Casson invariant of cyclic coverings via eta–invariant and Dedekind sums

by András Némethi

Department of Mathematics The Ohio State University 231 West 18th Avenue Columbus, OH 43210, USA. e-mail: nemethi@math.ohio-state.edu

§1. Introduction and the main result.

In this note we will show, that the Casson invariant of the 3–dimensional integer homology spheres behaves as a secondary invariant with respect to the n–fold cyclic (branched) coverings.

We start with some notations. If Σ is a 3–dimensional oriented manifold such that H_{∗}(Σ, Z) = H_{∗}(S^{3}, Z), we denote its Casson invariant (see [1]) by λ(Σ). If K ⊂ Σ is a knot in Σ, then the n–fold cyclic covering of Σ branched along K is denoted by Σ(K, n). We would like to compute the expression λ(Σ(K, n)) − n · λ(Σ) in terms of homological invariants of the covering, provided that Σ(K, n) is an integer homology sphere as well.

In this note, if the group of coefficients of a homology group is not specified, then it is C. Set X = Σ \ T, where T is an open tubular neighborhood of K, and let X̃ be the infinite cyclic covering of X determined by the natural homomorphism π_{1}(X) → H_{1}(X, Z) = Z. In particular Z acts freely as the group of covering transformations of X̃. Its generator 1_{Z} induces the monodromy transformation t_{∗} : H_{1}(X̃) → H_{1}(X̃). On the other hand, there exists a natural skew–symmetric non–singular cup product pairing H^{1}(X̃, ∂X̃)^{⊗2} → H^{2}(X̃, ∂X̃) = C [9]. By the duality H^{1}(X̃, ∂X̃) = H_{1}(X̃), we can consider its dual b : H_{1}(X̃) ⊗ H_{1}(X̃) → C. Moreover b(t_{∗}x, t_{∗}y) = b(x, y) for any x and y, in particular, the system I = (H_{1}(X̃), b, t_{∗}) constitutes an isometric structure. Notice also that t_{∗} has no eigenvalue equal to one. The isometric structure I defines a flat hermitian bundle over the circle S^{1} with hermitian form \sqrt{−1} · b and monodromy t_{∗}. In particular, following Atiyah–Patodi–Singer [2], we can define its eta– invariant. Actually, for any integer n with the property det(t_{∗}^{n} − 1) ≠ 0 (or equivalently H_{1}(Σ(K, n)) = 0), we define η(I; n) in the following way. Consider the spectral decomposition (H_{1}(X̃), b) = ⊕_{χ}(H_{1}(X̃)_{χ}, b_{χ}) provided by the operator t_{∗}^{n} (i.e. (H_{1})_{χ} is the generalized χ–eigenspace of t_{∗}^{n}). Then the eta–invariant η(I; n) is defined by the sum ∑_{χ} η(I; n)_{χ}, where

η(I; n)_{χ} = \{ \begin{matrix}(1 − 2c) · signature(ib_{χ}) & if χ = e^{2πic}; 0 < c < 1, \\ 0 & if |χ| ≠ 1. \end{matrix}

(In the above definition we used the fact that χ = 1 is not an eigenvalue of t_{∗}^{n}. For the definition of η in the general case, see [14].)

Our leading conjecture is the following:

Conjecture. Let (Σ, K) be as above and assume that the n-fold cyclic covering of Σ, branched along K, is integer homology sphere as well. Then

(∗) λ(Σ(K, n)) − n · λ(Σ) = \frac{1}{8}(η(I; n) − n · η(I; 1)).

In this note we prove the following:

Theorem 1. The above conjecture is true if (Σ, K) has a plumbing representation (or equivalently, if (Σ, K) is a graph knot, in the sense of Eisenbud-Neumann’s book [5]).

In the plumbing representation we allow non–connected plumbing graphs as well, or equivalently, we allow disjoint union of splice diagrams.

Remarks.

1) Using surgery, it is not difficult to prove that the conjecture is true if and only if it is true for Σ = S^{3} (and for arbitrary knot K ⊂ S^{3}).

2) The above theorem gives in homological terms the Casson invariant of any cyclic covering of S^{3} branched along a knot K ⊂ S^{3}, which can be represented by plumbing (or splice) diagram (since λ(S^{3}) = 0).

3) The formula from Theorem 2 shows that the function n ↦ η(I; n) (n with det(t_{∗}^{n} − 1) ≠ 0) is periodic (cf. [14], section 5). Therefore, if (Σ, K) is a graph knot, then n ↦ λ(Σ(K, n)) is quasi–periodic (i.e. is a sum of a linear function and a periodic function).

4) The relation (∗) makes sense even in the case of rational homology spheres (for the left hand side, see generalization by Walker [24]). Hence, one can expect that it is true even in that case.

The first part of the proof of the theorem contains the reduction to the Seifert graph case. The key steps of the proof are the relations (1) and (3) (proved in [6] and [14] respectively). The first one relates the Casson invariant to the signature of the Milnor fiber of a hypersurface singularity. The second one expresses this signature in terms of eta–invariant, this is an index–theoretical result about the signature defect.

In section 3 and 4, we emphasize the arithmetical aspects. If a 3–manifold Σ can be represented by a splice diagram, then there is an arithmetical formula of λ(Σ) in terms of the weights (see, for example, [13]). In this paper we give a different set of relations (see below).

In section 3, we express the eta–invariant η(I; n) in terms of generalized Dedekind sums [20, 25] associated with the multiplicities of the plumbing graph of the pair (Σ, K) (Theorem 2). This relation makes the connection between the Casson invariant, eta–invariant and the Dedekind sums. In section 4 we apply the Rademacher–Dedekind reciprocity law in order to transform our relation into a deeper formula which describes λ(Σ(K, n)) in terms of the combinatorics of the graph and Dedekind sums (Theorem 3). Here we discuss also the particular case of algebraic knots (Σ, K) and we reprove the main result of [13]. This makes the connection with the signature of hypersurface singularities and the number of lattice points in a (three dimensional) tetrahedron (cf. [18]). In particular, we reprove a result of Mordell [10] about the number of lattice points in a tetrahedron.

§2. Proof of Theorem 1.

In the case of integer homology spheres, the plumbing representation is equivalent to the representation of (Σ, K) in terms of splice diagrams [5]. Since the latter one is more concise, in this proof we prefer this representation. Reduction to the irreducible case. Assume that the splice diagram Γ of (Σ, K) is not connected: Γ is the disjoint union Γ_{1} ∪ Γ_{2}, where Γ_{1} is connected and contains the arrow corresponding to the knot K. This means that Σ is the disjoint sum Σ_{1}#Σ_{2}, where K ⊂ Σ_{1}. Moreover, Σ(K, n) = Σ_{1}(K, n)#Σ_{2}# ⋯ #Σ_{2} (n copies of Σ_{2}), hence by the additivity theorem of Casson (see, [1]): λ(Σ(K, n)) − nλ(Σ) = λ(Σ_{1}(K, n)) − nλ(Σ_{1}). On the other hand, the isometric structure I(Σ, K) associated with the pair (Σ, K) is isomorphic to the isometric structure I(Σ_{1}, K), hence the relation (∗) is true if and only if it is true for connected diagrams.

In the sequel we assume that the splice diagram Γ of (Σ, K) is connected and n > 1. The irreducible case. Call a weight on an edge of Γ “near” or “far” according as it is on the end of the edge nearest to or further from the (unique) arrowhead of Γ (representing K). We recall the following results. (In the following two facts we assume that Γ is minimal.)

Fact 1. [13] The n–fold cyclic cover of Σ branched along K is an integer homology sphere Σ(K, n) if and only if n is prime to all near weights in Γ, and Σ(K, n) is then represented by the splice diagram obtained from Γ by multiplying each far weight by n.

Fact 2. [5, (11.2)] (Σ, K) is a fibered knot if and only if the multiplicities of all nodes are non–zero, or equivalently, all the near weights in Γ are non–zero.

These give the following:

Corollary 1. The irreducible graph knot (Σ, K) is fiberable if and only if there exists n > 1 such that the n–fold cyclic cover Σ(K, n) is an integer homology sphere. In particular, we obtain the fiberability of (Σ, K) in the main theorem.

If F denotes the fiber of Σ \ T → S^{1} (which is isotopic to the minimal Seifert surface of K), then X̃ ∼ F × R, hence: I = (H_{1}(F); <, >; h), where <, > is the skew-symmetric intersection form on the 2–dimensional manifold F, and h is the monodromy transformation of the fibration.

Notice that the graph Γ is not unique, but equivalent graphs ([5], 8.1) give the same invariants λ and η, so we can choose (by [5], 8.1, property 6) a diagram Γ such that each vertex has a degree less than or equal to three (see also [13], p. 60).

Now, if we have a multilink (Σ′, K′(m)), represented by the graph Γ′:

[figure: Γ′′_{1}; Γ′′_{2}; Γ′ :; ϵ; (m) K′ (ϵ = ±1); a; c; b]

then we can consider its splice decomposition:

[figure: Γ′′_{1}; Γ′′_{2}; (bm) K′′_{1}; (am) K′′_{2}; (0); (0); ϵ; (m) K′; a; c; b]

and the splice diagram:

[figure: Γ_{a,b,c}(ϵ, m) :; ϵ; (m) K′; a; c; b]

(Sometimes it is convenient to write a = a_{1}, b = a_{2}. Above, if Γ′′_{i} (i = 1, 2) is only a vertex, then the diagram Γ′′_{i} → (a_{i}m) is empty.)

Using the additivity of the Casson invariant under splicing, proved independently by Ahbulut & McCarthy, Boyer & Nicas and Fukuhara & Maruyama (see [4] and [13]), one has: λ(Σ′(K′, n)) − nλ(Σ′) = λ(Σ(a, b, c)(K′, n)) − nλ(Σ(a, b, c)) + ∑_{i=1}^{2} λ(Σ′′(K′′_{i}, n)) − nλ(Σ′′_{i}).

On the other hand, by a simple Mayer–Vietoris argument, the isometric structure I(Γ′, K′(m)) splits in the direct (orthogonal) sum ⊕_{i=1}^{2}I(Γ′′_{i}, K′′_{i}(a_{i}m)) ⊕I(Γ_{a,b,c}(ϵ, m), K′(m)) (see, for example, [5], p. 114-116). Therefore, in order to prove (∗), it is enough to prove it for diagrams Γ_{a,b,c}(ϵ, m), where (m, n) = 1 (cf. Fact 1), and the isometric structure is provided by the fibration of the multilink structure K′(m).

Using [5] (8.1, property 2), we can assume that a ≥ 0, b ≥ 0, c ≥ 0. Fact 2 gives that a > 0 and b > 0. If c = 0, then (a, b, c) = (1, 1, 0) and (by the classification theorem of [5]) Σ is S^{3} and the link is trivially embedded. Hence both sides of (∗) are zero.

In the sequel assume that a > 0, b > 0, c > 0. By [5] (8.1, property 1), we can assume that m > 0 as well. In the diagram ϵ = ±1 denotes the orientation class of Σ, if we change ϵ into −ϵ then in (∗) both sides will change their sign, so we can assume that ϵ = +1 (cf. [5], p. 119).

Notice that the left hand side of (∗), applied for the diagram Γ_{a,b,c}(1, m) is independent of (the multilink structure) m. In the following lemma we prove the similar fact for the right hand side.

Let F be the fiber of (Γ_{a,b,c}(1, 1), K(1)), b the intersection form, h the monodromy operator, and set I = (H_{1}(F, k), b, h). Then the fiber of the multilink (Γ_{a,b,c}(1, m), K(m)) is the disjoint union F ∪⋯∪F (m copies) and the corresponding isometric structure I_{m} is ([5], p. 115): I_{m} = (H_{1}(F, k)^{⊕m}, b^{⊕m}, h_{m}), where h_{m}(x_{1}, …, x_{m}) = (x_{2}, x_{3}, …, h(x_{1})). Lemma 1. Assume that m > 0, (m, n) = 1, and det(h^{n} − 1) ≠ 0. Then η(I_{m}; n) = η(I; n), in particular:

η(I_{m}; n) − n · η(I_{m}; 1) = η(I; n) − n · η(I; 1).

Proof. Notice that the monodromy h has distinct eigenvalues hence, over C, the isometric structure I decomposes in a sum of one–dimensional hermitian isometric structures [8]. Therefore, it is enough to verify the above equality for a hermitian isometric structure I = (C, i, e^{2πic}), where 0 < c < 1, and nc ∉ Z. In this case: I_{m} = ⊕_{j=0}^{m−1}(C, i, e^{2πi(c+j)/m}), hence:

η(I_{m}; n) = ∑_{j=0}^{m−1}(1 − 2\{n · \frac{c + j}{m} \}) = 1 − 2\{nc\} = η(I; n)

(cf. (A.1) or Lemma 1 [21]. Above {x} denotes the fractional part of x.) □

Remark 1. It is not difficult to verify that for any Γ and m > 0 one has: η(I(Γ, K(−m)); n) = η(I(Γ, K(m)); n). Indeed, the modification m ↦ −m in the multilink structure provides the modification (b, t_{∗}) ↦ (−b, t_{∗}^{−1}) in the isometric structure. In particular, η(I(Γ, K(m)); n) is independent of the choice of m ∈ Z \ {0}.

By lemma 1, we can assume that m = 1, i.e. (Σ = Σ(a, b, c), K) is the Seifert knot given by the diagram Γ_{a,b,c}(1, 1), where a, b, c are relative prime integers. The n–fold cyclic covering Σ(K, n) is exactly Σ(a, b, cn) (where (a, n) = (b, n) = 1). Now, by [6, (2.10)]:

(1) \begin{matrix}8 · λ(Σ(a, b, N)) = & signature σ of the Milnor fiber of the hyper- \\ & surface singularity x^{a} + y^{b} + z^{N}. \end{matrix}

Let I_{a,b} = (H, b_{a,b}, h_{a,b}) be the isometric structure associated with the knot (S^{3} = Σ(a, b, 1), K_{a,b}) (given by the diagram Γ_{a,b,1}(1, 1)). Then (see, for example, [7]) (Σ(a, b, N), K) is the N–fold cyclic covering of (S^{3}, K_{a,b}), in particular:

(2) I(Σ(a, b, N), K) = (H, b_{a,b}, h_{a,b}^{N}).

Now, by [14] (5.22):

(3) σ(x^{a} + y^{b} + z^{N}) = η(I_{a,b}; N) − N · η(I_{a,b}; 1).

Therefore:

8[λ(Σ(K, n)) − nλ(Σ)] =^{(1)} σ(x^{a} + y^{b} + z^{nc}) − n · σ(x^{a} + y^{b} + z^{c})

=^{(3)} η(I_{a,b}; nc) − ncη(I_{a,b}; 1) − n[η(I_{a,b}; c) − cη(I_{a,b}; 1)]

= η(I_{a,b}; nc) − nη(I_{a,b}; c)

=^{(2)} η(I; n) − nη(I; 1).

This ends the proof of theorem 1.

§3. The eta–invariant via the plumbing graph

In this section we compute the eta–invariant η(I; n) of the isometric structure I = I(Σ, K) in terms of the combinatorical data of a plumbing graph G of (Σ, K).

First, we introduce some notations. Let W be the set of non–arrowhead vertices of G. The knot K is represented in G by an arrow attached to the vertex w_{0} ∈ W. Let V be the set of vertices: V = W ⊔ {v_{0}}, where v_{0} is the arrowhead corresponding to K. For any w ∈ W, we denote by V_{w} the set of vertices v ∈ V adjacent to w. Set δ_{w} = #V_{w} for any w ∈ W. If δ_{w} > 2, then w ∈ W is called “rupture point”. The set of rupture points is denoted by R. Let E be the set of edges (i.e. the set of non–ordered pairs (u, v), u, v ∈ V, such that u adjacent to v).

Since Σ is a homology sphere, G is a tree. It is decorated by the Euler– numbers e_{w} (w ∈ W). We recall that the plumbing construction provides a canonical orientation of Σ, and an orientation of K: K is a fiber of the oriented circle bundle corresponding to w_{0}.

Recall that any m ∈ H_{1}(Σ \ K, Z)^{∗} defines a multilink structure of the pair (Σ, K) [5]. If M = M_{v_{0}} is the standard topological meridian of K (i.e. l(M, K) = 1, where l denotes the linking number), then m(M) = m_{v_{0}} (which sometimes is denoted only by m) is the multiplicity of the knot K; and the map [S] ↦ m([S]) is defined by m([S]) = m · l(S, K). For any w ∈ W let M_{w} be an oriented fiber of the oriented circle bundle (used in the plumbing construction) corresponding to w. Then the “multiplicity of w” is:

(3.1) m_{w} = m([M_{w}]) = m · l(M_{w}, K).

They satisfy the following relations: for any w ∈ W one has

(3.2) e_{w} · m_{w} + ∑_{v∈V_{w}} m_{v} = 0.

Lemma 2. Let (Σ, K) and n be as in the introduction, G a plumbing graph of (Σ, K), and m = m_{v_{0}} = ±1. Then one has:

a) (n, m_{w}) = 1 for any w ∈ R;

b) (n, m_{u}, m_{v}) = 1 for any (u, v) ∈ E; and

c) if G is minimal (see, for example [5]), then m_{v} ≠ 0 for any v ∈ V. Proof.

a) The rupture points correspond to the Seifert components in a splice decomposition. By the algorithm given in [5] (p. 84) m_{w} is a product of near weights. Now apply Fact 1.

b) Consider a chain {u_{1}, u_{2}, …, u_{s}} of G (i.e. u_{i+1} ∈ V_{u_{i}} for i = 1, …, s− 1; δ_{u_{i}} = 2 for i = 2, …, s − 1) such that {u_{1}, u_{2}} = {u, v} and u_{s} ∈ R. Let d = (n, m_{u}, m_{v}). Then (3.2) applied for the nodes u_{2}, …, u_{s−1} gives that d|m_{u_{s}}. Therefore d = 1 by a).

c) This follows from the fiberability condition; it is a reformulation of Fact 2 for plumbing graphs. □

In the sequel it is convenient to use the following classical notation:

((x)) = \{ \begin{matrix}\{x\} − \frac{1}{2} & if x ∉ Z \\ 0 & if x ∈ Z. \end{matrix}

Theorem 2. Let (Σ, K) be as above, and I its isometric structure. Let G be a plumbing graph of (Σ, K) such that m_{v} ≠ 0 for any v ∈ V. Fix a multilink structure m with m ≠ 0. Let n be an integer such that Σ(K, n) is a rational homology sphere. Then:

(∗∗) η(I, n) = 4 · ∑_{w∈W} S_{w} where S_{w} = ∑_{v∈V_{w}} ∑_{k=1}^{m_{w}−1}((\frac{km_{v}}{m_{w}})) · ((\frac{kn | m |}{| m_{w} |})).

Remark 2.

a) Notice that by (3.1) S_{w} is independent of the choice of m. (This fact is consistent with lemma 1 and remark 1.)

b) Equivalent graphs (with the property m_{w} ≠ 0 for any w ∈ W) provide the same expression ∑_{w} S_{w}, i.e. the operations described on the page 140 in [5] do not alter the right hand side of (∗∗).

c) By (∗∗), the function n ↦ η(I; n) is periodic.

Proof of Theorem 2. The relations (3.2) imply that for w with δ_{w} ≤ 2 one has S_{w} = 0. Therefore, the right hand side R(G, n) = 4 ∑_{w∈W} S_{w} of (∗∗) is 4·∑_{w∈R} S_{w}. This shows that R(G, n) is additive with respect to splicing. On the other hand, we can repeat the additivity argument of the eta–invariant used in the proof of theorem 1 (namely [5], p. 114-116). These, and remark (2.a), imply that it is enough the verify the identity η(I, n) = R(G, n) only for plumbing graphs corresponding to Seifert knots (ϵΣ(a_{1}, …, a_{δ}), K(1)), where a_{2}, …, a_{δ} > 0, a_{1} ≥ 0, and a_{1} is the far–weight of the edge which has the arrow corresponding to K.

If a_{1} = 0, then a_{i} = 1 for i > 1 and η(I; n) = 0. But the multiplicity of the unique rupture point is ±1, so R(G, n) = 0 too.

In the sequel we assume that a_{1}, …, a_{δ} > 0. In the next paragraph we prove that the case ϵ = −1 follows from the case ϵ = +1.

Indeed, if G({e_{w}}_{w∈W}, K(1)) is the decorated plumbing graph associated with (+Σ(a_{1}, …, a_{δ}), K(1)), then G({−e_{w}}_{w∈W}, K((−1)^{r})) is a possible decorated graph of (−Σ(a_{1}, …, a_{δ}), K(1)), where r is the number of vertices in the chain (strict) between the rupture point and the arrow-head ([11] (3.3)). If m_{v}^{±} denotes the corresponding multiplicities for ϵ = ±1, then by (3.2) m_{v}^{−} ∈ {m_{v}^{+}, −m_{v}^{+}} for any v ∈ V, and

\frac{m_{u}^{−}}{m_{v}^{−}} = −\frac{m_{u}^{+}}{m_{v}^{+}} for any (u, v) ∈ E.

Therefore R(G(−ϵ), n) = −R(G(+ϵ), n). But η(I(−Σ, K); n) = −η(I(Σ, K); n) (because the change of the orientation provides the modification (b, t_{∗}) ↦ (b, t_{∗}^{−1})). Hence the reduction follows.

Now consider the knot (Σ(a_{1}, …, a_{δ}), K(1)). This is an algebraic knot (see, for example, [5], page 62): there exist an analytic normal surface singularity (X , x) and an analytic germ f : (X , x) → (C, 0) such that (X ∩S_{r}, f^{−1}(0)∩S_{r}) is diffeomorphic to (Σ, K), where S_{r} = ρ^{−1}(r) for a sufficiently small r > 0 and real analytic map ρ : X → [0, ∞) with ρ^{−1}(0) = {x}.

In the sequel we would like to apply the results of [14] (section 5.16-5.22) and [15]. These are formulated for hypersurface singularities. In their proofs we used two ingredients: the variation map associated with the germ f is an isomorphism (over real numbers), and the polarization properties of the limit mixed Hodge structure of the vanishing cohomology of f. In our new situation here, X ∩ S_{r} = Σ is an integer homology sphere, therefore the variation map of f (which is equivalent to the Seifert form of K ⊂ Σ by Alexander duality) is unimodular. On the other hand, all the polarization properties of the mixed Hodge structure are valid in this case as well, see, for example [23, 17, 15]. (Actually, in our case the monodromy operator has no eigenvalues =1, so the limit mixed Hodge structure of the vanishing cohomology has the same nice polarization properties as the limit mixed Hodge structure associated with degeneration of projective fibers.) In particular, the results described in [15] and [14] (5.16-5.22) are true for germs f : (X , x) → (C, 0) as well, provided that X ∩ S_{r} is a homology sphere.

Notice that det(t_{∗} − 1) ≠ 0 and det(t_{∗}^{n} − 1) ≠ 0, hence (5.20-5.21) in [14] reads as:

η(I; n) = 2 · ∑((nc))(Σp_{λ,−}(f) − Σp_{λ,+}(f)),

where the sum is over all eigenvalues λ = e^{−2πic} (0 < c < 1) of the monodromy operator; and

Σp_{λ,±}(f) = #{c : c is a spectral number of f with λ = e^{−2πic}, (−1)^{[c]} = ±1}.

On the other hand, the set of spectral numbers (or the characteristic numbers) Sp(f) ∈ Z[Q] associated with f is computed from the plumbing graph (or equivalently from the embedded resolution graph (X , f^{−1}(0))) in [22] and [19]. Proposition (6.5) and remark (2.11) of [19] give:

Sp(f) = ∑_{e∈E} ∑_{k=1}^{m_{e}−1}( (−\frac{k}{m_{e}}) + (\frac{k}{m_{e}}) )+

+ ∑_{w∈W} ∑_{k=1}^{m_{w}−1}( ∑_{v∈V_{w}}\{\frac{km_{v}}{m_{w}} \} − 1 )( (1 − \frac{k}{m_{w}}) + (−1 + \frac{k}{m_{w}}) ),

where m_{e} = (m_{u}, m_{v}) for any e = (u, v) ∈ E. Since

(3.3)  ∑_{k=1}^{m−1}((\frac{kn}{m})) = 0,

the contribution from E is zero in η(I; n), and by a computation:

η(I; n) = 4 · ∑_{w∈W} ∑_{v∈V_{w}} ∑_{k=1}^{m_{w}−1}\{\frac{km_{v}}{m_{w}} \}((\frac{kn}{m_{w}})).

This is equivalent to (∗∗) because of (3.3). □

Remark 3. Assume that (Σ, K) is represented by a splice diagram Γ. Then the right hand side of (∗∗) can be computed (without the construction of the whole plumbing graph) as follows [12]. We need for any w ∈ R the multiplicities {m_{v}}_{v∈V_{w}} (modulo m_{w}). The rupture points of the plumbing diagram, which can be constructed by the algorithm described in [5], corresponds to the Seifert components of Γ. If w ∈ R corresponds to the Seifert component (Σ(a_{1}, …, a_{δ}); m_{1}K_{1} ∪ ⋯ ∪ m_{δ}K_{δ}), then m_{w} = ∑_{j=1}^{δ} a_{1} ⋯ â_{j} ⋯ a_{δ}m_{j}, and the neighbor vertex corresponding to the edge marked with a_{j} (j = 1, …, δ) has a multiplicity, which is modulo m_{w} equal to (m_{j} − β_{j}m_{w})/a_{j}, where the numbers {β_{j}}_{j} satisfy β_{j}a_{1} ⋯ â_{j} ⋯ a_{δ} ≡ 1 (mod a_{j}).

§4. The Casson and eta invariant via generalized Dedekind sums

For arbitrary non–zero integers a, b, c, we consider the generalized Dedekind sum (cf. [20, 25]):

s(b, c; a) = ∑_{k=1}^{a}((\frac{kb}{a}))((\frac{kc}{a})).

Using this notation, theorem 2 reads as: Theorem 2’. With the choice m = 1 one has:

η(I; n) = 4 ∑_{w∈R} ∑_{v∈V_{w}} sign(m_{w}) · s(m_{v}, n; m_{w}).

Now, we recall the famous generalization of the reciprocity law of Dedekind given by Rademacher [20] (see also [25]). If a, b, c are strict positive, mutually coprime integers, then: s(b, c; a) + s(c, a; b) + s(b, a; c) = −1/4 + (a^{2} + b^{2} + c^{2})/(12abc). Now, for any a, b and c with (a, b, c) = 1:

s(b, c; a) = s(\frac{b}{(a, b)(b, c)}, \frac{c}{(a, c)(b, c)}; \frac{a}{(a, b)(a, c)}).

Therefore, Rademacher’s result reads as: Reciprocity Law. Let a, b, c be strict positive integers such that (a, b, c) = 1. Then the following relation holds:

(RL) s(b, c; a)+s(c, a; b)+s(a, b; c) = −\frac{1}{4}+\frac{a^{2}(b, c)^{2} + b^{2}(a, c)^{2} + c^{2}(b, a)^{2}}{12abc} .

Using this, in the next theorem we express the eta–invariant in terms of Dedekind sums with denominator n. Theorem 3. Let (Σ, K), I and n be as in the introduction. Fix a plumbing graph G with a multilink structure given by m_{v_{0}} = 1, such that m_{w} ≠ 0 for any m_{w} ∈ W (e.g. take G minimal). Then, with the notations of section 3, one has:

η(I; n) = −4 ∑_{(u,v)∈E} s(m_{u}, m_{v}; n) − ∑_{(u,v)∈E} sign(m_{u}m_{v}) + n · ∑_{(u,v)∈E} \frac{(m_{u}, m_{v})^{2}}{3m_{u}m_{v}} +

+\frac{1}{3n}( m_{w_{0}} + ∑_{w∈W}(n, m_{w})^{2}(−e_{w}) ).

Proof. First notice that (m_{u}, m_{v}, n) = 1 for any (u, v) ∈ E (cf. lemma 2b). Then:

η(I; n) = 4 ∑_{w∈W} ∑_{v∈V_{w}} sign(m_{w}m_{v}) · s(| m_{v} |, n; | m_{w} |) =

= 4 ∑_{(u,v)∈E} sign(m_{u}m_{v})( s(| m_{v} |, n; | m_{u} |) + s(| m_{u} |, n; | m_{v} |) ).

Now apply (RL), and notice the following:

∑_{(u,v)∈E} \frac{m_{u}^{2}(n, m_{v})^{2} + m_{v}^{2}(n, m_{u})^{2}}{m_{u}m_{v}} = \frac{(n, 1)^{2}m_{w_{0}}^{2}}{1 · m_{w_{0}}} + 4 ∑_{w∈W} ∑_{v∈V_{w}} \frac{(n, m_{w})^{2}m_{v}^{2}}{m_{w}m_{v}} =

= m_{w_{0}} + ∑_{w∈W} \frac{(n, m_{w})^{2}}{m_{w}} ∑_{v∈V_{w}} m_{v} =^{(3.2)} m_{w_{0}} + ∑_{w∈W}(n, m_{w})^{2}(−e_{w}). □

Corollary 3. With the above notations, one has:

λ(Σ(K, n)) − n · λ(Σ) = −4 ∑_{(u,v)∈E} s(m_{u}, m_{v}; n) − (1 − n) · ∑_{(u,v)∈E} sign(m_{u}m_{v})+

+\frac{1 − n^{2}}{3n} m_{w_{0}} + ∑_{w∈W} \frac{(n, m_{w})^{2} − n^{2}}{3n} (−e_{w}).

Example. If Σ(K, 2) is an integer homology sphere, then:

λ(Σ(K, 2)) − 2 · λ(Σ) = ∑_{(u,v)∈E} sign(m_{u}m_{v}) + \frac{1}{2}( −m_{w_{0}} + ∑_{{w∈W \atop mwodd}} e_{w} ).

Remark 4. η(I; n) even for n = 1 is important, because:

(−8) · lim_{n→∞} \frac{λ(Σ(K, n)) − n · λ(Σ)}{n} = η(I; 1).

(Notice that in the expression of η(I; 1) (in Theorem 3) s(m_{u}, m_{v}; 1) = 0.)

The algebraic case. If (Σ, K) is algebraic (i.e. it is the link of the pair (X , f^{−1}(0)), where (X , x) is a normal surface singularity, and f : (X , 0) → (C, 0) is an analytic germ), then the embedded resolution graphs of (X , f^{−1}(0)) provide nice plumbing diagrams. E.g.: they satisfy: e_{w} < 0 and m_{w} > 0 for any w ∈ W (here m = m_{v_{0}} = 1).

If X is smooth, then Σ = S^{3} and K ⊂ S^{3} is the link of an irreducible plane curve singularity f. Let I(f) be its isometric structure (i.e. I(f) = I(S^{3}, K)). By [7], Σ(K, n) is exactly the link of the singularity {f(x, y)+z^{n}}. By theorem 1: 8 · λ(Σ(K, n)) = η(I(f); n) − nη(I(f); 1). But, by [14], the right hand side of the last equality is exactly the signature σ(f + z^{n}) of the Milnor fiber of f(x, y) + z^{n}. In particular, we reproved the main result of [13], i.e.:

8 · λ(link of f + z^{n}) = signature of f + z^{n}.

Now, using this identification, all the results of [18] about the signature of f +z^{N} can be transformed for the Casson invariant of the link of {f +z^{N} = 0}.

Above, if f(x, y) = x^{a} + y^{b}, then the signature of x^{a} + y^{b} + z^{n} can be computed by Brieskorn formula [3] from the number of lattice points in the tetrahedron (0, 0, 0), (0, 0, a), (0, b, 0), (n, 0, 0). Now, if we apply for the graph of x^{a} + y^{b} (which has only one rupture point) the result of theorem 2’, we obtain the number of these lattice points in terms of Dedekind sums. This is exactly the famous Mordell’s formula [10].

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