Principal Analytic Link Theory in Homology Sphere Links

For the link M of a normal complex surface singularity (X, 0) we ask when a knot K ⊂M exists for which the answer to whether K is the link of the zero set of some analytic germ (X, 0) → (C, 0) affects the analytic structure on (X, 0). We show that if M is an integral homology sphere then such a knot exists if and only if M is not one of the Brieskorn homology spheres M(2, 3, 5), M(2, 3, 7), M(2, 3, 11). 1. Principal analytic link theory Let M be a normal surface singularity link. In particular, M is a closed 3–manifold which can be given by a negative definite plumbing. There may exist many different complex analytic structures on the cone C(M), i.e., many analytically different normal surface singularities (X, 0) whose links LX are homeomorphic to M . Our aim is to understand these different analytic structures from the point of view of the “principal analytic link theory” on M . A link or multilink L = m1K1∪ . . .∪mrKr ⊂M = LX is algebraic if (M,L) is the link (M,L) = (LX , LC) of a germ pair (X,C, 0) consisting of a normal surface germ and a (not necessarily reduced) complex curve through the singular point 0 ∈ X (this was called “analytic” in [3]). This is a topological property: L is algebraic if the Ki are S-fibres in a negative definite plumbing decomposition of M obtained by possibly applying blow-ups to the minimal negative definite plumbing of M . We say L is principal analytic for X if there exists a holomorphic germ f : (X, 0)→ (C, 0) such that the pair (M,L) is homeomorphic to the link (LX , Lf ) of the pair (X, f −1(0)), taking account of multiplicities. We say that L = m1K1 ∪ . . . ∪mrKr ⊂M is potentially principal if there exists a normal surface germ X with link LX = M for which L is principal analytic. According to ([3], Theorem 2.1), the potential principality of an algebraic multilink L ⊂ M is a topological property which is equivalent to any one of the following 1 2 A. NÉMETHI, WALTER D NEUMANN, AND A. PICHON • The multilink (M,L) is fiberable; • [L] = 0 in H1(M ; Z) (note that [L] is always torsion); • I−1b is an integral vector, where I is the intersection matrix for the plumbing and b the vector whose entry corresponding to a plumbing component is the sum of multiplicities of components of L that are fibres of that component. When M in the link of a rational singularity, then a potentially principal multilink (M,L) is principal analytic for every analytic structure (X, 0) ([1]). The same conclusion holds when M is the link of a minimally elliptic singularity and L is a knot ([6, Lemma p. 112]). In [3], we gave several examples of surface singularity links M whose principal analytic link theory is sensitive to the analytic structure in the following sense: for each analytic structure (X, 0) on C(M), there exists a potentially principal knot in M which is not principal analytic for this structure. In fact, we gave examples of pairs of potentially principal links, where the principality of each obstructed the principality of the other. The aim of this paper is to show that when M is an integral homology sphere (ZHS) this behaviour is general, except in the rational and minimally elliptic cases. Our technique consists of constructing a set of principal analytic knots K1, . . . , Kn which are not compatible, i.e., which cannot be realized by germs fi : (X, 0) → (C, 0) from the same analytic structure (X, 0). Example 1.1. Let V (p, q, r) := {(x1, x2, x3) ∈ C | xp1 + x q 2 + x r 3 = 0} with p, q, r pairwise coprime. Its link M = M(p, q, r) is a Z–homology sphere with three singular fibres K1, K2, K3 realized as principal analytic knots by Ki = M ∩ {xi = 0}. Let K be the (2, 1)-cable on K3 ⊂ M(2, 3, 13). It is a potentially principal knot in M = M(2, 3, 13). Let (Z, p) be an analytic structure on the cone C(M) such that K3 is realized by a holomorphic function f3 : (Z, p) → (C, 0). Then K is not realized by any f : (Z, p) → (C, 0) on (Z, p) ([3], 3,1). Before stating more precisely the main result, let us generalize the notion of principal analytic multilink, and say what we mean by the principal analytic link theory of a surface singularity link M . Definition 1.2. A coloured multilink in M is the data of an algebraic multilink L ⊂ M with a partition of its components: L = L1 ∐ . . . ∐ Ln. PRINCIPAL ANALYTIC LINK THEORY IN HOMOLOGY SPHERE LINKS 3 Definition 1.3. A coloured multilink L = L1 ∐ . . . ∐ Ln ⊂M is principal analytic for a normal surface singularity (X, 0) with link LX = M if there exist analytic germs fi : (X, 0)→ (C, 0), i = 1, . . . , n such that (1) the pair (M,L) is homeomorphic to (LX , Lf ), where f = f1 . . . fr; (2) each (M,Li) is homeomorphic to (LX , Lfi) (note that this does not imply (1)—see Remark 2.5). We say L is potentially principal if it is principal analytic for some analytic structure (X, p). Of course, the potential principality of each link Li is a necessary condition for the potential principality of L. But it is not sufficient when n ≥ 2, as shown by the examples of incompatible knots mentioned above: the coloured link K1 ∐ . . . ∐ Kn is not potentially principal, but each component is. That is, the knots K1, . . . , Kn can be realized by functions fi : (Xi, 0) → (C, 0), i = 1, . . . , n defined on some analytic structures (Xi, 0) on the cone C(M), but the (Xi, 0) cannot have the same analytical type. So, although the multilink L = K1 ∪ . . . ∪ Kn can be realized by a function f : (X, 0)→ (C, 0) for some (X, 0), there is no (X, 0) and f such that f splits into a product f = f1 . . . fn with fi : (X, 0)→ (C, 0) realizing the knot Ki. Given M , let us denote by PPL(M) the set of potentially principal coloured multilinks L in M ; we call PPL(M) the principal analytic theory on M . Given a normal surface singularity (X, 0) with link M , we denote by PAL(X) ⊂ PPL(M) the set of coloured links L in M which are principally analytic for (X, 0). So PPL(M) = ⋃ LX∼=M PAL(X). The study of the principal analytic link theory on M consists of the two following natural questions: (1) Describe the set PPL(M); (2) describe the subsets PAL(X) for (X, 0) realizing M . The unique rational singularity with ZHS link is (V (2, 3, 5), 0) with link M(2, 3, 5). There are only two ZHS links which belong to minimally elliptic singularities: M(2, 3, 7) and M(2, 3, 11). Our main result, which is a first important step in this program, is as follows: Theorem 1.4. Let M be a ZHS singularity link which is not homeomorphic to M(2, 3, 5), M(2, 3, 7) or M(2, 3, 11). Then there exists an algebraic coloured link L = K1 ∐ . . . ∐ Kn which is not in PPL(M) and such that: (1) Each Ki is a potentially principal knot (2) ∀i 6= j, (M,Ki) is not homeomorphic to (M,Kj). 4 A. NÉMETHI, WALTER D NEUMANN, AND A. PICHON (Of course, since M is a Z–homology sphere, the potential principality of Ki is automatic.) 2. Two constructions of non-PPL coloured links In this section, we present through examples two methods (Methods 1 and 2) to construct some coloured links L in a given M such that L / ∈ PPL(M) but each Li is in PPL(M). The first one, which was introduced in [3], could be used in any M , whereas the second is only available in a ZHS. Method 1 (Using the delta invariant of a reduced curve). Let K be a fibred knot in M , and let Φ: M \K → S be an open-book fibration with binding K. We set μ(K) = 1− χ(Φ−1(t)) , where t ∈ S and where χ denotes the Euler characteristic. Notice that μ(K) does not depend on the choice of Φ, and that it can be computed from any plumbing graph of (M,K) (or any splice diagram if M is a QHS). Let K1 ∐ · · · ∐ Kn, n ≥ 2 be a coloured link whose components Ki are potentially principal knots with multiplicity 1. For each i = 1, . . . , n− 1, let Φi : M \Ki → S be a fibration of Ki. We consider the coloured multilink L = K1 ∐ . . . ∐ Kn−1 and we define the semigroup Γ(L,Kn) as the semigroup generated by the degrees of the maps Φi on the knot Kn. Notice that these degrees do not depend on the Φi’s and can be computed from any plumbing graph of (M,K1 ∐ . . . ∐ Kn). We denote by δ(L,Kn) the number of gaps in Γ(L,Kn), i.e., the number of positive integers that are not in Γ(L,Kn). Lemma 2.1. If L ∐ Kn ∈ PPL(M) then μ(Kn) ≤ 2δ(L,Kn) . Proof. Let (X, 0) be such L ∐ Kn ∈ PAL(X) and let fj : (X, 0) → (C, 0) be a holomorphic germ with link Kj for j = 1, . . . , n. Then μ(Kn) = μ(fn), the Milnor number of fn. According to [2], one has μ(fn) = 2δ(fn), where δ(fn) denotes the δ-invariant of the curve f−1 n (0). Recall that δ(fn) counts the number of gaps in the semigroup Γ(fn) generated by the all the multiplicities of the holomorphic germs g : (X, 0) → (C, 0) along the curve f−1 n (0). Moreover, if g is such a germ then this multiplicity is the degree of the Milnor fibration g/|g| restricted to the link of f−1 n (0). Since K1, . . . , Kn−1 can be realized by germs f1, . . . , fn−1, we have Γ(L,Kn) ⊂ Γ(fn), so δ(fn) ≤ δ(L,Kn). PRINCIPAL ANALYTIC LINK THEORY IN HOMOLOGY SPHERE LINKS 5 Example 2.2 (Non-PPL coloured link). Let M be the link of the Brieskorn-Pham singularity z 1 + z 4 2 + z 5 3 = 0 and and let Ki, i = 1, 2, 3 be the end-knots in M corresponding to zi = 0. Let us consider the (2, 5)−cabling K on the link K3 of z3 = 0. Its splice diagram is as follows. K1 oo 3 ◦