by K_{2}^{(i)} (1 = 1, …, d), and their tubular neighborhood by T (K_{2}^{(i)}) as in 2.1.

The link-components {K_{2}^{(i)}}_{i} generate H_{1}(M_{2}) (see, e.g. [31], section 6); in fact H_{1}(M_{2}) has the following presentation (written additively):

H_{1}(M_{2}) = ⟨[K_{2}^{(1)}], …, [K_{2}^{(d)}] : a[K_{2}^{(i)}] for each i, and [K_{2}^{(1)}] + ⋯ + [K_{2}^{(d)}] = 0⟩. (1)

Besides K_{2}, there are two more special orbits in M_{2} = Σ, namely Z := {z = 0} and X := {x = 0}. Moreover, let O be the generic fiber of the Seifert fibration of Σ (i.e. g_{v} associated with the central vertex v). Then one has the following linking numbers (use 2.4(5) and [31] (5.5)(1)):

(a) Lk_{M_{2}}(K_{2}^{(i)}, K_{2}^{(j)}) = np/(d^{2}a)  for any i ≠ j;
(b) Lk_{M_{2}}(K_{2}^{(i)}, O) = np/d^{2}; Lk_{M_{2}}(K_{2}^{(i)}, Z) = p/d  for any i;
(c) Lk_{M_{2}}(O, O) = npa/d^{2} and Lk_{M_{2}}(O, Z) = pa/d;
(d) Lk_{M_{2}}(X, Z) = a.

(L#)

Notice that K_{2} ⊂ M_{2} is fibrable. Indeed, K_{2} = {y = 0} is the link associated with the algebraic germ y : (X_{2}, 0) → (ℂ, 0), hence one can take its Milnor fibration. Let F be the fiber with ∂F = K_{2} (equivalently, take any minimal Seifert surface F with ∂F = K_{2}). Then for each i = 1, …, d, we define the parallel ℓ_{2}^{(i)} in ∂T (K_{2}^{(i)}) by F ∩ ∂T (K_{2}^{(i)}).

Let λ_{2}^{(i)} be the longitude of K_{2}^{(i)} ⊂ M_{2}, and consider the invariants o_{2}^{(i)}, k_{2}^{(i)}, etc. as in 2.1 with the corresponding sub- and superscripts added.

4.4. Lemma. For each i = 1, …, d one has:

o_{2}^{(i)} = a and k_{2}^{(i)} = \frac{np(d − 1)}{d^{2}}.

Proof. The first identity is clear (cf. 4.3(1)). For the second, notice that (cf. 2.3(1))

−k_{2}^{(i)}/o_{2}^{(i)} = ⟨ℓ_{2}^{(i)}, λ_{2}^{(i)}⟩/o_{2}^{(i)} = Lk_{M_{2}}(ℓ_{2}^{(i)}, K_{2}^{(i)}).

Moreover,

Lk_{M_{2}}(∑_{j} ℓ_{2}^{(j)}, K_{2}^{(i)}) = 0,

hence

k_{2}^{(i)}/a = ∑_{j≠i} Lk_{M_{2}}(K_{2}^{(j)}, K_{2}^{(i)}).

Then use 4.3(L#a). □

4.5. The manifold M. Next, we consider d manifolds M_{1}^{(i)} with knots K_{1}^{(i)} ⊂ M_{1}^{(i)} (i = 1, …, d), each satisfying the assumption WA1 (i.e. o_{1}^{(i)} = 1, ℓ_{1}^{(i)} = λ_{1}^{(i)}, and k_{1}^{(i)} = 0, cf. 3.1). Then, for each i = 1, …, d, we consider the splicing identification of ∂T (K_{2}^{(i)}) with −∂T (K_{1}^{(i)}) (similarly as in 3.1(1)):

A^{(i)}(m_{2}^{(i)}) = λ_{1}^{(i)} and A^{(i)}(ℓ_{2}^{(i)}) = m_{1}^{(i)}.

Schematically:

[figure: …; ⋮; ⋮; ⋮; ·; ·; ·; ·; K_{2}^{(1)}; K_{2}^{(d)}; K_{1}^{(1)}; K_{1}^{(d)}]

In the sequel we denote by WA1’ the assumption which guarantees that the manifold M is constructed by this splicing procedure. Moreover, WA2’ guarantees that all the involved 3-manifolds have negative definite plumbing representations.

Here some comments are in order.

(1) Assume that for some i, M_{1}^{(i)} = S^{3}, and K_{1}^{(i)} is the unknot S^{1} in S^{3}. Then performing splicing along K_{2}^{(i)} with (S^{3}, S^{1}) is equivalent to put back T (K_{2}^{(i)}) unmodified, hence it has no effect. In this case, one also has Δ_{S^{3}}(S^{1})(t) ≡ Δ_{S^{3}}^{♮}(S^{1})(t) ≡ 1.

(2) Assume that we already had performed the splicing along the link-components K_{2}^{(i)} for i ≤ k − 1, but not along the other ones. Let us denote the result of this partial modification by M^{(k−1)}. Consider K_{2}^{(k)} in M^{(k−1)} (in a natural way). Then all the invariants (e.g. o_{2}^{(k)}, λ_{2}^{(k)}, k_{2}^{(k)}, etc.) associated with K_{2}^{(k)} in M_{2} or in M^{(k−1)} are the same. (This follows from the discussion in 3.6, and basically, it is a consequence of WA1’.)

In particular, performing splicing at place i does not effect the splicing data of the place j (j ≠ i). Therefore, using induction, the computation of the invariants can be reduced easily to the formulae established in the previous section.

4.6. Definitions/Notations.

(1) In order to simplify the exposition, for any 3-manifold invariant I, we write

O(I) := I(M) − I(M_{2}) − ∑_{i=1}^{d} I(M_{1}^{(i)}),

for the “additivity obstruction” of I (with respect to the splicing construction WA1’). E.g., using 3.6 and 4.5(2) one has O(log |H_{1}(·)|) = 0. Moreover, in all our Alexander invariant notations (e.g. in Δ_{M_{1}^{(i)}}(K_{1}^{(i)})(t)), we omit the link K_{1}^{(i)} (e.g. we simply write Δ_{M_{1}^{(i)}}(t)).

When we compare O(𝒯_{·,σ_{can}}) with O(λ_{W} (·)), the next terminology will be helpful.

(2) For any set of integers c_{1}, …, c_{r} define

D(c_{1}, …, c_{r}) := ∑_{i,j=1}^{r} c_{i}c_{j} min(i, j) − ∑_{i=1}^{r} i c_{i}.

(3) Define D(Δ^{♮}(t)) by D(c_{1}, …, c_{r}) for any symmetric polynomial

Δ^{♮}(t) = 1 + ∑_{i=1}^{r} c_{i}(t^{i} + t^{−i} − 2) (for some c_{i} ∈ ℤ).

(4) A set {c_{i}}_{i∈I} (I ⊂ ℕ) is called alternating if c_{i} ∈ {−1, 0, +1} for any i ∈ I; and if c_{i} ≠ 0 then c_{i} = (−1)^{n_{i}}, where n_{i} = #{j : j > i, and c_{j} ≠ 0}.

4.7. Corollary. Assume that M satisfies WA1’. Then

O(λ_{W}) = \frac{np(d − 1)}{ad^{2}} · ∑_{i=1}^{d} (Δ_{M_{1}^{(i)}}^{♮})′′(1).

Proof. Use 3.5, 4.4 and 4.5(2). □

4.8. Corollary. Assume that M satisfies WA1’ and WA2’, and M_{1}^{(i)} is an integral homology sphere for any i. Identify ℤ_{a} := {ξ ∈ ℂ : ξ^{a} = 1} and write ℤ_{a}^{∗} := ℤ_{a} \ {1}. Then

O(𝒯_{·,σ_{can}}(1)) = \frac{np}{ad^{2}} ∑_{i≠j} ∑_{ξ∈ℤ_{a}^{∗}} \frac{Δ_{M_{1}^{(i)}}(ξ) · Δ_{M_{1}^{(j)}}(ξ̄) − 1}{(ξ − 1)(ξ̄− 1)}.

Proof. We recall first how one computes the torsion for the manifold M_{2} (for a detailed presentation, see [31]). The point is (cf. also with the last sentence of 2.7), that for any character χ ∈ H_{1}(M_{2})ˆ \ {1}, one can choose the central vertex of the star-shaped graph for the vertex u in order to generate the weights in P̂. Then, by [31] or 4.3(L#), one gets:

P̂_{M_{2},χ,u}(t) = \frac{(t^{α} − 1)^{d}}{(t^{dα/n} − 1)(t^{dα/p} − 1) ∏_{i} (t^{α/a}χ(K_{2}^{(i)}) − 1)},

where α := npa/d^{2}. The limit of this expression, as t → 1, always exists. In particular #{i : χ(K_{2}^{(i)}) ≠ 1} ≥ 2 (cf. also with 4.3(1)). If this number is strict greater than 2, then the limit is zero. If χ(K_{2}^{(i)}) ≠ 1 exactly for two indices i and j, then using 4.3(1) clearly χ(K_{2}^{(i)})χ(K_{2}^{(j)}) = 1. Since there are exactly d(d − 1)/2 such pairs, one gets:

𝒯_{M_{2},σ_{can}}(1) = \frac{1}{|H_{1}(M_{2})|} · lim_{t→0} \frac{(t^{α} − 1)^{d}}{(t^{dα/n} − 1)(t^{dα/p} − 1)(t^{α/a} − 1)^{d−2}} · ∑_{i≠j} ∑_{ξ∈ℤ_{a}^{∗}} \frac{1}{(ξ − 1)(ξ̄− 1)}

= \frac{np}{ad^{2}} ∑_{i≠j} ∑_{ξ∈ℤ_{a}^{∗}} \frac{1}{(ξ − 1)(ξ̄− 1)} = \frac{np}{ad^{2}} \frac{d(d − 1)}{2} ∑_{ξ∈ℤ_{a}^{∗}} \frac{1}{(ξ − 1)(ξ̄− 1)} = \frac{np(d − 1)(a^{2} − 1)}{24ad}, (∗)

where the last equality follows from the identity

∑_{ξ∈ℤ_{a}^{∗}} \frac{1}{(ξ − 1)(ξ̄− 1)} = \frac{a^{2} − 1}{12}.  (∗∗)

Consider now the manifold M. Then using 3.9(B) (and/or its proof), by the same argument as above, one obtains:

𝒯_{M,σ_{can}}(1) = \frac{np}{ad^{2}} ∑_{i≠j} ∑_{ξ∈ℤ_{a}^{∗}} \frac{Δ_{M_{1}^{(i)}}(ξ) · Δ_{M_{1}^{(j)}}(ξ̄)}{(ξ − 1)(ξ̄− 1)}.  (∗ ∗ ∗)

Finally, to obtain the result, subtract (∗) from (∗ ∗ ∗). □

4.9. Example/Discussion. Assume that M satisfies WA1’ and WA2’, and additionally (M_{1}^{(i)}, K_{1}^{(i)}) = (M_{1}, K_{1}) for some integral homology sphere M_{1}. Then

O(λ_{W})/2 = \frac{np(d − 1)}{2ad} · (Δ_{M_{1}}^{♮})′′(1);

O(𝒯_{·,σ_{can}}(1)) = \frac{np(d − 1)}{2ad} · ∑_{ξ∈ℤ_{a}^{∗}} \frac{Δ_{M_{1}}(ξ) · Δ_{M_{1}}(ξ̄) − 1}{(ξ − 1)(ξ̄− 1)}.

Recall that the modified Seiberg-Witten invariants sw_{M}^{0}(σ_{can}) is defined by the difference 𝒯_{M,σ_{can}}(1) − λ_{W} (M)/2 (cf. 2.7(13)). Notice the remarkable fact that in the above expressions the coefficients before the Alexander invariants became the same. Hence

O(sw_{·}^{0}(σ_{can})) = \frac{np(d − 1)}{2ad}D_{a}, where D_{a} := ∑_{ξ∈ℤ_{a}^{∗}} \frac{Δ_{M_{1}}(ξ) · Δ_{M_{1}}(ξ̄) − 1}{(ξ − 1)(ξ̄− 1)} −(Δ_{M_{1}}^{♮})′′(1). (D)

Recall that Δ_{M_{1}}^{♮}(t) is a symmetric polynomial (cf. [21, 2.3.1]) with Δ_{M_{1}}^{♮}(1) = 1 (cf. 2.6(6)). In the sequel we will compute explicitly D_{a}, provided that a is sufficiently large, in terms of the coefficients {c_{i}}_{i=1}^{r} of Δ_{M_{1}}^{♮}(t) (cf. 4.6(3)).

The contribution (Δ_{M_{1}}^{♮})′′(1) is easy: it is ∑_{i=1}^{r} 2i^{2} c_{i}. By 2.6(9), det(V) = 1, hence by 2.6(7), Δ_{M_{1}}(t) = t^{r} · Δ_{M_{1}}^{♮}(t). In particular, Δ_{M_{1}}(ξ) · Δ_{M_{1}}(ξ̄) = Δ_{M_{1}}^{♮}(ξ) · Δ_{M_{1}}^{♮}(ξ̄). Then write

\frac{Δ_{M_{1}}^{♮}(t)}{1 − t} = \frac{1}{1 − t} − ∑_{i=1}^{r} c_{i}(1 + t + ⋯ + t^{i−1}) + ∑_{i=1}^{r} c_{i}(t^{−1} + ⋯ t^{−i}).

Using the identity ∑_{ξ∈ℤ_{a}^{∗}} 1/(1 − ξ) = (a − 1)/2, an elementary computation gives

∑_{ξ∈ℤ_{a}^{∗}} \frac{Δ_{M_{1}}^{♮}(ξ) · Δ_{M_{1}}^{♮}(ξ̄) − 1}{(1 − ξ)(1 − ξ̄)} = ∑_{i=1}^{r} 2i^{2} c_{i} + 2a · D(Δ_{M_{1}}^{♮}(t)), provided that a ≥ 2r.

In particular, if a ≥ 2r, then D_{a} = 2a · D(Δ_{M_{1}}^{♮}(t)). Hence

O(sw_{·}^{0}(σ_{can})) = \frac{np(d − 1)}{d} · D(Δ_{M_{1}}^{♮}(t)).

This raises the following natural question: for what Alexander polynomials the expression D(Δ_{M_{1}}^{♮}(t)) is zero? The next lemma provides such an example (the proof is elementary and it is left to the reader).

4.10. Lemma. If {c_{i}}_{i=1}^{r} is an alternating set then D(c_{1}, …, c_{r}) = 0.

The above discussions also have the following topological consequence.

Start with an integral homology sphere M and an oriented knot K ⊂ M. Fix two relative prime positive integers p, a, and let L be the primitive simple curve in ∂T (K) with homology class am + pλ. Fix a positive integer n. Now define,

• d := gcd(n, p);

• M̃ - the n-cyclic cover of M branched along L;

• M̂ - the (n/d)-cyclic cover of M branched along K; K̂ ⊂ M̂ the pre-image of K in M̂;

• Δ♮ ; M̂(t) - the normalized Alexander polynomial of (M̂, K̂).

4.11. Corollary. Consider the above data. Additionally, assume that WA2’ is satisfied and M̃ is a rational homology sphere. Then: (A) (d − 1) · (gcd(n, a) − 1) = 0. (B) If d = 1, then

sw0 ; M̃(σ_{can}) = sw0 ; M̂(σ_{can}) + sw_{Σ(p,a,n)}^{0}(σ_{can}).

(C) If gcd(n, a) = 1, a ≥ deg ΔM̂(t), and M̂ is an integral homology sphere, then

sw0 ; M̃(σ_{can}) = d · sw0 ; M̂(σ_{can}) + sw_{Σ(p,a,n)}^{0}(σ_{can}) + \frac{np(d − 1)}{d} · D(Δ♮ ; M̂(t)).

If the coefficients of Δ♮ ; M̂(t) form an alternating set, then D(Δ♮ ; M̂(t)) = 0.

Proof. Consider the following schematic splicing of splice diagrams (cf. 2.5):

[figure: M; K; K_{2}; L; K′_{2}; a; 1; p]

The result of the splicing can be identified with M (and under this identification K is identified with K′_{2}). The advantage of this splicing representation is that it emphasizes the position of the knot L in the Seifert component Σ(p, a, 1). If M̃ is a rational homology sphere then the n-cyclic cover of Σ(p, a, 1) branched along L (which is Σ(p, a, n)) should be rational homology sphere, hence (A) follows. If d = 1 then M̃ has a splice decomposition of the following schematic plumbing diagrams (where at the right M_{2} = Σ(p, a, n) and the dots mean gcd(n, a) arms):

[figure: M̂; K̂; K_{2}; ⋯]

Here o = o_{2} = 1 and k = k_{2} = 0, and A is the identification λ_{2} = m, m_{2} = λ. Therefore, part (B) follows from 3.3(4) and 3.9(A). The last case corresponds exactly to the situation treated in 4.9. □

4.12. Remarks. (1) Our final goal (see the following sections) is to prove the additivity result O(sw^{0}(σ_{can})) = 0 for any (M̂, K̂), which can be represented as a cyclic cover of S^{3} branched along the link K_{f} ⊂ S^{3} of an arbitrary irreducible (complex) plane curve singularity (even if M̂ is not an integral homology sphere), provided that a is sufficiently large. This means that from the above Corollary, part (C), we will need to eliminate the assumption about the vanishing of H_{1}(M̂). The assumption about a will follow from the special property 5.1(6)) of irreducible plane curve singularities (cf. also with 5.4, especially with (7)).

The proof of the vanishing of the D-correction will take up most of the last section of the paper. It relies in a crucial manner on the alternating nature of the Alexander polynomial Δ_{S^{3}}^{♮}(K_{f})(t) of any irreducible plane curve singularity f, fact which will be establish in Proposition 5.2.

(2) It is really interesting and remarkable, that the behavior of the modified Seiberg-Witten invariant with respect to (some) splicing and cyclic covers (constructions, which basically are topological in nature) definitely gives preference to the Alexander polynomials of some algebraic links. The authors hope that a better understanding of this phenomenon would lead to some deep properties of the Seiberg-Witten invariant.

4.13. Example. In general, in 4.11, the invariant D(Δ♮ ; M̂(K̂)(t)) does not vanish. In order

to see this, start for example with a pair (M, K) with non-zero D(Δ_{M}^{♮}(K)(t)), and consider the case when d|n. (If d ≠ 1, then the coefficient of D(Δ_{M}^{♮}(K)(t)) in 4.11(C) will be non-zero as well.)

Let us show now how one can construct such a pair (M, K) which satisfies WA2, H_{1}(M) = 0 but D(Δ_{M}^{♮}(K)(t)) ≠ 0. Notice first the following fact.

If the Alexander polynomial Δ_{M}^{♮}(K)(t) is realizable for some pair (M, K) (satisfying WA2 and H_{1}(M) = 0), then the k-power of this polynomial is also realizable for some pair (M^{k}, K^{k}) (satisfying WA2 and H_{1}(M^{k}) = 0). Indeed, assume that (M, K) has a schematic splice diagram of the following form:

[figure: ⋮; c; K; Γ]

Then let (M^{k}, K^{k}) be given by the following schematic splice diagram:

[figure: ⋮; q; Γ; ⋮; q; Γ; ⋮; ⋮; (k copies); 1; 1; 1; K^{k}]

Here, we take q sufficiently large (in order to assure that the new edges will also satisfy the algebraicity condition [10](9.4)), and also q should be relative prime with some integers which appear as decorations of Γ (see [loc. cit.]). Obviously, by construction, M^{k} is an integral homology sphere. Then, by [loc. cit.] 12.1, one can easily verify that

Δ_{M^{k}}^{♮}(K^{k})(t) = Δ_{M}^{♮}(K)(t)^{k}.

For example, if (M, K) = (S^{3}, K_{f}), where K_{f} is the (2, 3)-torus knot (or, equivalently, the knot of the plane curve singularity f = x^{2} + y^{3}, cf. 5.1), then Δ_{S^{3}}^{♮}(K_{f})(t) = t − 1 + 1/t (see 5.1(5)). Now, if we take k = 2 and q = 7 then (M^{2}, K^{2}) has the following splice, respectively plumbing graph:

[figure: 2; 7; 1; 1; 7; 2; 3; 1; 3; −2; −1; −13; −1; −2; −3; −3]

Then (M^{2}, K^{2}) is algebraic, H_{1}(M^{2}) = 0. But Δ_{M^{2}}^{♮}(K^{2})(t) = (t−1+1/t)^{2} whose coefficients are not alternating. In fact, r = 2, c_{1} = −2 and c_{2} = 1; in particular D(Δ_{M^{2}}^{♮}(K^{2})(t)) = 2.

We end this section with the following property which is needed in the last section.

4.14. Lemma. Assume that M satisfies WA1’ and WA2’ with (M_{1}^{(i)}, K_{1}^{(i)}) = (M_{1}, K_{1}). Let Γ denote the plumbing graph of M. Let v be the central vertex of M_{2} considered in M, and let Γ_{−} be that connected component of Γ \ {v} which contains the vertices of M_{1}^{(1)}. Then | det(Γ_{−})| = a · |H_{1}(M_{1})|.

Proof. If I denotes the intersection matrix of M, then

−I_{vv}^{−1} =^{2.4(5)} Lk_{M}(g_{v}, g_{v}) =^{3.6(12)} Lk_{Σ(p,a,n)}(O, O) =^{4.3(L#c)} npa/d^{2}.

On the other hand, I_{vv}^{−1} can be computed from the determinants of the components of Γ \ {v}, hence (cf. also with 4.2(a))

−|H_{1}(M)| · I_{vv}^{−1} = | det(Γ_{−})|^{d} · pn/d^{2}.

By 3.6(7) and 4.2(e) |H_{1}(M)| = |H_{1}(M_{1})|^{d} · a^{d−1}, hence the result follows. □

5. Properties of irreducible plane curve singularities

5.1. The topology of an irreducible plane curve singularity. Consider an irreducible plane curve singularity f : (ℂ^{2}, 0) → (ℂ, 0) with Newton pairs {(p_{k}, q_{k})}_{k=1}^{s} (cf. [10], page 49). Clearly gcd(p_{k}, q_{k}) = 1 and p_{k} ≥ 2 and q_{k} ≥ 2. Define the integers {a_{k}}_{k=1}^{s} by

a_{1} = q_{1} and a_{k+1} = q_{k+1} + p_{k+1}p_{k}a_{k} if k ≥ 1.  (1)

Then again, gcd(p_{k}, a_{k}) = 1 for any k. The minimal (good) embedded resolution graph of the pair (ℂ^{2}, {f = 0}) has the following schematic form:

[figure: v̄_{0}; v_{1}; v_{2}; v_{s−1}; v_{s}; v̄_{1}; v̄_{2}; v̄_{s−1}; v̄_{s}; K_{f}; ⋯]

This can be identified with the plumbing graph Γ(S^{3}, K_{f}), where K_{f} is the link of f (with only one component) in the Milnor sphere S^{3}. In the above diagram we emphasized only those vertices {v̄_{k}}_{k=0}^{s} and {v_{k}}_{k=1}^{s} which have δ̄ ≠ 2. We denote the set of these vertices by V^{∗}. The dash-line between two such vertices replaces a string [figure: ⋯] . In our discussion the corresponding self-intersection (or Euler) numbers will be not important (the interested reader can find the complete description of the graph in [10] section 22, or in [30]). The above numerical data {(p_{k}, a_{k})}_{k} and the set of vertices V^{∗} is codified in the splice diagram (cf. [10]):

[figure: K_{f}; a_{1}; a_{2}; a_{s−1}; a_{s}; 1; 1; 1; 1; p_{1}; p_{2}; p_{s−1}; p_{s}; ⋯]

The knot K_{f} ⊂ S^{3} defines a set of weights {w_{v}(u)}_{v∈V^{∗}} as in 2.4 (where K_{f} = g_{u}, and evidently o(u) = 1). In terms of the resolution, w_{v}(u) is exactly the vanishing order (multiplicity) of f ∘ π along the exceptional divisor codified by v, where π denoted the resolution map. In the sequel we write just w_{v} for it. Then (cf. [10], section 10) one has:

\begin{matrix}w_{v_{k}} = a_{k}p_{k}p_{k+1} ⋯ p_{s} & for 1 ≤ k ≤ s; \\ wv̄_{0} = p_{1}p_{2} ⋯ p_{s}; \\ wv̄_{k} = a_{k}p_{k+1} ⋯ p_{s} & for 1 ≤ k ≤ s. \end{matrix}  (2)

Recall that the characteristic polynomial Δ(f)(t) := Δ_{S^{3}}(K_{f})(t) of the monodromy acting on the first homology of the Milnor fiber of f is given by A’Campo’s formula 2.6(8):

\frac{Δ_{S^{3}}(K_{f})(t)}{t − 1} = ∏_{v∈V^{∗}} (t^{w_{v}} − 1)^{δ̄_{v}−2}.  (3)

In inductive proofs and constructions (over the number of Newton pairs of f), it is convenient to use the notation f_{(l)} for an irreducible plane curve singularity with Newton pairs {(p_{k}, q_{k})}_{k=1}^{l}, where 1 ≤ l ≤ s. Evidently f_{(s)} = f, and f_{(1)} can be taken as the Brieskorn singularity x^{p_{1}} + y^{a_{1}}. We write Δ(f_{(l)}) for the characteristic polynomial associated with f_{(l)}. Then from (2) and (3) one gets

Δ(f_{(l)})(t) = Δ(x^{p_{l}} + y^{a_{l}})(t) · Δ(f_{(l−1)})(t^{p_{l}}) for l ≥ 2,  (4)

where

Δ(x^{p} + y^{a})(t) = \frac{(t^{pa} − 1)(t − 1)}{(t^{p} − 1)(t^{a} − 1)}.  (5)

By induction, using the identities (1), one can prove (see e.g. [29, 5.2])

a_{l} > p_{l} · deg Δ(f_{(l−1)}) for any l ≥ 2.  (6)

5.2. Proposition. (a) Δ(f)(0) = Δ(f)(1) = 1, and the degree of Δ(f)(t) is even (say 2r). (b) If Δ(f)(t) = ∑_{i=0}^{2r} b_{i}t^{i}, then the set {b_{i}}_{i=0}^{2r} is alternating (cf. 4.6(4)).

(c) The coefficients {c_{i}}_{i=1}^{r} of Δ^{♮}(f)(t) := t^{−r}Δ(f)(t) (cf. 4.6(3)) are alternating as well.

Proof. (a) is clear from (4) and (5), and (c) follows easily from (b). We will prove (b) by induction over s. For each 1 ≤ l ≤ s we verify that there exist

(i) a_{l}-residue classes {r_{1}, …, r_{t}} ⊂ {1, 2, …, a_{l} − 1} (where t may depend on l); and (ii) integers n_{1}, …, n_{t} ∈ ℕ, such that

Δ(f_{(l)})(t) = 1 + ∑_{i=1}^{t} ∑_{j=0}^{n_{i}} t^{r_{i}+ja_{l}} · (t − 1).

It is clear that the coefficients of a polynomial of this form are alternating.

Let us start with the case l = 1. Write (p_{1}, a_{1}) = (p, a). Then (cf. (5))

Δ(x^{p} + y^{a})(t) = (t^{p(a−1)} + ⋯ + t^{p} + 1)/Q(t), Q(t) := t^{a−1} + ⋯ + t + 1.

For each i = 0, 1, …, a − 1 write pi in the form x_{i}a + r_{i} for some r_{i} ∈ {0, …, a − 1}. Since gcd(p, a) = 1, {r_{i}}_{i} = {0, …, a − 1}, and p|r_{i} if and only if x_{i} = 0. Therefore,

∑_{i=0}^{a−1} t^{pi} = Q(t) + ∑_{i: p ∤ r_{i}} t^{r_{i}}(t^{x_{i}a} − 1) = Q(t) · [1 + ∑_{i: p ∤ r_{i}} ∑_{j=0}^{x_{i}−1} t^{r_{i}+ja}(t − 1)].

Now we prove that Δ(f_{(l)}(t) has a similar form. By the inductive step, assume that

Δ(f_{(l−1)})(t) = 1 + ∑_{i=1}^{t} ∑_{j=0}^{n_{i}} t^{r_{i}+ja_{l−1}}(t − 1).

Then, using 5.1(4) and (5), one gets for Δ(f_{(l)})(t):

\frac{(t^{p_{l}a_{l}} − 1)(t − 1)}{(t^{p_{l}} − 1)(t^{a_{l}} − 1)} + ∑_{i=1}^{t} ∑_{j=0}^{n_{i}} t^{(r_{i}+ja_{l−1})p_{l}} · \frac{(t^{p_{l}a_{l}} − 1)(t − 1)}{t^{a_{l}} − 1}.

Let {s_{j}}_{j=0}^{a_{l}−1} be the set of a_{l}-residues classes. Then (using the result of case l = 1) the above expression reads as

1 + ∑_{j: p_{l} ∤ s_{j}} ∑_{k=0}^{x_{j}−1} t^{s_{j}+ka_{l}}(t − 1) + ∑_{i=1}^{t} ∑_{j=0}^{n_{i}} ∑_{k=0}^{p_{l}−1} t^{(r_{i}+ja_{l−1})p_{l}+ka_{l}} · (t − 1).

Notice that 5.1(6) guarantees that for each i and j one has the inequality (r_{i} +ja_{l−1})p_{l} < a_{l}, hence these numbers can be considered as (non-zero) a_{l}-residues classes. Moreover, they are all different from the residue classes {s_{j} : p_{l} ∤ s_{j}} since they are all divisible by p_{l}. □

The “alternating property” of the coefficients of the Alexander polynomial of any irreducible plane curve singularity will be crucial in the computation of the Reidemeister-Turaev torsion of {f + z^{n} = 0}. The key algebraic fact is summarized in the next property:

5.3. Algebraic Lemma. In the next expressions t is a free variable and a is a positive integer. ℤ_{a} is identified with the a-roots of unity. Assume that the coefficients of a polynomial Δ(t) ∈ ℤ[t] form an alternating set, Δ(1) = 1, and a ≥ deg Δ. Then:

(a) For an arbitrary complex number A one has:

\frac{1}{a} ∑_{ξ∈ℤ_{a}} \frac{Δ(ξt)}{1 − ξt} · \frac{Δ(ξ̄At)}{1 − ξ̄At} = \frac{(1 − A^{a}t^{2a}) · Δ(At^{2})}{(1 − t^{a})(1 − A^{a}t^{a})(1 − At^{2})}.

(b) For arbitrary integers d ≥ 2 and k ≥ 1 one has:

\frac{1}{a^{d−1}} ∑_{{ξ_{1},...,ξ_{d}∈ℤa \atop ξ_{1}···ξ_{d}=1}} \frac{Δ(ξ_{1}t)}{1 − ξ_{1}t} · · · · · \frac{Δ(ξ_{d−1}t)}{1 − ξ_{d−1}t} · \frac{Δ(ξ_{d}t^{k})}{1 − ξ_{d}t^{k}} = \frac{(1 − t^{a(d+k−1)}) · Δ(t^{d+k−1})}{(1 − t^{a})^{d−1}(1 − t^{ak})(1 − t^{d+k−1})}.

Proof. The assumption about Δ(t) guarantees that one can write Δ(t) = 1 − R(t)(1 − t) for some R(t) = ∑_{j≥1} b̃_{j}t^{j} with b̃_{j} ∈ {0, 1} for all j. Then the left hand side of (a) is

\frac{1}{a} ∑_{ξ∈ℤ_{a}} \frac{1}{(1 − ξt)(1 − ξ̄At)} − \frac{1}{a} ∑_{ξ∈ℤ_{a}} \frac{R(ξ̄At)}{1 − ξt} − \frac{1}{a} ∑_{ξ∈ℤ_{a}} \frac{R(ξt)}{1 − ξ̄At} + \frac{1}{a} ∑_{ξ∈ℤ_{a}} R(ξt) · R(ξ̄At).

The first sum (with the coefficient 1/a) can be written in the form

\frac{1}{a}(1 + ξt + ξ^{2}t^{2} + ⋯)(1 + ξ̄At + ξ̄^{2}A^{2}t^{2} + ⋯) = \frac{1}{a} ∑_{n≥0} ∑_{j=0}^{n} ξ^{n−2j}A^{j}t^{n}.

This, by an elementary computation gives:

\frac{1 − A^{a}t^{2a}}{(1 − t^{a})(1 − A^{a}t^{a})(1 − At^{2})}.

The second term gives R(At^{2})/(1 − t^{a}). In order to prove this, first notice that the formula is additive in the polynomial R, hence it is enough to verify for R(t) = t^{k} for all 0 ≤ k < a. The case k = 0 is easy, it is equivalent with the identity

\frac{1}{a} ∑_{ξ} \frac{1}{1 − ξt} = \frac{1}{1 − t^{a}} .  (∗)

If 1 ≤ k < a, then write

\frac{1}{a} ∑_{ξ} \frac{ξ̄^{k}A^{k}t^{k}}{1 − ξt} = \frac{A^{k}t^{2k}}{a} [ ∑_{ξ} \frac{1}{1 − ξt} + ∑_{ξ}(ξt)^{−1} + ⋯ + (ξt)^{−k}].

Since k < a the last sum is zero (here k < a is crucial !), hence (∗) gives the claimed identity.

By a similar method, the third term is R(At^{2})/(1 − A^{a}t^{a}). Finally, the forth is R(At^{2}) (here one needs to apply the alternating property, namely that b̃_{j}^{2} = b̃_{j} for any j).

For part (b), use (a) and induction over d. For this, write ξ_{d} as ξ̄_{1} ⋯ ξ̄_{d−1} and use (a) for ξ = ξ_{d−1} and A = ξ̄_{1} ⋯ ξ̄_{d−2}t^{k−1}. Then apply the inductive step. □

5.4. Remarks. (1) Let Δ(t) and a be as in 5.3. The expression in 5.3(a), with A = 1, has a pole of order 2 at t = 1. This comes from the pole of the summand given by ξ = 1. Therefore:

\frac{1}{a} ∑_{ξ∈ℤ_{a}^{∗}} \frac{Δ(ξ)}{1 − ξ} · \frac{Δ(ξ̄)}{1 − ξ̄} = lim_{t→1} [ \frac{(1 − t^{2a}) · Δ(t^{2})}{(1 − t^{a})^{2}(1 − t^{2})} − \frac{Δ(t)^{2}}{a(1 − t)^{2}} ]

= \frac{a^{2} − 1}{12a} + \frac{1}{a}[Δ′(1) − Δ′(1)^{2} + Δ′′(1)],

where the first equality follows from 5.3, the second by a computation.

(2) Assume that Δ(t) is an arbitrary symmetric polynomial of degree 2r, and write Δ^{♮}(t) = t^{−r}Δ(t). Then it is easy to show that

Δ′(1) = rΔ(1), (Δ^{♮})′(1) = 0, and (Δ^{♮})′′(1) = (r − r^{2}) + Δ′′(1)/Δ(1).

(3) If one combines (1) and (2), then for a symmetric polynomial Δ(t) with alternating coefficients and with Δ(1) = 1 one gets

∑_{ξ∈ℤ_{a}^{∗}} \frac{Δ(ξ)}{1 − ξ} · \frac{Δ(ξ̄)}{1 − ξ̄} = \frac{a^{2} − 1}{12} + (Δ^{♮})′′(1) (for a ≥ deg Δ).

This reproves the vanishing of D_{a} in 4.9(D) for such polynomials (cf. also with 4.8(∗∗)).

(4) Although the Alexander polynomial Δ(f) of the algebraic knot (S^{3}, K_{f}) (f irreducible plane curve singularity) is known since 1932 [7, 43], and it was studied intensively (see e.g. [20, 1, 10]), the property 5.2 remained hidden (to the best of the author’s knowledge).

On the other hand, similar properties were intensively studied in number theory: namely in 40’s, 50’s and 60’s a considerable large number of articles were published about the coefficients of cyclotomic polynomials. Here we mention only a few results. If ϕ_{n} denotes the n^{th}-cyclotomic polynomial, then it was proved that the coefficients of ϕ_{n} have values in {−1, 0, +1} for n = 2^{α}p^{β}q^{γ} (p and q distinct odd primes) (result which goes back to the work of I. Schur); if n is a product of three distinct primes pqr (p < q < r and p + q > r) then the coefficient of t^{r} in ϕ_{n} is −2 (result of V. Ivanov); later Erdős proved interesting estimates for the growth of the coefficients; and G.S. Kazandzidis provided exact formula for them. The interested reader can consult [22], pages 404-411, for a large list of articles about this subject. (Reading these reviews, apparently the alternating property was not perceived in this area either.)

Clearly, the above facts are not independent of our problem: by 5.1(3) the Alexander polynomial Δ(f) is a product of cyclotomic polynomials.

(5) In fact, there is a recent result [14] in the theory of singularities, which implies the alternating property 5.2(b). For any irreducible curve singularity, using its normalization, one can define a semigroup S ⊂ ℕ, with 0 ∈ S and ℕ\S finite. Then, in [14], based on some results of Zariski, for an irreducible plane curve singularities is proved that Δ(f)(t)/(1−t) = ∑_{i∈S} t^{i}. This clearly implies the alternating property.

(6) Are the irreducible plane curve singularities unique with the alternating property? The answer is negative. In order to see this, consider the Seifert integral homology sphere Σ = Σ(a_{1}, a_{2}, …, a_{k+1}) (where {a_{i}}_{i} are pairwise coprime integers). Let K be the special orbit associated with the last arm (with Seifert invariant a_{k+1}). Then the Alexander polynomial Δ_{Σ}(K) has alternating coefficients. Indeed, write a := a_{1} ⋯ a_{k}, a′_{i} := a/a_{i} for any 1 ≤ i ≤ k, and let S ⊂ ℕ be the semigroup (with 0 ∈ S) generated by a′_{1}, ⋯, a′_{k}. Then

Δ_{Σ}(K)(t) = \frac{(1 − t^{a})^{k−1}(1 − t)}{∏_{i=1}^{k}(1 − t^{a′_{i}})} = (1 − t) ∑_{i∈S} t^{i}.

The first equality follows e.g. by [10], section 11; the second by an induction over k. This implies the alternating property as above.

(7) We can ask the following natural question: what is that property which distinguishes (M, K_{f}) (where f is an irreducible plane curve singularity), or (Σ, K) given in (6), from the example described in 4.13 ? Why is in the first case the D-invariant zero and in the second case not ? Can this be connected with some property of the semigroups S associated with the curve whose link is K ? (We believe that the validity of an Abhyankar-Azevedo type theorem for this curve plays an important role in this phenomenon.)

(8) Examples show that the assumptions of 5.3 are really essential (cf. also with 6.8(2)).

6. The link of {f(x, y) + z^{n} = 0}

6.1. Preliminaries. The present section is more technical than the previous ones, and some of the details are left to the reader.

Fix an irreducible plane curve singularity f : (ℂ^{2}, 0) → (ℂ, 0) and let K_{f} ⊂ S^{3} be its link as in the previous section. Fix an integer n ≥ 1, and consider the “suspension” germ g : (ℂ^{3}, 0) → (ℂ, 0) given by g(x, y, z) = f(x, y) + z^{n}. Its link (i.e. {g = 0} ∩ S_{ϵ}^{5} for ϵ ≪ 1) will be denoted by M. We will assume that M is a rational homology sphere, cf. 6.2(c).

First, we recall/fix some numerical notations. We set:

• the Newton pairs {(p_{k}, q_{k})}_{k=1}^{s} of f;

• the integers {a_{k}}_{k=1}^{s} defined as in 5.1(1); recall that gcd(p_{k}, a_{k}) = 1 for any k;

• d_{k} := gcd(n, p_{k+1}p_{k+2} ⋯ p_{s}) for 0 ≤ k ≤ s − 1, and d_{s} := 1; • h_{k} := d_{k−1}/d_{k} = gcd(p_{k}, n/d_{k}) for 1 ≤ k ≤ s;

• h̃_{k} := gcd(a_{k}, n/d_{k}) for 1 ≤ k ≤ s.

For any integer 1 ≤ l ≤ s, let M_{(l)} be the link of the suspension singularity g_{(l)}(x, y, z) := f_{(l)}(x, y) + z^{n/d_{l}}. Evidently, M_{(s)} = M, and M_{(1)} = Σ(p_{1}, a_{1}, n/d_{1}).

6.2. Some properties of the 3-manifolds {M_{(l)}}_{l}.

(a) [9, 16, 33, 39] For each 1 ≤ l ≤ s, M_{(l)} is the (n/d_{l})-cyclic cover of S^{3} branched along {f_{(l)} = 0}. Let K_{(l)} ⊂ M_{(l)} be the preimage of {f_{(l)} = 0} with respect to this cover.

(b) (M_{(l)}, K_{(l)}) can be represented by (a “canonical”) plumbing graph (or resolution graph) which is compatible with the above cover. This is made explicitly in [28] (based on the idea of [19]), see also 6.3 here. Using this one obtains the following inductive picture.

For any 2 ≤ l ≤ s, M_{(l)} can be obtained by splicing, as it is described in section 4, the 3-manifold M_{2} = Σ(p_{l}, a_{l}, n/d_{l}) along K_{2} = {y = 0} with h_{l} copies of M_{(l−1)} along the link K_{(l−1)} (with the same splicing data {A^{(i)}}_{i} as in section 4). (In order to prove this, one needs to determine the invariant M_{w} used in [28]; this is done in [29], in the proof of (3.2).)

(c) Part (b) assures that M is a rational homology sphere if and only if for each 1 ≤ l ≤ s the Seifert 3-manifold Σ(p_{l}, a_{l}, n/d_{l}) is a rational homology sphere. Since gcd(p_{l}, a_{l}) = 1, this is happening if and only if

(h_{l} − 1)(h̃_{l} − 1) = 0, for any l (cf. [6] or [28])

(d) Using (b) and 3.6, one has:

|H_{1}(M_{(l)})| = |H_{1}(Σ(p_{l}, a_{l}, n/d_{l}))| + h_{l} · |H_{1}(M_{(l−1)})| for any 2 ≤ l ≤ s.

or

|H_{1}(M)| = ∑_{l=1}^{s} d_{l} · |H_{1}(Σ(p_{l}, a_{l}, n/d_{l}))|.

In fact, one can give a complete description of the group H_{1}(M) and the character group H_{1}(M)ˆ using 3.6 (we will come back to this in 6.3).

(e) As a parallelism, let us recall some similar formulae for other numerical invariants: let μ_{(l)}, respectively σ_{(l)}, be the Milnor number, respectively the signature of the Milnor fiber of g_{(l)}. Similarly, let μ(p_{l}, a_{l}, n/d_{l}) and σ(p_{l}, a_{l}, n/d_{l}) be the Milnor number and the signature of the Brieskorn singularity x^{p_{l}} + y^{a_{l}} + z^{n/d_{l}}. Then, by [29], for any 2 ≤ l ≤ s:

σ_{(l)} = σ(p_{l}, a_{l}, n/d_{l}) + h_{l} · σ_{(l−1)} or σ_{(s)} = ∑_{l=1}^{s} d_{l} · σ(p_{l}, a_{l}, n/d_{l}).

By contrast, for the Milnor numbers one has μ_{(l)} = μ(p_{l}, a_{l}, n/d_{l}) + p_{l} · μ_{(l−1)} (involving p_{l} versus h_{l}, fact which follows e.g. from 5.1(4)).

Our goal is to establish an inductive formula for sw_{M}^{0}(σ_{can}), similar to |H_{1}| or to the signature σ_{(l)}. (For λ_{W} or 𝒯_{M,σ_{can}}(1) such a formula does not hold, see below.)

(f) [9, 16, 33, 39] (S^{3}, K_{f_{(l)}}) is fibrable. Let ℳ_{geom,(l)} : F_{(l)} → F_{(l)} (respectively ℳ_{(l)}) be a geometric (respectively the algebraic) monodromy acting on the Milnor fiber F_{(l)} (respectively on H_{1}(F_{(l)}). Then (M_{(l)}, K_{(l)}) is also fibrable, whose open book decomposition has the same fiber F_{(l)} and geometric monodromy ℳ_{geom,(l)}^{n/d_{l}}. In particular, the (normalized) Alexander polynomial of (M_{(l)}, K_{(l)}) is (the normalization of)

Δ_{M_{(l)}}(t) := Δ_{M_{(l)}}(K_{(l)})(t) = det(1 − tℳ_{(l)}^{n/d_{l}}).

Therefore, using 2.6(7) and 2r_{l} := rank H_{1}(F_{(l)}) we deduce

Δ_{M_{(l)}}^{♮}(t) = \frac{1}{|H_{1}(M_{(l)})|} · t^{−r_{l}} · det(1 − tℳ_{(l)}^{n/d_{l}}) with |H_{1}(M_{(l)})| = | det(1 − ℳ_{(l)}^{n/d_{l}})|.

Notice that Δ_{M_{(l)}}^{♮}(t) can be deduced from the Alexander polynomial

Δ(f_{(l)})(t) = det(1 − tℳ_{(l)}).

of f_{(l)} (cf. section 5). Indeed, for any polynomial Δ(t) of degree 2r and of the form Δ(t) = ∏_{v} (1 − t^{m_{v}})^{n_{v}}, and for any positive integer k, define

Δ^{c(k)}(t) := ∏_{v}(1 − t^{m_{v}/gcd(m_{v},k)})^{n_{v}·gcd(m_{v},k)}.

Let Δ^{c(k),♮}(t) = t^{−r}Δ^{c(k)}(t)/Δ^{c(k)}(1) denote the normalization of Δ^{c(k)}(t). An eigenvalueargument then proves that

Δ_{M_{(l)}}(t) = Δ(f_{(l)})^{c(n/d_{l})}(t) and Δ_{M_{(l)}}^{♮}(t) = Δ(f_{(l)})^{c(n/d_{l}),♮}(t).

(g) The inductive formula 5.1(4) reduces the computation of the Alexander invariants to the Seifert case. Clearly (from 5.1(5), 4.2(c) and (f) above):

Δ(x^{p_{l}} + y^{a_{l}})^{c(n/d_{l}),♮}(t) = \frac{1}{a_{l}^{h_{l}−1}p_{l}^{h̃−1}} · t^{−(a_{l}−1)(p_{l}−1)/2} · \frac{(t^{p_{l}a_{l}/(h_{l}h̃_{l})} − 1)^{h_{l}h̃_{l}}(t − 1)}{(t^{p_{l}/h_{l}} − 1)^{h_{l}}(t^{a_{l}/h̃_{l}} − 1)h̃_{l}}.

(Recall that (h_{l} − 1)(h̃_{l} − 1) = 0 for any l.) Then, by a computation, one can show that

(Δ(x^{p_{l}} + y^{a_{l}})^{c(n/d_{l}),♮})′′(1) = \frac{1}{12}(\frac{a_{l}^{2}}{h̃_{l}} − 1)(\frac{p_{l}^{2}}{h_{l}} − 1).

(h) Using (f) and 5.1(4) one gets:

Δ_{M_{(l)}}^{♮}(t) = Δ(x^{p_{l}} + y^{a_{l}})^{c(n/d_{l}),♮}(t) · [Δ_{M_{(l−1)}}^{♮}(t^{p_{l}/h_{l}})]^{h_{l}}.

Then, using (Δ^{♮})′(1) = 0 (cf. 5.4(2)) and the result from (g), one obtains:

(Δ_{M_{(l)}}^{♮})′′(1) = \frac{1}{12}(\frac{a_{l}^{2}}{h̃_{l}} − 1)(\frac{p_{l}^{2}}{h_{l}} − 1) + \frac{p_{l}^{2}}{h_{l}} · (Δ_{M_{(l−1)}}^{♮})′′(1).

Therefore:

(Δ_{M_{(l)}}^{♮})′′(1) = ∑_{k=1}^{l} \frac{1}{12} (\frac{a_{k}^{2}}{h̃_{k}} − 1)(\frac{p_{k}^{2}}{h_{k}} − 1) · \frac{(p_{k+1} ⋯ p_{l})^{2}}{h_{k+1} ⋯ h_{l}} .

(i) For any 2 ≤ l ≤ s, one has:

λ_{W} (M_{(l)}) = λ_{W} (Σ(p_{l}, a_{l}, n/d_{l})) + h_{l} · λ_{W} (M_{(l−1)})

+\frac{np_{l}(h_{l} − 1)}{d_{l}a_{l}h_{l}} · ∑_{k=1}^{l−1} \frac{1}{12} (\frac{a_{k}^{2}}{h̃_{k}} − 1)(\frac{p_{k}^{2}}{h_{k}} − 1) · \frac{(p_{k+1} ⋯ p_{l−1})^{2}}{h_{k+1} ⋯ h_{l−1}} .

Indeed, if h_{l} = 1 then we have additivity as in 3.3(4), cf. also with the proof of 4.11, part B. If h̃_{l} = 1, then apply 4.7 (see also 4.9) and (h) above.

For the value of the Casson-Walker invariant λ_{W} (Σ(p, a, n)) of a Seifert manifold, see [21, 6.1.1] or [31, 5.4].

(j) Below in (k), we will compute (Δ_{M_{(l)}}^{♮})′′(1) in terms of { (Δ(f_{(k)})^{♮})′′(1) }_{k≤l}. Clearly,

one can obtain similar inductive formula for these (Δ(f_{(k)})^{♮})′′(1) as that one in (h) by taking n = 1. More precisely, for any 2 ≤ l ≤ s:

(Δ(f_{(l)})^{♮})′′(1) = (a_{l}^{2} − 1)(p_{l}^{2} − 1)/12 + p_{l}^{2} · (Δ(f_{(l−1)})^{♮})′′(1).

(k) The next (rather complicated) identity looks very artificial, but it is one of the most important formulae in this list. Basically, it validity is equivalent with the fact that the two correction terms O(𝒯) and O(λ_{W} /2) are the same, cf. 6.13 and 6.14. For any 2 ≤ l ≤ s one has:

(Δ_{M_{(l)}}^{♮})′′(1) = (Δ(f_{(l)})^{♮})′′(1) − ∑_{k=1}^{l} \frac{a_{k}^{2}p_{k}^{2}}{h̃_{k}^{2}h_{k}^{2}} · \frac{p_{k+1}^{2} ⋯ p_{l}^{2}}{h_{k+1} ⋯ h_{l}} · A_{k},

where

A_{k} := \frac{h_{k}(h_{k} − 1)}{a_{k}^{2}} · [ \frac{a_{k}^{2} − 1}{12} + (Δ(f_{(k−1)})^{♮})′′(1) ] + \frac{h̃_{k}(h̃_{k} − 1)}{p_{k}^{2}} · \frac{p_{k}^{2} − 1}{12}.

For the proof proceed as follows. Let E_{(l)} be the difference between the left and the right hand side of the identity. Then, using the inductive formulae (h) and (j) and the property (h_{k}−1)(h̃_{k}−1) = 0, by an elementary computation one can verify that E_{(l)}−(p_{l}^{2}/h_{l})·E_{(l−1)} = 0, and E_{(1)} = 0. Then E_{(l)} = 0 by induction.

(l) The last invariant we wish to determine is 𝒯_{M_{(l)},σ_{can}}(1). The computation is more involved and it is separated in the next subsections. The inductive formula for 𝒯_{M_{(l)},σ_{can}}(1) is given in 6.13.

6.3. Characters of H_{1}(M). In the computation of 𝒯_{M,σ_{can}}(1) we plan to use 2.7(16). For this, we need to describe the characters χ ∈ Ĥ = H_{1}(M)ˆ .

The group H can be determined in many different ways. For example, using the monodromy operator ℳ = ℳ_{(s)} of f, H can be identified, as an abstract group, with coker(1 − ℳ^{n}). The homology of the Milnor fiber of f and ℳ have a direct sum decomposition with respect to the splicing (see e.g. [10] or [29]). This can be used to provide an inductive description of H.

Nevertheless, we prefer to use 3.6. The main reason is that, in fact, we have to understand Ĥ (rather than H) together with the description of the supports

S_{χ} := {v : χ(g_{v}) ≠ 1}

for each character χ ∈ Ĥ. For this, the description in section 3 is more suitable.

We consider again the “canonical” plumbing graph Γ(M) of M provided by the algorithm [28], cf. 6.2(b) here. In fact, that algorithm provides Γ := Γ(M, K_{z}), the plumbing graph of the 3-dimensional link M = {f(x, y) + z^{n} = 0} with the knot K_{z} := {z = 0} in it. Again, if we replace the strings by dash-lines, then one can represent Γ as a covering graph of Γ(S^{3}, K_{f}); for details, see [loc. cit.], cf. also with 6.2(b). If we denote this graph-projection by π, then

#π^{−1}(v_{k}) = h_{k+1} ⋯ h_{s}, 1 ≤ k ≤ s,
#π^{−1}(v̄_{k}) = h̃_{k}h_{k+1} ⋯ h_{s}, 1 ≤ k ≤ s, and
#π^{−1}(v̄_{0}) = h_{1} ⋯ h_{s}

(see 5.1 for notations about Γ(S^{3}, K_{f})). In fact, there is a ℤ_{n}-action on Γ which acts transitively on each fiber of π, hence all the vertices above a given vertex v ∈ V^{∗} of Γ(S^{3}, K_{f}) are symmetric in Γ. In particular, their decorations and their numerical invariants (computed out of the graph Γ) are the same. Therefore, Γ has the following schematic form:

[figure: {z = 0}; ⋮; h_{1}; ⋮; h_{1}; ⋮; h_{1}; ⋯; h̃_{1}; ⋯; h̃_{1}; ⋯; h̃_{1}; ⋮; ⋮; h_{s−1}; ⋮; ⋮; h_{s}; ⋯; ⋯; h̃_{s−1}; ⋯; h̃_{s}; ⋯; ⋯; ⋯; ⋯]

Now, we consider the Seifert manifold Σ := Σ(p, a, n) with gcd(p, a) = 1.

If gcd(a, n) = 1 and gcd(p, n) = d, by 4.3, H_{1}(Σ) is generated by the homology classes of {K_{2}^{(i)}}_{i=1}^{d}. For an arbitrary character χ, we write χ(K_{2}^{(i)}) = ξ_{i}. Here ξ_{i} is an a-root of unity in ℂ (shortly ξ_{i} ∈ ℤ_{a}). Then χ is completely characterized by the collection {ξ_{i}}_{i=1}^{d} which satisfy ξ_{i} ∈ ℤ_{a} for any i and ξ_{1} ⋯ ξ_{d} = 1.

Notice that χ(O) = 1, and χ is supported by those d “arms” of the star-shaped graph which have Seifert invariant a (i.e. χ(g_{v}) = 1 for the vertices v situated on the other arms). In [31] is proved that for each non-trivial character χ, in P̂_{Σ,χ,u}(t) one can take for u the central vertex O. Moreover, lim_{t→1} P̂_{Σ,χ,u}(t) can be non-zero only if χ is supported exactly on two arms, i.e. ξ_{i} ≠ 1 exactly for two values of i, say for i_{1} and i_{2} (hence ξ_{i_{1}} = ξ̄_{i_{2}}).

If gcd(n, p) = 1, but gcd(n, a) ≠ 1, then clearly we have a symmetric situation; in this case we use the notation η instead of ξ.

In both situation, for any character χ, χ(g_{v}) = 1 for any v situated on the arm with Seifert invariant n/gcd(n, ap).

These properties proved for the building block Σ will generate all the properties of H = H_{1}(M) via the splicing properties 3.6 and linking relations 4.3 (L#).

For example, one can prove by induction, that for any character χ of H, χ(g_{v}) = 1 for any vertex v situated on the string which supports the arrow of K_{z}.

Consider the splicing decomposition

M = M_{(s)} = h_{s}M_{(s−1)} ∐ Σ(p_{s}, a_{s}, n).

As in the previous inductive arguments, assume that we understand the characters χ of H_{1}(M_{(s−1)}). By 3.6, these can be considered in a natural way as characters of H_{1}(M_{(s)}) satisfying additionally χ(g_{v}) = 1 for any vertex v of Σ(p_{s}, a_{s}, n) (cf. also with the first paragraph of the proof of 3.9). We say that these characters do not “propagate” from M_{(s−1)} into Σ(p_{s}, a_{s}, n).

In the “easy” case when h_{s} = 1 (even if h̃_{s} ≠ 1), the splicing invariants are o_{1} = o_{2} = 1 and k_{1} = k_{2} = 0, hence H (together with its linking form) is a direct sum in a natural way, hence the characters of Σ(p_{s}, a_{s}, n) (described above) will not propagate into M_{(s−1)} either.

On the other hand, if h_{s} > 1, then the non-trivial characters of Σ(p_{s}, a_{s}, n) do propagate into the h_{s} copies of M_{(s−1)}. If χ is such a character, with χ(K_{2}^{(i)}) = ξ_{i}, then analyzing the properties of the linking numbers in 3.6 we deduce that χ can be extended into M in such a way that

χ(g_{v^{(i)}}) = ξ_{i}^{l}, l = Lk_{M_{(s−1)}^{(i)}}(g_{v^{(i)}}, K_{1}^{(i)}),

for any vertex v^{(i)} of M_{(s−1)}^{(i)}. This fact can be proved as follows.

Assume that χ = exp(Lk_{Σ}(L, ·)) for some fixed L ⊂ Σ := Σ(p_{s}, a_{s}, n). Then

ξ_{i} = χ(K_{2}^{(i)}) = exp(Lk_{Σ}(L, K_{2}^{(i)})).

Therefore, χ extended into M as exp(Lk_{M}(L, ·)) satisfies (cf. 3.6(11))

χ(g_{v^{(i)}}) = exp(Lk_{M}(L, g_{v^{(i)}})) = exp(Lk_{Σ}(L, K_{2}^{(i)}) · Lk_{M_{s−1}^{(i)}}(K_{1}^{(i)}, g_{v^{(i)}})) = ξ_{i}^{l}.

Clearly, all these linking numbers Lk_{M_{(s−1)}^{(i)}}(g_{v^{(i)}}, K_{1}^{(i)}) can be determined inductively by the

formulae provided in 3.6 and 4.3 (L#).

Moreover, if we multiply such an extended character χ with a character χ′ supported by M_{(s−1)}, then χχ′(g_{v}) = χ(g_{v}) for any vertex v situated above the vertices v_{s} or v_{s−1} or above any vertex on the edge connecting v_{s−1} with v_{s} (since the support of χ′ does not contain these vertices). We will say that such a character χχ′ is “born at level s” (provided that the original χ of Σ was non-trivial).

(The interested reader can reformulate the above discussion in the language of an exact sequence of dual groups, similar to one used in the first paragraph of the proof of 3.9.)

Below we provide some examples. In this diagrams, for any fixed character χ, we put on the vertex v the complex number χ(g_{v}).

6.4. Example. Assume that s = 2. Then basically one has two different cases: h̃_{1} = h_{2} = 1, or h̃_{1} = h̃_{2} = 1 (since the first pairs (p_{1}, a_{1}) can be permuted).

In the first (“easy”) case, the schematic diagram of the characters is:

[figure: ⋮; ⋯; 1; 1; 1; 1; η_{1}; ηh̃_{2}; ξ_{11}; ξ_{h_{1}1}; η_{i_{2}} ∈ ℤ_{p_{2}} for 1 ≤ i_{2} ≤ h̃_{2}, and ∏_{1≤i_{2}≤h̃_{2}} η_{i_{2}} = 1;; ξ_{i_{1}1} ∈ ℤ_{a_{1}} for 1 ≤ i_{1} ≤ h_{1}, and ∏_{1≤i_{1}≤h_{1}} ξ_{i_{1}1} = 1.]

In the second case h̃_{1} = h̃_{2} = 1, with the notation p′_{1} := p_{1}/h_{1}, the characters Ĥ are as follows.

[figure: ⋮; ⋮; ⋮; ξ_{1}^{a_{1}p′_{1}}; ξ_{1}^{a_{1}}; ξ_{h_{2}}^{a_{1}p′_{1}}; ξ_{h_{2}}^{a_{1}}; 1; 1; 1; ξ_{11}ξ_{1}^{p′_{1}}; ξ_{h_{1}1}ξ_{1}^{p′_{1}}; ξ_{1h_{2}}ξ_{h_{2}}^{p′_{1}}; ξ_{h_{1}h_{2}}ξ_{h_{2}}^{p′_{1}}; ξ_{i_{2}} ∈ ℤ_{a_{2}} for 1 ≤ i_{2} ≤ h_{2}, and ∏_{1≤i_{2}≤h_{2}} ξ_{i_{2}} = 1;; and for any fixed i_{2}:; ξ_{i_{1}i_{2}} ∈ ℤ_{a_{1}} for 1 ≤ i_{1} ≤ h_{1}, and ∏_{1≤i_{1}≤h_{1}} ξ_{i_{1}i_{2}} = 1.]

6.5. Using the discussion 6.3, the above example for s = 2 can be generalized inductively to arbitrary s. For the convenience of the reader, we make this explicit. In order to have a uniform notation, we consider the case h̃_{k} = 1 for any 1 ≤ k ≤ s. The interested reader is invited to write down similar description of the characters in those cases when h̃_{k} ≠ 1 for some k, using the present model and 6.4.

More precisely, for any character χ, we will indicate χ(g_{v′}) for any vertex v′ ∈ π^{−1}(V^{∗}). It is convenient to introduce the index set (i_{1}, i_{2}, …, i_{s}), where 1 ≤ i_{l} ≤ h_{l} for any

1 ≤ l ≤ s. As we already mentioned, this set can be considered as the index set of π^{−1}(v̄_{0}). Moreover, for any 1 ≤ k ≤ s − 1, (i_{k+1}, …, i_{s}) (where 1 ≤ i_{l} ≤ h_{l} for any k + 1 ≤ l ≤ s) is the index set of π^{−1}(v_{k}) (respectively of π^{−1}(v̄_{k}) since h̃_{k} = 1). Moreover, for any l we write p′_{l} := p_{l}/h_{l}.

Next, we consider a system of roots of unity as follows:

• ξ_{i_{s}} ∈ ℤ_{a_{s}} with ∏_{1≤i_{s}≤h_{s}} ξ_{i_{s}} = 1;

• for any fixed i_{s} a collection ξ_{i_{s−1}i_{s}} ∈ ℤ_{a_{s−1}} with ∏_{1≤i_{s−1}≤h_{s−1}} ξ_{i_{s−1}i_{s}} = 1; and, more generally, if 1 ≤ k ≤ s − 1:

• for any fixed (i_{k+1}, ⋯, i_{s}) a collection ξ_{i_{k} ··· i_{s}} ∈ ℤ_{a_{k}} with ∏_{1≤i_{k}≤h_{k}} ξ_{i_{k} ··· i_{s}} = 1.

Then, any character χ can be characterized by the following properties:

• π^{−1}(v_{s}) contains exactly one vertex, say v′. Then χ(g_{v′}) = 1. The same is valid for π^{−1}(v̄_{s}) and for that vertex in Γ which supports the arrow {z = 0}.

• For any 1 ≤ k ≤ s − 1, if v′(i_{k+1}, …, i_{s}) is the vertex in π^{−1}(v_{k}) corresponding to the index (i_{k+1}, …, i_{s}), then

χ(g_{v′(i_{k+1},...,i_{s})}) = ξ_{i_{k+1} ··· i_{s}}^{a_{k} p′_{k}} · ξ_{i_{k+2} ··· i_{s}}^{a_{k} p′_{k} p′_{k+1}} ⋯ ξ_{i_{s}}^{a_{k} p′_{k} ··· p′_{s−1}}.

• Similarly, if v̄′(i_{k+1}, …, i_{s}) is the vertex in π^{−1}(v̄_{k}) corresponding to the index (i_{k+1}, …, i_{s}), then

χ(g_{v̄′(i_{k+1},...,i_{s})}) = ξ_{i_{k+1} ··· i_{s}}^{a_{k}} · ξ_{i_{k+2} ··· i_{s}}^{a_{k} p′_{k+1}} ⋯ ξ_{i_{s}}^{a_{k} p′_{k+1} ··· p′_{s−1}}.

• Finally, if v̄′(i_{1}, …, i_{s}) is the vertex in π^{−1}(v̄_{0}) corresponding to the index (i_{1}, …, i_{s}), then

χ(g_{v̄′(i_{1},...,i_{s})}) = ξ_{i_{1} ··· i_{s}} · ξ_{i_{2} ··· i_{s}}^{p′_{1}} ⋯ ξ_{i_{s}}^{p′_{1} ··· p′_{s−1}}.

If ξ_{i_{s}} ≠ 1 for some i_{s} then χ is “born at level s”. If ξ_{i_{s}} = 1 for all i_{s}, but χ_{i_{s−1}i_{s}} ≠ 1 for some (i_{s−1}, i_{s}), then χ is “born at level s − 1”, etc.

In general, a character χ is “born at level k” (1 ≤ k ≤ s) if for any l ≥ k and v′ ∈ π^{−1}(v_{l}), one has χ(g_{v′}) = 1, but there exists at least one vertex v′ ∈ π^{−1}(v_{k}) which is adjacent (in the graph Γ) with the support of χ.

The next result will be crucial when we apply the Fourier inversion formula 2.7(14). It is a really remarkable property of the links associated with irreducible plane curve singularities. It is the most important qualitative ingredient in our torsion computation (see also 6.8 for another powerful application).

6.6. Proposition. Consider (S^{3}, K_{f}) associated with an irreducible plane curve singularity f. Let Δ_{S^{3}}(K_{f})(t) be its Alexander polynomial. For any integer n ≥ 1, consider (M, K_{z}), i.e. the link M of {f(x, y) + z^{n} = 0} and the knot K_{z} := {z = 0} in it. Let Δ_{M}^{H}(K_{z})(t) be the Alexander invariant defined in 2.6 (10-11) with H = H_{1}(M). Then

Δ_{M}^{H}(K_{z})(t) = Δ_{S^{3}}(K_{f})(t).

Proof. First we notice that in 2.6(10), g_{u} = K_{z} and o(u) = 1. Then, by 2.4(5), m_{v′}(u) = Lk_{M}(g_{v′}, K_{z}) for any vertex v′ of Γ. By the algorithm [28], for any vertex v′ ∈ π^{−1}(v), where v ∈ V^{∗}(Γ(S^{3}, K_{f})), the linking number Lk_{M}(g_{v′}, K_{z}) is given by w_{v′} = w_{v}/gcd(w_{v}, n). Recall, that for any v ∈ V^{∗}(Γ(S^{3}, K_{f})), the corresponding weights w_{v} are given in 5.1(2). In particular, this discussion provides all the weight w_{v′}(u) needed in the definition 2.6(10) of Δ_{M,χ}(K_{z})(t). One the other hand, for characters χ ∈ Ĥ we can use the above description. Then the proposition follows (by some computation) inductively using the Algebraic Lemma, part (b), for the Alexander polynomials Δ(f_{(l)}(0)) (0 ≤ l ≤ s − 1, where Δ(f_{(0)})(t) ≡ 1). Notice that this lemma can be applied thanks to the proposition 5.2 (which assures that the coefficients of Δ(f_{(l)}(t) are alternating), and to the inequality 5.1(6) (which assures that a “is sufficiently large”). (For an expression of Δ(f_{(l)}(t), see 5.1).)

In the next example we make this argument explicit for the case s = 2 and h̃_{1} = h̃_{2} = 1. Using this model, the reader can complete the general case easily. □

6.7. Example. Assume that s = 2 and h̃_{1} = h̃_{2} = 1. Then, with the notation of 6.4, Δ_{M,χ}(K_{z})(t) equals

∏_{i_{1},i_{2}} \frac{1}{1 − t^{p′_{1} p′_{2}}ξ_{i_{1}i_{2}} ξ_{i_{2}}^{p′_{1}}} · ∏_{i_{2}} \frac{(1 − t^{a_{1} p′_{1} p′_{2}}ξ_{i_{2}}^{a_{1} p′_{1}})^{h_{1}}}{1 − t^{a_{1}p′_{2}}ξ_{i_{2}}^{a_{1}}} · \frac{(1 − t^{a_{2} p′_{2}})^{h_{2}} · (1 − t)}{1 − t^{a_{2}}}.

First, for each fixed index i_{2}, we make a sum over ξ_{i_{1}i_{2}} ∈ ℤ_{a_{1}}. Using 5.3(b) for Δ ≡ 1, t = t^{p′_{1} p′_{2}}ξ_{i_{2}}^{p′_{1}} and a = a_{1} and d = h_{1}, the above expression transforms (after some simplifications) into:

∏_{i_{2}} \frac{1 − t^{a_{1} p_{1} p′_{2}}ξ_{i_{2}}^{a_{1} p_{1}}}{(1 − t^{p_{1} p′_{2}}ξ_{i_{2}}^{p_{1}}) · (1 − t^{a_{1}p′_{2}}ξ_{i_{2}}^{a_{1}})} · \frac{(1 − t^{a_{2} p′_{2}})^{h_{2}} · (1 − t)}{1 − t^{a_{2}}}.

The expression in the product is exactly Δ(f_{(1)})(t^{p′_{2}})/(1 − t^{p′_{2}}). Therefore, 5.3(b) can be applied again, now for Δ = Δ(f_{(1)}), t = t^{p′_{2}}, a = a_{2} and d = h_{2}. Then the expression transforms into Δ(f_{(2)}).

6.8. Remarks. (1) If the link M of {f(x, y) + z^{n} = 0} is a rational homology sphere, then in [26] we prove the following facts. Using the combinatorics of the plumbing graph of M, one can recover the knot K_{z} in it. Then, by the above proposition, from the pair (M, K_{z}) one can recover the Alexander polynomial Δ_{S^{3}}(K_{f}) of f. It is well-known that this is equivalent with the equisingular type of the plane singularity f. Moreover, analyzing again the graph of (M, K_{z}), one can recover the integer n as well. In particular, from M, we can recover not only the geometric genus of {f + z^{n} = 0} (fact which is proved in this article), but also its multiplicity, and in fact, any numerical invariant which can be computed from the Newton (or Puiseux) pairs of f and from the integer n (e.g. even all the equivariant Hodge numbers associated with the vanishing cohomology of the hypersurface singularity g = f + z^{n}, or even the embedded topological type (S^{5}, M) of g with its integral Seifert matrix). (In fact, in [26], one obtains the Newton pairs by a more direct argument.)

(2) 6.6 suggests the following question. Let N be an integral homology sphere, L ⊂ N a knot in it such that (N, L) can be represented by a (negative definite) plumbing. Let (M, K) be the n-cyclic cover of (N, L) (branched along L) such that M is a rational homology sphere with H_{1}(M) = H. Then is it true that Δ_{M}^{H}(K)(t) = Δ_{N}(L)(t) ?

The answer is negative: one can construct easily examples (satisfying even the algebraicity condition) when the identity 6.6 fails. For example, consider (N, L) given by the following splice, respectively plumbing diagram:

[figure: 2; 7; 3; 1; 5; 2; −2; −1; −4; −5; −2]

Then one can show that e.g. for n = 2 the identity Δ_{M}^{H}(K)(t) = Δ_{N}(L)(t) fails.

This example also shows that in the Algebraic Lemma 5.3 the assumption a ≥ deg Δ is crucial. Indeed, in this example H = ℤ_{3}; and in order to determine Δ_{M}^{H}(K)(t), one needs to compute a sum like in 5.3(a) with a = 3, A = 1 and Δ = t^{4} − t^{3} + t^{2} − t + 1 (i.e. with a < deg Δ). But for these data, the identity in 5.3(a) fails.

6.9. The Reidemeister-Turaev sign-refined torsion. Now we will start to compute 𝒯_{M,σ_{can}}(1) associated with M = M_{(s)} and the canonical spin^{c}-structure σ_{can} of M. Similarly as above, we write H = H_{1}(M). Using 2.7(14), 𝒯_{M,σ_{can}}(1) can be determined by the Fourier inversion formula from {𝒯̂_{M,σ_{can}}(χ) : χ ∈ Ĥ \ {1}}. On the other hand, each 𝒯̂_{M,σ_{can}}(χ̄) is given by the limit lim_{t→1} P̂_{M,χ,u}(t) for some convenient u, cf. 2.7(16).

In the next discussion, the following terminology is helpful. Fix an integer 1 ≤ k ≤ s and a vertex v′(I) := v′(i_{k+1}, …, i_{s}) ∈ π^{−1}(v_{k}). Consider the graph Γ \ {v′(I)}. If h̃_{k} = 1 then it has h_{k} + 2 connected components: h_{k} (isomorphic) subgraphs Γ_{−}^{i_{k}}(v′(I)) (1 ≤ i_{k} ≤ h_{k}) which contain vertices at level k − 1, a string Γ_{st}(v′(I)) containing a vertex above v̄_{k}, and the component Γ_{+}(v′(I)) which supports the arrow {z = 0}. Similarly, if h_{k} = 1, then Γ \ {v′(I)} has h̃_{k} + 2 connected components, Γ_{−}(v′(I)) contains vertices at level k − 1, Γ_{+}(v′(I)) supports the arrow {z = 0}, and h̃_{k} other (isomorphic) components Γ_{st}^{j_{k}}(v′(I)) (1 ≤ j_{k} ≤ h̃_{k}), which are strings, and each of them contains exactly one vertex staying above v̄_{k}.

Similarly as for the Seifert manifold Σ(p, n, n) (see the discussion in 6.3 after the diagram), for a large number of characters χ, the limit lim_{t→1} P̂_{M,χ,u}(t) is zero. Analyzing the structure of the graph of M and the supports of the characters, one can deduce that a non-trivial character χ, with the above limit nonzero, should satisfy one of the following structure properties.

E(asy) case: The character χ is born at level k (for some 1 ≤ k ≤ s) with h̃_{k} > 1. For any vertex v′ ∈ π^{−1}(v_{k}) one has χ(g_{v′}) = 1, but there is exactly one vertex v′(I) := v′(i_{k+1}, …, i_{s}) ∈ π^{−1}(v_{k}) which is adjacent with the support of χ. Moreover, χ is supported by exactly two components of type Γ_{st}^{j_{k}}(v′(I)), say for indices j′_{k} and j′′_{k}. Let v′(j′_{k}) be the unique vertex in Γ_{st}^{j_{k}}(v′(I)) ∩ π^{−1}(v̄_{k}) (similarly for j′′_{k}). Then v′(j′_{k}) and v′(j′′_{k}) are the only vertices v′ of the graph of M with δ_{v′} ≠ 2 and χ(g_{v′}) ≠ 1. Moreover, χ(g_{v′(j′_{k})}) = χ̄(g_{v′(j′′_{k})}) = η ∈ ℤ_{p_{k}}^{∗}. Therefore, with fixed (i_{k+1}, …, i_{s}) and (j′_{k}, j′′_{k}), there are exactly p_{k} − 1 such characters. D(ifficult) case: The character χ is born at level k (for some 1 ≤ k ≤ s) with h_{k} > 1. For any vertex v′ ∈ π^{−1}(v_{k}), χ(g_{v′}) = 1, but there is exactly one vertex v′(I) := v′(i_{k+1}, …, i_{s}) ∈ π^{−1}(v_{k}) which is adjacent with the support of χ. The character χ is supported by exactly two components of type Γ_{−}^{i_{k}}(v′(I)), say for indices i′_{k} and i′′_{k}. Using the previous notations, this means that ξ_{i_{k} i_{k+1}···i_{s}} = 1 excepting for i_{k} = i′_{k} or i_{k} = i′′_{k}. (Evidently, ξ_{i_{t}···i_{s}} = 1 for any t > k.) For t < k, the values ξ_{i_{t}···i_{k+1}···i_{s}} are un-obstructed. In particular, with indices (i_{k+1}, …, i_{s}) and (i′_{k}, i′′_{k}) fixed, there are exactly (a_{k} − 1) · |H_{1}(M_{(k−1)})|^{2} such characters. Here, a_{k}−1 stands for ξ_{i′_{k} i_{k+1}···i_{s}} = ξ̄_{i′′_{k} i_{k+1}···i_{s}} ∈ ℤ_{a_{k}}^{∗}, and |H_{1}(M_{(k−1)})|^{2} for the un-obstructed characters born at level < k on the two branches corresponding to (i′_{k}, i_{k+1}, …, i_{s}) and (i′′_{k}, i_{k+1}, …, i_{s}).

In both cases (E) or (D), if such a character χ is born at level k (i.e. if satisfies the above characterization for k), then we write χ ∈ B_{k}.

Now, we fix a nontrivial character χ. Let S(χ) be the support of χ and S̄(χ) its complement. Then

\frac{1}{|H|} · 𝒯̂_{M,σ_{can}}(χ̄) = Loc(χ̄) · Reg(χ̄),

where

Loc(χ̄) := ∏_{v′∈S(χ)} (χ(g_{v′}) − 1)^{δ_{v′}−2} and Reg(χ̄) := \frac{1}{|H|} · lim_{t→1} ∏_{v′∈S̄(χ)} (t^{w_{v′}(u)} − 1)^{δ_{v′}−2}.

We will call Loc(χ̄) the “local contribution”, while Reg(χ̄) the “regularization contribution”.

By the above discussion, Reg(χ̄) = 0 unless χ is not of the type (E) or (D) described above. If χ is of type (E) or (D) described as above, then in P̂_{M,χ,u}(t) (cf. 2.7(15)) one can take v′(I). Moreover, if χ ∈ B_{k}, then by the symmetry of the plumbing graph of M, Reg(χ̄) does not depend on the particular choice of χ, but only on the integer k. We write Reg(k) for Reg(χ) for some (any) χ ∈ B_{k}.

In particular,

𝒯_{M,σ_{can}}(1) = ∑_{k=1}^{s} Reg(k) · ∑_{χ∈B_{k}} Loc(χ̄).  (𝒯)

6.10. Proposition. For any fixed 1 ≤ k ≤ s one has:

(E) If h_{k} = 1 then

∑_{χ∈B_{k}} Loc(χ̄) = d_{k} · \frac{h̃_{k}(h̃_{k} − 1)}{2} · \frac{p_{k}^{2} − 1}{12}.

(D) If h̃_{k} = 1 then

∑_{χ∈B_{k}} Loc(χ̄) = d_{k} · \frac{h_{k}(h_{k} − 1)}{2} · |H_{1}(M_{(k−1)})|^{2} · [ \frac{a_{k}^{2} − 1}{12} + (Δ(f_{(k−1)})^{♮})′′(1) ].

Proof. (E) d_{k} = h_{k+1} ⋯ h_{s} is the cardinality of the index set (i_{k+1}, …, , i_{s}), h̃_{k}(h̃_{k} − 1)/2 is the number of possibilities to choose the indices (j′_{k}, j′′_{k}). The last term comes from a formula of type 4.8(∗∗), where the sum is over η ∈ ℤ_{p_{k}}^{∗}.

In the case (D), d_{k}h_{k}(h_{k}−1)/2 has the same interpretation. Fix the branch (i′_{k}, i_{k+1}, …, i_{s}) and consider the sum over all the characters born at level < k. Then 6.6, applied for (M_{(k−1)}, K_{(k−1)}) as a covering of (S^{3}, f_{(k−1)} = 0), provides |H_{1}(M_{(k−1)})|^{2} ·Δ(f_{(k−1)})(t)/(t− 1) evaluated at t = ξ_{i′_{k} i_{k+1}···i_{s}}. The same is true for the other index i′′_{k}. Then apply 5.4(3) for Δ = Δ(f_{(k−1)}) and a = a_{k}. This can be applied because of 5.2 and 5.1(6). □

6.11. The “regularization contribution” Reg(k). Fix a character χ ∈ B_{k} of type (E) or (D) as in 6.9. Recall that one can take u = v′(I). Consider the connected components of Γ \ {v′(I)} (as in 6.9), where we add to each component an arrow corresponding to the edge which connects the component to v′(I). For these graphs, if one applies 2.6(12), one gets that Reg(k) is

Reg(k) = \{ \begin{matrix}− det(Γ_{−}) · det(Γ_{st})^{h̃_{k}−2} · det(Γ_{+})/ det(Γ) & case (E) \\ − det(Γ_{−})^{h_{k}−2} · det(Γ_{st}) · det(Γ_{+})/ det(Γ) & case (D). \end{matrix}

Let I be the intersection matrix of M, and I_{k}^{−1} := I_{v′v′}^{−1} for any v′ ∈ π^{−1}(v_{k}). Then, by the formula which provides the entries of an inverse matrix, one gets that

det Γ · I_{v′v′}^{−1} = \{ \begin{matrix}det(Γ_{−}) det(Γ_{st})h̃_{k} det(Γ_{+}) & case (E) \\ det(Γ_{−})^{h_{k}} det(Γ_{st}) det(Γ_{+}) & case (D). \end{matrix}

These two facts combined show that

Reg(k) = \{ \begin{matrix}−I_{k}^{−1}/ det(Γ_{st})^{2} & = −I_{k}^{−1}/p_{k}^{2} & case (E) \\ −I_{k}^{−1}/ det(Γ_{−})^{2} & = −I_{k}^{−1}/(a_{k} · |H_{1}(M_{(k−1)})|)^{2} & case (D) \end{matrix}

since in case (E) | det(Γ_{st})| = p_{k} e.g. from 4.2(a)), and in case (D) | det(Γ_{−})| = a_{k} · |H_{1}(M_{(k−1)})| by 4.14.

This, together with 6.9 (𝒯) and 6.10 transform into

𝒯_{M,σ_{can}}(1) = − ∑_{k=1}^{s} I_{k}^{−1} · d_{k} · A_{k}/2,  (∗)

where A_{k} is defined in 6.2(k) in terms of the numerical invariants of f_{(k)}.

6.12. The computation of I_{k}^{−1}. For any l ≥ k, let I_{k}^{−1}(M_{(l)}) be the (v′, v′)-entry of the inverse of the intersection form I(M_{(l)}) associated with M_{(l)}, where v′ is any vertex above v_{k}. E.g. I_{k}^{−1}(M_{(s)}) is I_{k}^{−1} used above.

By 2.4(5), −I_{k}^{−1}(M_{(l)}) = Lk_{M_{(l)}}(g_{v′}, g_{v′}). If l = k, by 3.6(12) this is Lk_{Σ(p_{k},a_{k},n/d_{k})}(O, O), hence by 4.3 (L#) it is np_{k}a_{k}/(d_{k}h_{k}^{2}h̃_{k}^{2}).

Next, assume that l > k. If h_{l} = 1, then the splicing

M_{(l)} = h_{l}M_{(l−1)} ∐ Σ(p_{l}, a_{l}, n/d_{l})

is trivial (with o_{1} = o_{2} = 1 and k_{1} = k_{2} = 0), hence by 3.6(10) one gets

−I_{k}^{−1}(M_{(l)}) = −I_{k}^{−1}(M_{(l−1)}).

If h_{l} > 1, then 4.4 and an iterated application of 3.6(10) and 4.3 (L#) give

−I_{k}^{−1}(M_{(l)}) = −I_{k}^{−1}(M_{(l−1)}) − (\frac{a_{k}p_{k}}{h̃_{k}h_{k}} · \frac{p_{k+1}}{h_{k+1}} ⋯ \frac{p_{l−1}}{h_{l−1}} )^{2} · \frac{np_{l}(h_{l} − 1)}{d_{l}a_{l}h_{l}^{2}}.

Indeed, by 2.4(5) and 3.6

−I_{k}^{−1}(M_{(l)}) = −I_{k}^{−1}(M_{(l−1)}) − (Lk_{M_{(l−1)}}(g_{v′}, K_{(l−1)}))^{2} · \frac{np_{l}(h_{l} − 1)}{d_{l}a_{l}h_{l}^{2}},

and, again by 3.6, Lk_{M_{(l−1)}}(g_{v′}, K_{(l−1)}) equals

Lk_{Σ(p_{k},a_{k}n/d_{k})}(O, Z) · Lk_{Σ(p_{k+1},a_{k+1}n/d_{k+1})}(K_{2}^{(i)}, Z) ⋯ Lk_{Σ(p_{l−1},a_{l−1}n/d_{l−1})}(K_{2}^{(i)}, Z).

6.13. The splicing formula for 𝒯_{Mσ_{can}}(1). Using 6.11(∗) and 6.12 (and d_{s} = 1), we can write

𝒯_{M_{(s)},σ_{can}}(1) − h_{s} · 𝒯_{M_{(s−1)},σ_{can}}(1) = \frac{na_{s}p_{s}}{2h̃_{s}^{2}h_{s}^{2}} · A_{s}

+ ∑_{k=1}^{s−1}(−I_{k}^{−1}(M_{(s)})) · h_{k+1} ⋯ h_{s} · A_{k}/2 − h_{s} ∑_{k=1}^{s−1}(−I_{k}^{−1}(M_{(s−1)})) · h_{k+1} ⋯ h_{s−1} · A_{k}/2

= \frac{na_{s}p_{s}}{2h̃_{s}^{2}h_{s}^{2}} · A_{s} − ∑_{k=1}^{s−1} h_{k+1} ⋯ h_{s} · \frac{a_{k}^{2}p_{k}^{2} ⋯ p_{s−1}^{2}}{h̃_{k}^{2}h_{k}^{2} ⋯ h_{s−1}^{2}} · \frac{np_{s}(h_{s} − 1)}{a_{s}h_{s}^{2}} · A_{k}/2.

But, by 4.2(e) one has

\frac{na_{s}p_{s}}{2h̃_{s}^{2}h_{s}^{2}} · A_{s} = 𝒯_{Σ(p_{s},a_{s},n),σ_{can}}(1) + \frac{na_{s}p_{s}}{2h̃_{s}^{2}h_{s}^{2}} · \frac{h_{s}(h_{s} − 1)}{a_{s}^{2}} · (Δ(f_{(s−1)})^{♮})′′(1).

Therefore, (using also (h_{s} − 1)/h̃^{2} = h_{s} − 1) one gets

O(𝒯_{·,σ_{can}}(1)) = 𝒯_{M_{(s)},σ_{can}}(1) − h_{s} · 𝒯_{M_{(s−1)},σ_{can}}(1) − 𝒯_{Σ(p_{s},a_{s},n),σ_{can}}(1)

= \frac{np_{s}(h_{s} − 1)}{2h_{s}a_{s}}[(Δ(f_{(s−1)})^{♮})′′(1) − ∑_{k=1}^{s−1} \frac{a_{k}^{2}p_{k}^{2} ⋯ p_{s−1}^{2}}{h̃_{k}^{2}h_{k}^{2}h_{k+1} ⋯ h_{s−1}} · A_{k}].

Notice that this inductive formula (together with 4.2(e)) shows that the numerical function n ↦ 𝒯(n) (where 𝒯(n) denotes the sign-refined Reidemeister-Turaev torsion of the link of f + z^{n}, associated with σ_{can}) can be written in the form n ↦ np_{1}(n) + p_{2}(n), where p_{i}(n) are periodic functions. Similar result for the signature of the Milnor fiber was obtained by Neumann [33]

Now, using 4.9 for M = M_{(s)}, and 6.2(k) for l = s−1, one has the following consequences:

6.14. Theorem. The additivity obstruction O(sw_{·}^{0}(σ_{can})) = 0, in other words:

sw_{M_{(s)}}^{0}(σ_{can}) = h_{s} · sw_{M_{(s−1)}}^{0}(σ_{can}) + sw_{Σ(p_{s},a_{s},n)}^{0}(σ_{can}).

In particular, by induction for M = M_{(s)} one gets:

sw_{M}^{0}(σ_{can}) = ∑_{k=1}^{s} d_{k} · sw_{Σ(p_{k},a_{k},n/d_{k})}^{0}(σ_{can}).

36 András Némethi and Liviu I. Nicolaescu

6.15. Corollary. Consider the hypersurface singularity g(x, y, z) = f(x, y) + z^{n}, where f is an irreducible plane curve singularity. Assume that its link M is a rational homology sphere. If σ(g) denotes the signature of the Milnor fiber of g, then

−sw_{M}^{0}(σ_{can}) = σ(g)/8.

In particular, the geometric genus of {g = 0} is topological and it is given by

p_{g} = sw_{M}^{0}(σ_{can}) − (K^{2} + #V)/8,

where the invariant K^{2} + #V (associated with any plumbing graph of M) is defined in the introduction.

Proof. By 6.14 an [29](3.2) (cf. also with 6.2(e)), we only have to show that

−sw_{Σ(p_{k},a_{k},n/d_{k})}^{0}(σ_{can}) = σ(p_{k}, a_{k}, n/d_{k})/8

for any k. But this follows from [31] (section 6). □

For an explicit formula of sw_{Σ(p_{k},a_{k},n/d_{k})}^{0}(σ_{can}), see [31] or [32].

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Department of Mathematics, Ohio State University, Columbus, OH 43210

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

University of Notre Dame, Notre Dame, IN 46556

   E-mail address: nicolaescu.1@nd.edu
   URL: http://www.nd.edu/ ~lnicolae/