ON THE CASSON INVARIANT CONJECTURE OF NEUMANN–WAHL

ANDRÁS NÉMETHI AND TOMOHIRO OKUMA

Abstract. In the article we prove the Casson Invariant Conjecture of Neumann– Wahl for splice type surface singularities. Namely, for such an isolated complete intersection we show that the Casson invariant of the link is one-eighth the signature of the Milnor fiber, provided that the link is an integral homology sphere.

1. Introduction

At the end of 80’s Neumann and Wahl formulated the following conjecture:

Casson Invariant Conjecture. [16] Let (X, o) be a complex analytic isolated complete intersection surface singularity whose link Σ is an integral homology 3–sphere. Then the Casson invariant λ(Σ) of the link is one-eighth the signature of the Milnor fiber of (X, o).

The conjecture is true for Brieskorn hypersurface singularities by a result of Fintushel and Stern [8]. This and additivity properties (with respect to splice decomposition of Σ) lead to the verification of the conjecture for Brieskorn complete intersections, done independently by Neumann–Wahl [16] and Fukuhara–Matsumoto–Sakamoto [9]. For suspension hypersurface singularities it was proved in [16]. Some iterative generalizations, related with cyclic coverings and equivariant Casson invariant, were covered by Collin and Saveliev (cf. [3, 4, 5]).

Recently, Neumann and Wahl have introduced an important family of complete intersection surface singularities, the splice type singularities. In [19] they treated the case when the link is an integral homology sphere (the reader may consult in [17, 18, 20] the case of rational homology sphere links too). In [19], they have also verified the above conjecture for special splice type singularities when the nodes of the splice diagram ‘are in a line’.

The goal of the present article is to verify the conjecture in the general case:

Theorem. The Casson Invariant Conjecture is true for any splice type singularity with integral homology sphere link.

This theorem can also be judged in the light of the following expectation (cf. [19, 20] or [23, §6]): any complete intersection surface singularity with integral homology sphere link is of splice type.

The article is organized as follows. In §2 we review splice type singularities. The proof of Theorem is broken into two parts: §3 contains the analytic part, while the topological/combinatorial part is in the last section 4.

2000 Mathematics Subject Classification. Primary. 14B05, 14J17, 32S25, 57M27, 57R57. Secondary. 14E15, 32S45, 57M25.

Key words and phrases. normal surface singularities, complete intersection singularities, geometric genus, signature, Neumann-Wahl conjecture, ℤ-homology spheres, Casson invariant.

The first author is partially supported by NSF grant DMS-0605323, Marie Curie grant and OTKA grants; the second author by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture, Sports, Science and Technology, Japan.

2 A. Némethi and T. Okuma

The conjecture can be reformulated in terms of the geometric genus p_{g} of (X, o) as well (see below in 4.1). In fact, in the proof we use essentially an inductive formula for p_{g}, proved in [21], which is readjusted here to the splice decomposition of Σ. But we wish to emphasize that the study of any analytic invariant with respect to the splice decomposition is rather delicate since the splice decomposition (gluing), rewritten in the language of plumbing, contains a purely topological step (namely the 0–absorption, cf. [7], §22) which cannot be represented in the world of negative definite plumbing graphs. The new construction of section 4 surmounts this difficulty.

Acknowledgements. The second author thanks Marie Curie Fellowship for Transfer of Knowledge for supporting his visit at the Rényi Institute of Mathematics, Budapest, Hungary, where the work on this paper was done. He is also grateful to the members of the Rényi Institute for the warm hospitality.

2. Splice type surface singularities (review)

This section contains a brief introduction of splice type singularities in terms of “monomial cycles” (cf. [22, §3], [20, §13]). We assume that the links are integral homology spheres (for the general case see [loc. cit.]).

The splice type singularities, which generalize Brieskorn complete intersections, were introduced by Neumann and Wahl (see [17, 19, 20]). To any resolution (or, negative definite plumbing) graph, associated with a fixed plumbed 3–manifold, one first associates a new weighted tree, called the “splice diagram”. From this one writes down the system of (leading forms of) splice diagram equations. They define a splice type singularity whose link is the original 3-manifold. (In the next presentation the splice diagram is almost ‘hidden’, the interested reader may consult the above references for more details.)

2.1. Basic notations. Let (X, o) be a germ of a normal complex surface singularity and π: X̃ → X a good resolution, i.e., its exceptional divisor E := π^{−1}(o) has only simple normal crossings. Let {E_{v}}_{v∈V} denote the set of irreducible components of E. The link Σ of (X, o) is an integral homology sphere if and only if E is a tree of rational curves and the intersection matrix I(E) := (E_{v} · E_{w})_{v,w∈V} is unimodular. Let (m_{vw})_{v,w∈V} = −I(E)^{−1}. Then every m_{vw} is a positive integer. We call an element of a group ∑_{v∈V} ℤE_{v} a cycle. For any cycle D = ∑_{w∈V} a_{w}E_{w}, we write m_{w}(D) = a_{w}.

Let δ_{v} = (E − E_{v}) · E_{v}, the number of irreducible components of E intersecting E_{v}. A curve E_{v} (or its index v) is called an end (resp. a node) if δ_{v} = 1 (resp. δ_{v} ≥ 3). We denote by E (resp. N) the set of indices of ends (resp. nodes). A connected component of E − E_{v} is called a branch of v.

2.2. Monomial cycle. For any v ∈ V, let E_{v}^{∗} = ∑_{w∈V} m_{vw}E_{w}. Then E_{v}^{∗} · E_{w} = −δ_{vw} for every w ∈ V, where δ_{vw} denotes the Kronecker delta. An element of the semigroup ∑_{w∈E} ℤ_{≥0}E_{w}^{∗}, where ℤ_{≥0} is the set of nonnegative integers, is called a monomial cycle. Let ℂ{z} := ℂ{z_{w}; w ∈ E} be the convergent power series ring in #E variables. Then for a monomial cycle D = ∑_{w∈E} α_{w}E_{w}^{∗}, we associate a monomial z(D) := ∏_{w∈E} z_{w}^{α_{w}} ∈ ℂ{z}.

2.3. Degree and order associated with v. We fix v ∈ V. One defines the v-weight of any variable z_{w}, w ∈ E, by m_{vw}. Note that gcd{m_{vw}}_{w∈V} = 1. If D is a monomial cycle, then the v-degree of z(D) is equal to m_{v}(D).

For any f ∈ ℂ{z} write f = f_{0} + f_{1} ∈ ℂ{z}, where f_{0} is a nonzero quasihomogeneous polynomial with respect to the v-weight and f_{1} is a series in monomials of higher v-degrees. Then we call f_{0} the v-leading form of f, and denote it by LF_{v}(f). We define the v-order of f by v-ord(f) := v-deg(LF_{v}(f)).

2.4. Monomial Condition. We say that E satisfies the monomial condition if for any node v and any branch C of v, there exists a monomial cycle D such that D − E_{v}^{∗} is an effective cycle supported on C. In this case, z(D) is called an admissible monomial belonging to the branch C. Note that a chain C always admits a monomial cycle.

On the Casson Invariant Conjecture 3

The monomial condition is equivalent to the semigroup and congruence conditions of Neumann-Wahl (see [20, §13]) (although the congruence condition is trivial in the present integral homology sphere case).

2.5. Neumann-Wahl system. Assume that the monomial condition is satisfied. For any fixed node v let C_{1}, …, C_{δ_{v}} be the branches of v. Suppose {m_{1}, …, m_{δ_{v}} } is a set of admissible monomials such that m_{i} belongs to C_{i} for i = 1, …, δ_{v}. Let F = (c_{ij}), c_{ij} ∈ ℂ, be any ((δ_{v} −2)×δ_{v})-matrix such that all the maximal minors have rank δ_{v} −2. We define polynomials f_{1}, …, f_{δ_{v}−2} by

( f_{1} ; ⋮ ; f_{δ_{v}−2}) = F ( m_{1} ; ⋮ ; m_{δ_{v}} ) .

We call the set {f_{1}, …, f_{δ_{v}−2}} a Neumann-Wahl system at v. If we have a Neumann-Wahl system F_{v} at every node v then we call the set F := ⋃_{v∈N} F_{v} a Neumann-Wahl system associated with E. Note that #F = #E − 2.

2.6. Splice diagram equations. Consider a finite set of germs

{f_{vj_{v}} | v ∈ N , j_{v} = 1, …, δ_{v} − 2 } ⊂ ℂ{z}.

If the set {LF_{v}(f_{vj_{v}}) | v ∈ N , j_{v} = 1, …, δ_{v} − 2 } is a Neumann-Wahl system associated with E then a system of equations

f_{vj_{v}} = 0, v ∈ N , j_{v} = 1, …, δ_{v} − 2,

is called the splice diagram equations and the germs {f_{vj_{v}} } splice diagram functions. The germ of singularity defined by the splice diagram equations in (ℂ^{#E}, o) is called a splice type singularity (associated with the combinatorics of E).

We note that a splice type singularity is an equisingular deformation of a singularity defined by a Neumann-Wahl system associated with E (cf. [22, (4.)]). Notice also that the validity of the monomial condition and the construction of splice equations depend only on the dual graph Γ associated with E, hence only on a fixed plumbing graph of the link Σ. For a fixed Γ, the family of splice singularities associated with Γ form an equisingular family depending only on this graph.

2.7. Theorem. [19, (2.1)] Suppose that (Z, o) is a splice type singularity associated with Γ. Then (Z, o) is an isolated complete intersection surface singularity whose link is Σ.

2.8. End-Curve Condition. We keep the notations of (2.1). We say that X̃ satisfies the end-curve condition if for each w ∈ E there exists an irreducible curve H_{w} ⊂ X̃, not an exceptional curve, and a function f ∈ H^{0}(OX̃(−E_{w}^{∗})) such that div(f) = E_{w}^{∗} + H_{w}. We call such f an end-curve function of E_{w}. The end-curve condition is equivalent to the fact that for every w ∈ E the linear system of OX̃(−E_{w}^{∗}) has no fixed component in E.

Notice that if X̃ satisfies the end-curve condition then so do the minimal good resolution and any resolution obtained from X̃ by blowing-up the singular points of E.

If (X, o) is defined by splice diagram equations, then its resolution X̃ satisfies the endcurve condition (since the coordinate functions serve as end-curve functions). The converse is guaranteed by the following result, proved for integral homology sphere links in [19, (4.1)]; for a singularity with rational homology sphere link the corresponding statement was announced in [15].

2.9. Theorem. Let π : X̃ → X be a good resolution of a singularity (X, o). If X̃ satisfies the end-curve condition, then E satisfies the monomial condition and X is of splice type. In fact, if

ψ: ℂ{z_{w}; w ∈ E} → O_{X,o}

is a homomorphism of ℂ-algebras which maps each z_{w} = z(E_{w}^{∗}) to an end-curve function of E_{w}, then ψ is surjective and Ker ψ is generated by splice diagram functions.

3. Filtration and p_{g}-formula

Assume that (X, o) is of splice type, hence its resolution satisfies the end-curve condition. We also assume the presence of a node in E. The main result of this section presents an inductive formula for the geometric genus p_{g}(X) of X with respect to a modification which removes an arbitrary fixed component E_{v} from E. It is convenient to choose v in such a way that it isolates either a ‘star-shaped subgraph’ or a ‘string’. The inductive formula involves a filtration of O_{X,o} associated with v. It can be proved by repeating the line of arguments of [21] for our slightly more general situation (in [21] v is node); for the convenience of the reader we provide the main points of the proof emphasizing the differences.

Let v_{1} be an end-node, i.e., E_{v_{1}} is an end of the minimal reduced connected cycle containing all nodes of E. Let v_{2} be the node which is nearest to v_{1}. We fix v ∈ V in that branch of v_{2} which contains v_{1} such that v is not an end. (If v_{1} is the only node, let v be any vertex which is not an end.) Let C_{1}, …, C_{δ_{v}} denote the branches of E_{v}.

3.1. Filtration with respect to E_{v}. The v-order defines a filtration {F_{n}} of ℂ{z} by

F_{n} = {f ∈ ℂ{z} | v-ord(f) ≥ n} (n ∈ ℤ_{≥0}).

This induces a filtration of ideals I_{n} ⊂ O_{X,o} (n ∈ ℤ_{≥0}) by I_{n} := ψ(F_{n}) (for ψ : ℂ{z} → O_{X,o} see (2.9)). Let G denote the associated graded algebra ⨁_{n≥0} G_{n}, where G_{n} = I_{n}/I_{n+1}.

3.2. Proposition. (Cf. [20, (2.6)].) Let {f_{wj_{w}}} ⊂ ℂ{z} be a set of splice diagram functions as in (2.6), and I the ideal of the polynomial ring ℂ[z] generated by v-leading forms {LF_{v}(f_{wj_{w}})}. Then G ≅ ℂ[z]/I and it is a reduced complete intersection ring.

Proof. The statement can be reduced to the case when v is a node considered in [20, (2.6)] (or [22, (4.4)]) as follows. Let D_{1}, …, D_{δ_{v_{1}}} denote the branches of v_{1}; suppose that D_{j} is a chain of curves and the variable z_{j} corresponds to the end of D_{j} for 1 ≤ j ≤ δ_{v_{1}} − 1. If E_{v} is on one of these branches, let D_{1} be that branch. Then we may assume that the Neumann-Wahl system satisfies the following properties:

LF_{w}(f_{wj_{w}}) = m_{wj_{w}} + a_{wj_{w}}m_{wδ_{w}−1} + b_{wj_{w}}m_{wδ_{w}},  w ∈ N , 1 ≤ j_{w} ≤ δ_{w} − 2,
m_{v_{1}j} = z_{j}^{α_{j}} ,  1 ≤ j ≤ δ_{v_{1}} − 1.

Note that LF_{v}(f_{wj_{w}}) = LF_{v}(LF_{w}(f_{wj_{w}})) by [22, (3.8)]; while one gets LF_{v} of LF_{w}(f_{wj_{w}}) by deleting the monomials belonging to a branch containing v. Suppose the admissible monomial M_{v_{1}δ_{v_{1}}} contains z_{t}. Then we get that

{LF_{v}(f_{wj_{w}})} ⋃{z_{1}, z_{t}}

is a regular sequence if and only if

{LF_{v_{1}}(f_{wj_{w}})} ⋃{z_{1}, z_{t}}

is so. In [20, (3.4)] it is shown that the second set is a regular sequence and {LF_{v_{1}}(f_{wj_{w}})} ⋃{z_{1}} defines a curve in ℂ^{#E} smooth off the origin. Hence {LF_{v}(f_{wj_{w}})} is also a regular sequence, and G ≅ ℂ[z]/I by [20, (3.3)]. For the reducedness, see the argument at the end of section 3 of [20]. □

3.3. Hilbert series. Let H(t) denote the Hilbert series of the graded ring G, i.e.,

H(t) = ∑_{i≥0} (dim G_{i}) t^{i}.

From (3.2), by a well–known formula valid for complete intersections, one has:

3.4. Proposition.

H(t) = ∏_{w∈V} (1 − t^{m_{vw}})^{δ_{w}−2} .

From (3.2) one may also read the a–invariant of Goto and Watanabe associated with G (namely ∑_{w∈N ,j_{w}} v-deg(LF_{v}(f_{wj_{w}})) − ∑_{w∈E} v-deg(z_{w}), cf. [10, (3.1.4)]):

a(G) = ∑_{w∈V}(δ_{w} − 2)m_{vw}.

3.5. Let π′: X̃ → X′ be the morphism which contracts each connected component of the divisor E−E_{v} to a normal point and π_{1} : X′ → X the natural morphism. Let E′ = π′(E_{v}).

3.6. Lemma.

(1) There exists k ∈ ℤ_{>0} such that OX̃(−kE_{v}^{∗}) is π-generated.

(2) For any n ∈ ℤ_{≥0}, I_{n} = H^{0}(O_{X′}(−nE′)) = H^{0}(OX̃([−nπ^{′∗}E′])).

(3) −E′ is π_{1}-ample and π_{1} coincides with the filtered blowing up

Projan_{X} (⨁_{n≥0} π_{1∗}O_{X′}(−nE′)) → X.

(4) H^{1}(O_{X′}(−nE′)) = 0 for n > a(G).

Proof. (1) Let C be one of the branches of v, and E_{u} be that component of C which intersects E_{v}. Fix any end E_{w} of E in C, and set A := − ∑_{E_{v}⊂C} det(−C) I(C)_{vw}^{−1} E_{v}. This is an effective cycle supported by C such that m_{u}(A)E_{v}^{∗} + A is a monomial cycle, namely det(−C)E_{w}^{∗}. Hence, if we fix two branches C_{i} (i = 1, 2), we can construct two positive integers k_{i} and effective cycles A_{i} supported by C_{i} such that k_{i}E_{v}^{∗}+A_{i} is monomial. Set k = k_{1}k_{2} and f_{i} ∈ H^{0}(O_{X}) the image of the monomial z(kE_{v}^{∗} +k_{j}A_{i}), where i, j = 1, 2 and i ≠ j. Then it is easy to see that f_{1}, f_{2} ∈ H^{0} (OX̃(−kE_{v}^{∗})) generate OX̃(−kE_{v}^{∗}).

The proof of (2)–(4) is same as the proof of [21, (3.3–3.4)]. It uses part (1) and results of Tomari and Watanabe from [25] regarding filtered rings. Once (1) is known, (2)–(4) follows by the very same argument as in [21]. □

Since m_{vv}π^{′∗}E′ = E_{v}^{∗} is integral, from (2) we get for any m ∈ ℤ_{>0}

(3.1)  I_{mm_{vv}} = H^{0}(OX̃(−mE_{v}^{∗})).

3.7. p_{g}-formula. Let (X_{i}, x_{i}) denote the singularity germ obtained by contracting the branch C_{i} (hence X_{i} ⊂ X′). Fix k ∈ ℤ_{>0} such that π′_{∗}(kE_{v}^{∗}) is Cartier. Its existence is guaranteed by the proof of (3.6)(1) (namely by the existence of functions f_{i}).

3.8. Theorem. Set P (n) = ∑_{i=0}^{n−1}(dim G_{i}) for n ∈ ℤ_{>0}. Then for any integer m > a(G)/km_{vv}

p_{g}(X, o) = P (mkm_{vv}) − χ(mkE_{v}^{∗}) + ∑_{i=1}^{δ_{v}} p_{g}(X_{i}, x_{i}).

Proof. We write D = mkE_{v}^{∗} and D′ = mkm_{vv}E′. Consider the exact sequence

0 → OX̃(−D) → OX̃ → O_{D} → 0.

By equation (3.1) one has

dim_{ℂ} H^{0}(OX̃)/H^{0}(OX̃(−D)) = dim_{ℂ} I_{0}/I_{mkm_{vv}} = P (mkm_{vv}).

Therefore it suffices to show that

(3.2)  h^{1}(OX̃(−D)) = ∑ p_{g}(X_{i}, x_{i}).

From the spectral sequence

E_{2}^{i,j} = H^{i}(R^{j}π′_{∗}OX̃(−D)) ⇒ H^{n}(OX̃(−D))

one gets the exact sequence

H^{1}(π′_{∗}OX̃(−D)) ⟶ H^{1}(OX̃(−D)) ⟶^{α} H^{0}(R^{1}π′_{∗}OX̃(−D)) → 0.

Since π′_{∗}D = D′ is Cartier and π^{′∗}E′ = E_{v}^{∗}/m_{vv}, by projection formula and (3.6)(4),

H^{1}(π′_{∗}OX̃(−D)) = H^{1}(π′_{∗}OX̃(−π^{′∗}D′)) = H^{1}(O_{X′}(−D′)) = 0,

hence α is an isomorphism. Clearly the support of R^{1}π′_{∗}OX̃(−D) is in the set {x_{i}}_{i}. Since D = π^{′∗}D′ and D′ is Cartier, we have

(R^{1}π′_{∗}OX̃(−D))_{x_{i}} ≅ (R^{1}π′_{∗}OX̃ ⊗ O_{X′}(−D′))_{x_{i}} ≅ (R^{1}π′_{∗}OX̃)_{x_{i}} .

This proves (3.2). □

3.9. Periodic constants. In the formula of (3.8), P (mkm_{vv}) − χ(mkE_{v}^{∗}) is independent of m ≫ 0. Let us formulate this property. Let F (t) = ∑_{i≥0} a_{i}t^{i} be a formal power series. Set P_{F} (n) = ∑_{i=0}^{n−1} a_{i}. Suppose that P_{F} (kn) is a polynomial function of n for some k ∈ ℤ_{>0}. Then for any other k′ ∈ ℤ_{>0} satisfying the same property, the constant terms of P_{F} (kn) and P_{F} (k′n) are the same. We call this constant the periodic constant of F (t) and denote it by F |_{pc}. For example, the formula of Theorem 3.8 is

p_{g}(X, o) = H|_{pc} + ∑ p_{g}(X_{i}, x_{i}).

Note that if two formal power series F_{1}(t) and F_{2}(t) have periodic constants then

(F_{1} + F_{2})|_{pc} = F_{1}|_{pc} + F_{2}|_{pc}.

3.10. Remarks. (1) The message of theorem (3.8) is the following. Fix a negative definite resolution graph Γ (or its plumbed 3-manifold Σ) which satisfies the monomial condition. Then there exists a splice type singularity (defined by a Neumann-Wahl system) whose link is Σ. Its geometric genus is independent of the choice of the Neumann-Wahl system (by [22, 4.3] and [20, 10.1]). Hence, it depends only on Γ, and it will be denoted by p_{g}(Γ). Moreover, the geometric genus of any such analytic structure satisfies the inductive formula (3.8).

(2) The end-curve condition and splice diagram equations are considered for any surface singularity with rational homology sphere links, and the statements of part (1) are valid in this situation too. For this case, and under the assumption that v is a node, a p_{g}-formula is established in [21].

4. The proof of the main theorem

4.1. In this section we prove the main Theorem stated in the introduction. For this, we will apply theorem (3.8) in several different situations.

We start with (X, o), a splice type singularity (whose minimal good resolution X̃ satisfies the end-curve condition), and whose link Σ is an integral homology sphere. Γ will denote the dual resolution graph associated with X̃; V stays for the set of vertices of Γ.

It is well-known that in the conjecture one may replace the signature σ of the Milnor fiber by the geometric genus p_{g} of (X, o) via the formula σ + 8p_{g} + c(Γ) = 0 of Durfee and Laufer [6], where c(Γ) := K2 ; X̃ + b_{2}(X̃) depends only on Γ (here K denotes the canonical class and b_{2} the second Betti number).

If Γ has only one node then Theorem follows from [19, (7.7)]. A different argument runs as follows. Let v be the node of the graph. The above p_{g} formula (namely p_{g} = H|_{pc}, cf. 3.9) is valid for any splice type analytic structure (supported by the same topological type). On the other hand, splice diagram functions {f_{vj}}_{j=1}^{δ_{v}−2} with f_{vj} = LF_{v}(f_{vj}) define a Brieskorn complete intersection (in particular, we may assume that (X, o) admits a good ℂ^{∗}–action). Hence we may apply [16] too.

Therefore, in the sequel we assume that Γ has at least two nodes.

4.2. In our inductive procedure the following fact is crucial (cf. 3.10): for any splice type singularity with fixed (not necessarily minimal) good resolution graph Γ, consider any connected subgraph Γ′ ⊂ Γ, which is not necessarily unimodular. Let E′ denote the reduced connected cycle corresponding to Γ′. Then a neighborhood of E′ satisfies the endcurve condition (see [21, (2.15)]); hence E′ (or Γ′) satisfies the monomial condition too (cf. (2.9)). Therefore (cf. 3.10, or [21]) if (X′, x′) denotes the normal surface singularity obtained by contracting E′, then p_{g}(X′, x′) is topologically computable from Γ′. We write p_{g}(X′, x′) = p_{g}(Γ′).

4.3. Starting the induction. Let v_{1} ∈ N be an end-node of Γ and v_{2} ∈ N the node which is nearest to v_{1}. Write Γ′ ⊂ Γ for the branch of v_{1} containing v_{2}, and denote by w that vertex in Γ′ which is connected by v_{1} in Γ. Let H_{Γ,v_{1}}(t) be the Hilbert series of the associated graded ring of the filtration with respect to E_{v_{1}}. Since p_{g}(Γ′′) = 0 for all the branches Γ′′ of v_{1} which are chains, by (3.8) we have the following.

4.4. Proposition. p_{g}(Γ) = H_{Γ,v_{1}}|_{pc} + p_{g}(Γ′).

4.5. One needs to modify this inductive step in order to fit Γ′ into a ‘good’ inductive procedure which is compatible with the splice decomposition of Σ (see also (4.11) for more motivation).

Let Δ be the splice diagram of Σ; for the correspondence between Γ and Δ see [7, §22]. We decompose Δ as the splice of two splice diagrams Δ_{1} and Δ_{2}, where Δ_{i} ∋ v_{i}.

[figure: w_{1}; w_{2}; w_{r}; v_{1}; b; v_{2}; c; w_{1}; w_{2}; w_{r}; ⋮; a_{1}; a_{2}; a_{r}; ⋮; d_{1}; d_{2}; d_{s}; ; Δ_{1}; ; Δ_{2}; Δ:]

Let Γ_{i} be the minimal plumbing graph associated with Δ_{i} (i = 1, 2) (cf. [7]). Next we review how one can recover from Δ_{1} and Δ_{2} the maximal chain Γ_{0} in Γ which connects v_{1} and v_{2}, and we relate Γ_{2} with Γ′. Set a := ∏_{i=1}^{r} a_{i} and d := ∏_{i=1}^{s} d_{i}. Express a/b as a continued fraction [α_{0}, …, α_{m}], i.e. a/b = α_{0} − 1/(α_{1} − 1/(… − 1/α_{m})) with α_{i} ∈ ℤ_{>0} and α_{i} ≥ 2 for i ≥ 1. Similarly, write d/c = [β_{0}, …, β_{n}], and consider the chain:

[figure: …; …; −α_{m}; −α_{0}; −β_{0}; −β_{n}; v_{1}; v_{2}; Γ̃_{0} :]

This is not a ‘minimal’ graph, and it is equivalent via (topological) plumbing calculus with the (minimal) chain Γ_{0}. This calculus runs as follows: by the positivity of the edge determinant bc > ad (cf. [19, §1]), α_{0} = 1 or β_{0} = 1. Then one successively blows down the vertices whose weight are −1, and at some moment inevitably a vertex with weight 0

appears; then one makes a 0-absorption by the rule: [figure: −e_{1}; 0; −e_{2}; ⟶; −e_{1} − e_{2}] . In this way one gets Γ_{0}. On the other hand, the corresponding maximal chain (with determinant c) in Γ_{2} has the following form:

[figure: …; −β_{1}; −β_{n}; v_{2}; Γ_{2} :]

4.6. Lemma. Consider the following resolution graph Γ̃_{2} which satisfies (1)–(4) below:

[figure: Γ^{0}; …; Γ^{m}; −1; v′_{1}; w; Γ′; Γ̃_{2} :]

(1) the subgraph on the left hand side of w is a chain; (2) Γ^{0} consists of (α_{0} − 1)-vertices; the right-end has weight −3, all the others −2;

(3) Γ^{m} consists of (α_{m} − 2)-vertices, all of them with weight −2;

(4) for 0 < i < m, Γ^{i} consists of (α_{i} − 2)-vertices, the right-end has weight −3 and all the others −2.

Then Γ̃_{2} is a negative definite plumbing graph whose associated minimal graph is Γ_{2}.

Proof. First replace in Γ̃_{2} the chain Γ_{0} of Γ′ by Γ̃_{0}. Hence, the above chain of Γ̃_{2} from the left hand side of w is glued to Γ̃_{0}. Then blow down repeatedly the (−1)–vertices and use

the 0–absorption which identifies [figure: −β_{0}; 0; Γ_{2}]  with Γ_{2}.  □

4.7. Corollary. There exists a (non-minimal) resolution graph, namely Γ̃_{2}, which represents the splice diagram Δ_{2}, it has Γ′ as a subgraph, and supports a splice type singularity.

Proof. We only have to show that Γ̃_{2} satisfies the monomial condition. Let u be a node of Γ_{2}, let C′′ a branch of it which contains the distinguished chain involved in the splicing (otherwise the condition is trivial). Let C be the branch of u in Γ which contains C′′. Then there exists an effective cycle A supported on C so that E_{u}^{∗} + A is monomial in Γ. Then the restriction A|Γ′ to Γ′ has the property that E_{u}^{∗}(Γ′) + A|Γ′ is monomial and has negative intersection with E_{w}. Then A|Γ′ can be extended easily on the chain considered in (4.6) to an effective cycle B on Γ̃_{2} such that E_{u}^{∗}(Γ̃_{2}) + B is monomial in Γ̃_{2}. □

Let H_{Γ̃_{2},v′_{1}}(t) be the Hilbert series of associated graded ring of the filtration with respect

to E_{v′_{1}} of Γ̃_{2}. Then, again by (3.8), we have:

4.8. Proposition. p_{g}(Γ_{2}) = H_{Γ̃_{2},v′_{1}}|_{pc} + p_{g}(Γ′).

4.9. Finally we consider the star–shaped graph Γ_{1} (which automatically satisfies the monomial condition), hence (again by 3.8):

4.10. Proposition. p_{g}(Γ_{1}) = H_{Γ_{1},v_{1}}|_{pc}.

4.11. As a consequence, from the above three propositions one gets:

(4.3)  p_{g}(Γ) − p_{g}(Γ_{1}) − p_{g}(Γ_{2}) = (H_{Γ,v_{1}} − H_{Γ̃_{2},v′_{1}} − H_{Γ_{1},v_{1}})|_{pc}.

The main point in this ‘additivity’ formula is the following. By the relation σ+8p_{g} +c(Γ) = 0 (cf. 4.1), the identity σ/8 = λ(Γ) transforms into 8λ(Γ) + 8p_{g} + c(Γ) = 0. Two terms in this expression satisfy some additivity properties with respect to the splicing. Namely, the additivity λ(Γ) = λ(Γ_{1}) + λ(Γ_{2}) (here we identify a plumbing graph with its plumbed 3-manifold) was proved independently by Akbulut-McCarthy, Boyer-Nicas and Fukuhara-Maruyama (according to [2]). Let b_{1}(G_{i}) be the first Betti number of the fiber G_{i} of the fibered knot determined by Γ_{i} and the splicing data (i = 1, 2). Then, by [19, (6.4)] (cf. [12, (5.20)]), one has c(Γ) = c(Γ_{1}) + c(Γ_{2}) − 2b_{1}(G_{1})b_{2}(G_{2}). In particular, in order to run the induction of our proof, we only need to verify

p_{g}(Γ) = p_{g}(Γ_{1}) + p_{g}(Γ_{2}) + b_{1}(G_{1})b_{1}(G_{2})/4,

a fact already noticed in [19, (6.3)]. Notice also that the splice components inherit the monomial condition of Γ. Therefore the following proposition, together with equation (4.3), implies the theorem of the introduction.

4.12. Proposition. (H_{Γ,v_{1}} − H_{Γ̃_{2},v′_{1}} − H_{Γ_{1},v_{1}})|_{pc} = b_{1}(G_{1})b_{1}(G_{2})/4.

Proof. Note that in (4.12), each periodic constant independently is a very complicated (Dedekind sum) expression. Nevertheless, their combination provides the simple expression from the right hand side. The proof is based on an explicit computation of the Hilbert series using (3.4). The corresponding entries m_{vw} might be determined from the splice diagrams using identities of [7, §10], or [19, (9.1)]). Note that the ‘unreduced’ splice diagram associated with Γ̃_{2}, having v′_{1} as a node, is the following:

[figure: ⋮; v′_{1}; b; v_{2}; a; ΔΓ̃_{2}:]

We will use the notation of (4.5) and let V_{2} be the set of vertices of Γ_{2}. Then m_{v_{1}v_{1}} = ab and m_{v_{1}w_{i}} = ab/a_{i}. Consider the integers p_{i} := m_{v_{1}w_{i}}/b = a/a_{i} and q_{w} := m_{v_{1}w}/a for all 1 ≤ i ≤ r and w ∈ V_{2} respectively. Finally, set g(t) = ∏_{w∈V_{2}}(1 − t^{q_{w}})^{δ_{w}−2}. Then

(4.4)  H_{Γ,v_{1}}(t) = \frac{g(t^{a})(1 − t^{ab})^{r−1}}{∏_{i=1}^{r}(1 − t^{p_{i}b})} .

Since for any w ∈ V_{2}, m_{v_{1}w} in Γ and m_{v′_{1}w} in Γ̃_{2} are the same, one has:

(4.5)  H_{Γ̃_{2},v′_{1}}(t) = \frac{g(t^{a})}{1 − t^{b}} .

Clearly

(4.6)  H_{Γ_{1},v_{1}}(t) = \frac{(1 − t^{ab})^{r−1}}{(1 − t^{a}) ∏_{i=1}^{r}(1 − t^{p_{i}b})} .

We define the functions Q_{1}(t) and Q_{2}(t) by

Q_{1}(t) = \frac{(1 − t^{a})^{r−1}}{∏_{i=1}^{r}(1 − t^{p_{i}})} − \frac{1}{1 − t} , Q_{2}(t) = g(t) − \frac{1}{1 − t} .

Then from (4.4), (4.5) and (4.6) we have

(4.7)  H_{Γ,v_{1}}(t) − H_{Γ̃_{2},v′_{1}}(t) − H_{Γ_{1},v_{1}}(t) = Q_{1}(t^{b})Q_{2}(t^{a}) − \frac{1}{(1 − t^{a})(1 − t^{b})} .

On the other hand, consider the characteristic polynomial P_{i}(t) = det(I − th_{1,i}) (where I denotes the identity matrix) of the monodromy h_{1,i}: H^{1}(G_{i}) → H^{1}(G_{i}) of the fibered knot with Milnor fiber G_{i} (i = 1, 2). Using A’Campo’s formula [1], or [7, §11–12] one has

P_{1}(t) = (1 − t) · \frac{(1 − t^{a})^{r−1}}{∏_{i=1}^{r}(1 − t^{p_{i}})} ,
P_{2}(t) = (1 − t)g(t).

It is known for fibered links in integral homology 3–spheres (or using A’Campo’s formula is easy to check) that P_{i}(1) = 1 and the derivative also satisfies P′_{i}(1) = b_{1}(G_{i})/2.

Notice also that Q_{i}(t) = (P_{i}(t) − 1)/(1 − t), hence Q_{i}(1) = −P′_{i}(1). Therefore, since Q_{i}’s are polynomials, (Q_{1}(t^{b})Q_{2}(t^{a}))|_{pc} = Q_{1}(1)Q_{2}(1) = b_{1}(G_{1})b_{1}(G_{2})/4. We end the proof with the remark that (1 − t^{a})^{−1}(1 − t^{b})^{−1}|_{pc} = 0. □

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

Rényi Institute of Mathematics, Budapest, Hungary

E-mail address: nemethi@math.ohio-state.edu;  nemethi@renyi.hu

Department of Education, Yamagata University, Yamagata 990-8560, Japan.

E-mail address: okuma@e.yamagata-u.ac.jp