Invariants of Newton non-degenerate surface singularities

Gábor Braun and András Némethi

Dedicated to J. Steenbrink on his 60th birthday.

Abstract

We recover the Newton diagram (modulo a natural ambiguity) from the link for any surface hypersurface singularity with non-degenerate Newton principal part whose link is a rational homology sphere. As a corollary, we show that the link determines the embedded topological type, the Milnor fibration, and the multiplicity of such a germ. This proves (even a stronger version of) Zariski’s Conjecture about the multiplicity for such a singularity.

1. Introduction

In general, it is a rather challenging task to connect the analytic and topological invariants of normal surface singularities. The program which aims to recover different discrete analytic invariants from the abstract topological type of the singularity (i.e. from the oriented homeomorphism type of the link K, or from the resolution graph) can be considered as the continuation of the work of Artin, Laufer, Tomari, S. S.-T. Yau (and the second author) about rational and elliptic singularities. It includes the efforts of Neumann and Wahl to recover the possible equations of the universal abelian covers [NW02], and the efforts of the second author and Nicolaescu about the possible connections of the geometric genus with the Seiberg–Witten invariants of the link [NN02]. See [Ném04] for a review of this program.

In order to have a chance for this program, one has to consider a topological restriction (the weakest one for which we still hope for positive results maybe that the link is a rational homology sphere), and a restriction about the analytic type of the singularity, also. By [LVMHN05], the Gorenstein condition is not sufficient. We expect pathologies even for hypersurface singularities.

For isolated hypersurface singularities a famous conjecture was formulated by Zariski [Zar71], which predicts that the multiplicity is determined by the embedded topological type. For hypersurface germs with rational homology sphere links, Mendris and the second author in [MN05] formulated (and verified for suspension singularities) an even stronger conjecture, namely that already the abstract link determines the embedded topological type, the multiplicity and equivariant Hodge numbers (of the vanishing cohomology).

The goal of the present article is to verify this stronger conjecture for isolated singularities with non-degenerate Newton principal part. In fact, we will prove that from the link (provided that it is a rational homology sphere) one can recover the Newton boundary (up to a natural ambiguity, see Theorem 1.0.1 below, and up to a permutation of coordinates), and hence the equation of the germ (up to an equisingular deformation). This is the maximum what we can hope for.

2000 Mathematics Subject Classification Primary: 14J17, 14Q10; Secondary: 52B20 Keywords: hypersurface singularities, links of singularities, resolution graphs, Newton boundary, Newton polyhedrons.

The first author is partially supported by Hungarian National Research Fund, grant No. T 042 769.

The second author is partially supported by NSF grant DMS-0304759, Marie Curie and OTKA grants.

Gábor Braun and András Némethi

The reader is invited to consult [AGZV88, Ném04] for general facts about singularities. §2 reviews the terminology and some properties of germs with non-degenerate Newton principal part. In §3 we define the equivalence relation ∼ of Newton boundaries characterizing the above-mentioned ambiguities. It may also be generated by the following elementary step: two diagrams Γ_{1} and Γ_{2} are equivalent if both define isolated singularities and Γ_{1} ⊂ Γ_{2}. (At the level of germs, this can be described by a linear deformation.) Although the structure of an equivalence class is not immediate from the definition, we define an easily recognizable representative in every class, which we call the d-minimal representative.

In §4.2 we review Oka’s algorithm which provides a possible resolution graph G(Γ) (or equivalently, a plumbing graph of the link) from the Newton boundary Γ [Oka87]. (Equivalent graphs provide plumbing graphs related by blowing ups/downs, and hence determine the same link.) Our main result says that Oka’s algorithm can be essentially inverted:

1.0.1 Theorem. Assume that the Newton diagrams Γ_{1} and Γ_{2} determine isolated singularities with non-degenerate Newton principal part whose links are rational homology spheres. Assume that the good minimal resolution graphs associated with G(Γ_{1}) and G(Γ_{2}) are isomorphic. Then (up to a permutation of coordinates) Γ_{1} ∼ Γ_{2}. In particular, from the link K, one can identify the ∼-equivalence class of the Newton boundary (up to a permutation of coordinates) or, equivalently, the d-minimal representative of this class.

In fact, we prove an even stronger result: one can recover the corresponding class of Newton diagrams (or its distinguished representative) already from the orbifold diagram G^{o} associated with the good minimal resolution graph. This diagram, a priori, contains less information then the resolution graph, because it codifies only its shape and some subgraph-determinants, see 4.4 for details. (Although G^{o} has a different decoration, it is comparable with the ‘splice diagram’ considered in [NW02].)

Since most of the invariants of the germs are stable under the deformations defining the equivalence relation ∼ (see §3.2), one has the following

1.0.2 Corollary. Let f be an isolated germ with non-degenerate Newton principal part whose link is a rational homology sphere. Then the oriented topological type of its link determines completely its Milnor number, geometric genus, spectral numbers, multiplicity, and, finally, its embedded topological type.

Such a statement is highly non-trivial for any of the above invariants. For the history of the problem regarding the Milnor number and the geometric genus, the reader is invited to consult [Ném04]. Here we emphasize only the following:

– Regarding the embedded topological type, Corollary 1.0.2 shows that if a rational homology sphere 3-manifold can be embedded into S^{5} as the embedded link of an isolated hypersurface singularity with non-degenerate Newton principal part, then this embedding is unique. (Notice the huge difference to the case of plane curves, and also to the higher dimensional case, where already the Brieskorn singularities provide a big variety of embeddings S^{2n−1} ⊂ S^{2n+1}, n ≠ 2.)

– Such a link can be realized by a germ f with non-degenerate Newton principal part in an essentially unique way, i.e. up to a sequence of linear μ-constant deformations (corresponding to ∼) and permutation of coordinates, see 5.1.3(ii).

Regarding the main theorem, some more comments are in order.

– The assumption that the link is a rational homology sphere is necessary: the germs {z_{1}^{a}+z_{2}^{b}+z_{3}^{c} = 0} with exponents (3, 7, 21) and (4, 5, 20) share the same minimal resolution graph.

– The proof of 1.0.1 is, in fact, a constructive algorithm which provides the d-minimal representatives of the corresponding class of diagrams from the orbifold diagram G^{o}.

Hence, one may check effectively whether an arbitrary resolution graph can be realized by a hypersurface singularity with non-degenerate Newton principal part. Indeed, if one runs our algorithm and it fails, then it is definitely not of this type. If the algorithm goes through and provides some candidate for a Newton diagram, then one has to compute the graph (orbifold diagram) of this candidate (by Oka’s procedure) and compare with the initial one. If they agree then the answer is yes; if they are different, the answer again is no (this may happen since our algorithm uses only a part of the information of G^{o}).

E.g., one can check that the following resolution graph cannot be realized by an isolated singularity with non-degenerate Newton principal part (although it can be realized by a suspension {z_{3}^{2} + g(z_{1}, z_{2}) = 0}, where g is an irreducible plane curve singularity with Newton pairs (2, 3) and (1, 3)).

[figure: −7; −3; −3; −1; −3; −2]

Figure 1: A resolution graph not coming from a singularity with non-degenerate Newton principal part

We mention that, in general, there is no procedure to decide whether a graph is the resolution graph of a hypersurface isolated singularity (this is one of the open problems asked by Laufer [HR89, p. 122]; for suspension singularities, it is solved in [MN05]).

2. Singularities with non-degenerate Newton principal part

2.1 The Newton boundary [Kou76]. Criterion for isolated singularities. 2.1.1. For any set S ⊂ ℕ^{3} denote by Γ_{+}(S) ⊂ ℝ^{3} the convex closure of ⋃_{p∈S}(p + ℝ_{+}^{3}). We call the 1-faces of any polytope edges, and face will simply mean a 2-face. The collection of all boundary faces of Γ_{+}(S) is denoted by F. The set of compact faces of Γ_{+}(S) is denoted by F_{c}. By definition, the Newton boundary (or diagram) Γ(S) associated with S is the union of compact boundary faces of Γ_{+}(S). Let ∂Γ denote the union of those edges of Γ(S) which are not intersection of two faces of Γ(S). Let Γ_{−}(S) denote the cone with base Γ(S) and vertex 0.

Let f: (ℂ^{3}, 0) → (ℂ, 0) be an analytic function germ defined by a convergent power series ∑_{p} a_{p}z^{p} (where p = (p_{1}, p_{2}, p_{3}) and z^{p} = z_{1}^{p_{1}}z_{2}^{p_{2}}z_{3}^{p_{3}}). By definition, the Newton boundary Γ(f) of f is Γ(supp(f)), where supp(f) is the support {p : a_{p} ≠ 0} of f, and we write Γ_{−}(f) for Γ_{−}(supp(f)). The Newton principal part of f is ∑_{p∈Γ(f)} a_{p}z^{p}. Similarly, for any q-face △ of Γ(f) (of any dimension q), set f_{△}(z) := ∑_{p∈△} a_{p}z^{p}. We say that f is non-degenerate on △ if the system of equations ∂f_{△}/∂z_{1} = ∂f_{△}/∂z_{2} = ∂f_{△}/∂z_{3} = 0 has no solution in (ℂ^{∗})^{3}. When f is non-degenerate on every q-face of Γ(f), we say (after Kouchnirenko [Kou76]) that f has a non-degenerate Newton principal part. The diagram Γ(f) and the function f are called convenient if Γ(f) intersects all the coordinate axes.

2.1.2. In this article we will assume that f is singular, i.e. ∂f(0) = 0.

2.1.3. If we fix a Newton boundary Γ (i.e. Γ = Γ(S) for some S), then the set of coefficients {a_{p} : p ∈ Γ} for which f(z) = ∑_{p∈Γ} a_{p}z^{p} is Newton non-degenerate (as its own principal part) form a non-empty Zariski open set (cf. [Kou76, 1.10(iii)]). Nevertheless, even for generic coefficients {a_{p}}_{p∈Γ}, the germ f = ∑_{p∈Γ} a_{p}z^{p} (or any f with Γ(f) = Γ), in general, does not define an isolated singularity. The germ f (with generic {a_{p}}_{p∈Γ}) defines an isolated singularity if and only if Γ satisfies the next additional properties ([Kou76, 1.13(ii)]):

– {(0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0)} ∩ Γ = ∅ (cf. (2.1.2)),

– the diagram Γ has a vertex on every coordinate plane, and

– for every coordinate axis, Γ has a vertex at most 1 far from the axis.

(2.1)

E.g., a convenient f with generic coefficients defines an isolated singularity.

2.1.4 Example. Notice that (2.1) cannot be satisfied by one vertex. Moreover, if Γ satisfies (2.1) and has no faces then (modulo a permutation of the coordinates) it is the segment [(0, 1, 1) , (n, 0, 0)] for some n ⩾ 2.

2.1.5 Remark. Assume that Γ is not an edge. Then (2.1) implies that every edge of ∂Γ should lie either on a coordinate plane or be (after permuting coordinates) of the form AB = [(a, 0, c) , (0, 1, b)] with a > 0 and b + c > 0. The number of edges of second type coincides with the number of coordinate axes not intersected by Γ. (Indeed, assume that the z_{3} axis does not meet Γ. Project Γ to the z_{1}z_{2} plane by ψ(z_{1}, z_{2}, z_{3}) = (z_{1}, z_{2}). Then, by (2.1), the boundary of ψ(Γ) contains an edge of type [(a, 0), (0, 1)].)

2.1.6. If one tries to analyze the invariants of a germ in terms of its Newton diagram (see e.g. the references cited in §2.2), one inevitably faces the arithmetical properties of integral polytopes. In Appendix 8.1, we collect those which will be used in the body of the paper. The relevant notations and terminologies are listed below:

2.1.7. Notations/Definitions. Fix a Newton diagram. Set △ ∈ F. Let ▽ ∈ F be an adjacent face with a common (compact) edge AB := △ ∩ ▽. Then one defines:

\overrightarrow{a}_{△}

the normal vector of △, i.e. the primitive integral vector with non-negative entries, normal to △,

t_{△,▽}  the number of components of AB \ ℕ_{>0}^{3},
n_{△,▽}

the determinant of \overrightarrow{a}_{△} and \overrightarrow{a}_{▽}, namely, the greatest common divisor of the entries of the cross product \overrightarrow{a}_{△} × \overrightarrow{a}_{▽}, (n_{△,▽} ⩾ 1),

\overrightarrow{e_{1}} , \overrightarrow{e_{2}} , \overrightarrow{e_{3}} the three coordinate normal vectors.

The number n_{△,▽} is also called the determinant of the edge AB. Since it depends only on the corresponding normal vectors, sometimes we put the normal vectors in the index instead of the faces. E.g., if \overrightarrow{a}_{▽} = \overrightarrow{e_{i}} and △ ∈ F_{c}, then we may also write n_{△,\overrightarrow{e_{i}}} for n_{△,▽}. The number t_{△,\overrightarrow{e_{i}}} has a similar meaning. In fact, with the notation \overrightarrow{a}_{△} = (a_{1}, a_{2}, a_{3}), one has:

n_{△,\overrightarrow{e_{i}}} = gcd (a_{j}, a_{k}) , where \{i, j, k\} = \{1, 2, 3\}.  (2.2)

Similarly, for any lattice polygon △, the vector \overrightarrow{a}_{△} denotes the primitive integral vector normal to △ (well-defined up to a sign). The combinatorial area, by definition (cf. [Oka87, (6.2)]), is

g(△) := 2#{inner lattice points} + #{border lattice points} − 2.  (2.3)

Clearly, g(△) is additive. The face △ is called empty if its only lattice points are its vertices.

2.2 Some discrete invariants determined from the Newton boundary. If f defines an isolated singularity and has a non-degenerate Newton principal part, then its Newton boundary Γ(f) determines almost all its discrete analytic and embedded topological invariants. E.g.:

(a) the Milnor number μ(f) of f is given by Kouchnirenko [Kou76]. For any Γ let V_{3} be the 3dimensional volume of Γ_{−}, and for 1 ⩽ q ⩽ 2, let V_{q} be the sum of the q-dimensional volumes of all the intersections of Γ_{−} with q-dimensional coordinate planes. Set ν(Γ) := 6V_{3} − 2V_{2} + V_{1} − 1. Then, by [Kou76], the Milnor number μ(f) of any convenient germ f with non-degenerate Newton principal part is given combinatorially via Γ(f) by:

μ(f) = ν(Γ(f)).  (2.4)

In fact, the same formula is valid for non-convenient isolated singularities as well. Indeed, assume e.g. that the diagram Γ(f) does not intersect the z_{3} axis, and let AB be an edge as in 2.1.5. Then the deformation f_{d} := f + tz_{3}^{d} with d ⩾ μ(f) + 2 has a uniform stable radius for the Milnor fibration [Oka79], hence μ(f) = μ(f_{d}). Moreover, Γ_{−}(f_{d}) = Γ_{−}(f) ∪ W_{d}, where W_{d} is the 3-simplex with vertices 0, A, B and (0, 0, d). Since (6V_{3} − 2V_{2} + V_{1})(W_{d}) = 0, one gets that ν(Γ(f_{d})) = ν(Γ(f)). (Since f is finitely determined, f and f_{d} are right-equivalent for d ≫ 0 and their other invariants listed in this subsection agree too. Hence, being convenient, in many cases present in the literature, is not really essential for us, see also [Wal99].)

(b) the characteristic polynomial of the algebraic monodromy is determined in [Var76]; the geometric genus of the surface singularity ({f = 0}, 0) is given by #(Γ_{−}(f) ∩ ℕ_{>0}^{3}), cf. [MT80, Sai88]; the set of spectral numbers (or characteristic exponents) is computed in [Dan79, Sai88, Ste77, VK85]; the multiplicity of f by min_{p∈Γ(f)} ∑ p_{i};

(c) the embedded topological type and the Milnor fibration of f (with its homological ‘package’ including the Seifert form) is determined from Γ(f) uniquely by [Oka79, 2.1];

(d) an explicit construction of the dual resolution graph G(f) of the surface singularity ({f = 0}, 0) is given in [Oka87] (we review this in §4.2).

2.3 The structure of Newton polytopes in the case of rational homology sphere links 2.3.1. An important assumption of the main result of the present article is that the link K(f) of f is a rational homology sphere, i.e. H_{1}(K(f), ℚ) = 0. This additional assumption (besides (2.1), which says that f with non-degenerate Newton principal part is an isolated singularity) imposes serious restrictions on the Newton boundary Γ(f), cf. [Sai88]:

K(f) is a rational homology sphere ⟺ Γ(f) ∩ ℕ_{>0}^{3} = ∅.  (2.5)

In this subsection we assume that Γ(f) satisfies these two restrictions, namely (2.1) and (2.5). Our goal is to derive the structure theorem 2.3.9 for Newton diagrams.

We fix a diagram Γ. We start by classifying the non-triangular faces:

2.3.2 Lemma. If a face of Γ is not a triangle then it is a trapezoid. By permuting coordinates, its vertices are: A = (p, 0, n), B = (0, q, n), C = (r_{1}, r_{2} + tq, 0) and D = (r_{1} + tp, r_{2}, 0), where p, q > 0, gcd (p, q) = 1, t ⩾ 1 and r_{1}, r_{2} ⩾ 0. The only side which can have inner lattice points is the base lying on the z_{1}z_{2} plane (with t − 1 of them).

Proof. The idea of the proof is the following: if a lattice polygon △ is not a triangle or a trapezoid, then there exists a parallelogram in △ with three vertices on the boundary of △ and one in its interior, which contradicts (2.5). The details are left to the reader.

2.3.3 Terminology. The edges of a trapezoid have asymmetric roles. For future reference we give names to them. The bottom edge always lies on a coordinate plane. If two (ore more) edges lie on coordinate planes, the bottom edge is the one which has internal lattice points, if such exists. Otherwise, we choose one of them arbitrarily.

Opposite to the bottom edge lies the top edge, and the others are called side edges.

2.3.4 Terminology/Discussion. An edge crosses, say, the z_{3} axis if it is of the form [(p, 0, a) , (0, q, b)], where p > 0, q > 0, and a + b > 0. There are two types of edges on Γ: those lying on a coordinate plane and those crossing a coordinate axis.

While edges of the first type do not ‘cut’ Γ, edges of the second type usually cut Γ into two nonempty parts, one of which has a particularly simple structure. In order to see this, project ℝ_{⩾0}^{3} \ 0 from the origin to the triangle 𝕋 := {z_{1} + z_{2} + z_{3} = 1 : z_{i} ⩾ 0 (i = 1, 2, 3)}. The restriction ϕ: Γ → 𝕋 is one-to-one and preserves segments. An edge lying on a coordinate plane projects into ∂𝕋, while a crossing edge projects into a segment with only its end points on ∂𝕋 and cutting 𝕋 into two parts such that at least one of them, say 𝕋_{0}, is a triangle. By 2.3.2, the projection of a trapezoid hits the interior of all the sides of 𝕋, hence ϕ^{−1}(𝕋_{0}) may contain only a ‘sequence of triangles’. Therefore, one has:

2.3.5 Lemma. An edge of Γ crossing (say) the z_{3} axis, which is not on ∂Γ, cuts Γ into two non-empty parts. Consider the plane π formed by the edge and the origin. Then that part of Γ, which is on the same side of π as the positive z_{3} axis, consists only of triangular faces with vertices lying on the z_{1}z_{3} and z_{2}z_{3} planes. They form a sequence △_{1}, …, △_{k}; where △_{i} is adjacent with △_{i+1}, (and these are the only adjacent relations).

2.3.6 Corollary/Definition. Fix a coordinate axis.

First, assume that there is at least one triangular face whose vertices are on the two coordinate planes adjacent to the axis. Then the collection of such triangular faces form a sequence as in 2.3.5, and their union is called the arm of the diagram in the direction of that axis. The arm also contains all the crossing edges whose vertices lie on the two coordinate planes. Let the hand be the triangle of the arm which is nearest to the axis (in the ϕ-projection, say). Let the shoulder be the crossing edge of the arm which is most distant from the axis (in the same sense).

Next, assume that there is no triangular face whose vertices are on these two coordinate planes. Then we distinguish two cases:

(a) If there exists a crossing edge of the coordinate axis, then it is unique; in this case we say that the arm in that direction is degenerate, and the degenerate arm (and its shoulder too) is this unique crossing edge.

(b) If there is no crossing edge either, then we say that there is no arm in the direction of the axis.

2.3.7 Terminology. A triangular face of Γ is called central if its vertices are not situated on the union of two coordinate planes. A face of Γ is called central if it either is a central triangle or it is a trapezoid. An edge of Γ is central if (modulo a permutation of the coordinates) it has the form [(0, 0, a) , (p, q, 0)].

Using the projection ϕ: Γ → 𝕋, one may easily verify:

2.3.8 Lemma. Γ has at most one central face. Γ has a central face if and only if it has no central edge.

These facts can be summarized in the next result on structure of Newton diagrams:

2.3.9 Proposition. Every Newton diagram Γ (which satisfies (2.1) and (2.5)) sits in exactly one of the three disjoint families characterized as follows:

(1) Γ has a unique central trapezoid with at most 3 disjoint (possibly degenerate) arms. The arms correspond to those sides of the trapezoid which are crossing edges.

(2) Γ has a unique central triangle with 3 disjoint (possibly degenerate) arms.

(3) Γ has (at least one) central edge.

Moreover, if Γ has a central edge, then there are two cases. If Γ has only one face, this face is triangular with all vertices on coordinate axes, then all edges are central. Otherwise, all central edges have a common intersection point (say P) sitting on a coordinate axis; and the diagram has two (possibly degenerate) arms in the direction of the other two axes. The arms may overlap each other, i.e. have common triangles. P is a vertex of all the triangles in the intersection of the arms, and all those edges of these triangles which contain P are central (and these are all the central edges).

3. Equivalent Newton boundaries. Deformations.

3.1 The equivalence relation 3.1.1. Our aim is to recover the Newton boundary (up to a permutation of coordinates) of an isolated singularity with non-degenerate Newton principal part from the link K(f), provided that K(f) is a rational homology sphere. Strictly speaking, this is not possible: one can easily construct pairs of such germs having identical links but different boundaries. E.g., take an isolated non-convenient germ f and f_{d} = f + ∑ z_{i}^{d} with d ≫ 0. This motivates to define a natural equivalence relation of Newton boundaries. By definition, it will be generated by two combinatorial ‘steps’.

3.1.2. Fix a Newton boundary Γ = Γ(S) which satisfies (2.1). Let AB be an edge of ∂Γ which is not contained in any coordinate plane. By 2.1.5, up to a permutation of coordinates, A = (a, 0, c) (with a > 0) and B = (0, 1, b).

Move 1. We add a new vertex C = (a′, 0, c′) to Γ in such a way that Γ(S ∪ C) = Γ(S) ∪ △_{ABC}. Here △_{ABC}, the 2-simplex spanned by the points A, B, C appears as a new face. (In particular, 0 ⩽ a′ < a and c′ must be sufficiently large.)

Move 2. Assume that AB is in the face △ whose supporting plane is H. The line through AB cuts out the open semi-plane H_{+} of H which does not contain △. Set S′ := H_{+} ∩ ℕ^{3}. Then by adding a non-empty subset S′′ of S′ to S, we create a new Newton boundary Γ(S ∪ S′′). By this move, all faces of Γ(S) are unmodified, except △, which is replaced by a larger face containing △.

3.1.3 Definition. We denote Move 1 and Move 2 by M1_{+} and M2_{+}, respectively. We denote their inverses by M1_{−} and M2_{−}, respectively. The segment AB will be called the axis of the corresponding move.

Two Newton diagrams Γ_{1} and Γ_{2}, both satisfying (2.1), are equivalent (and we write Γ_{1} ∼ Γ_{2}), if they can be connected by a sequence of elementary moves (M1_{±} or M2_{±}), such that all the intermediate Newton boundaries satisfy (2.1) as well.

3.1.4 Example. Using 2.1.5 and induction, one can show that if Γ_{1} and Γ_{2} are Newton diagrams, Γ_{1} ⊂ Γ_{2}, both satisfying (2.1), then they are equivalent. In fact, the inclusion of Newton boundaries with (2.1) generates the same equivalence relation.

3.1.5 Example. The segments [(0, 1, 1) , (n, 0, 0)] and [(1, 0, 1) , (0, n, 0)] (considered as diagrams) are equivalent. Indeed, add to Γ_{1} = [(0, 1, 1) , (n, 0, 0)] the vertex (1, 0, 1) (by M1_{+}), then add (0, n, 0) (by M2_{+}), then remove the end points of Γ_{1} (cf. 3.1.4).

3.1.6. Sometimes it is more convenient to specify the deformation of the corresponding germs instead of the modification of Newton diagrams: adding a new vertex p to S translates into adding a new monomial ta_{p}z^{p} to f, with t ∈ [0, ϵ] a deformation parameter. (The fact that these deformations are linear in t is crucial in the proof of 3.2.1(c)).

3.1.7 Example. The number of ‘essential’ deformation parameters can be as large as we wish. E.g., for m, n ≫ 0, all the different Newton diagrams associated with the family

z_{3}(z_{1}^{p} + z_{2}^{q} + z_{3}^{r}) + ∑_{i} t_{i}z_{1}^{m−ip}z_{2}^{n+iq} (m − ip ⩾ 0, n + iq ⩾ 0)

satisfy (2.1), and are equivalent (via repeated M2_{±}) as soon as ∑_{i}|t_{i}| > 0. We call the ‘ambiguity’ of the choice of the monomials z_{1}^{m−ip}z_{2}^{n+iq} the moving triangle ambiguity.

More generally, a moving triangle of a Newton diagram Γ is a triangular face with vertices: P := (p, 0, 1), Q := (0, q, 1) and R := (m, n, 0), where the edge P Q is in some other face as well. Consider the line through R parallel to P Q. Then (the moving vertex) R can be replaced by any of the lattice points S on this line with non-negative coordinates (or any collection of them). If Γ satisfies (2.5), then gcd (p, q) = 1, and by (8.2) \overrightarrow{a}_{△} = (a_{1}, a_{2}, a_{3}) = (q, p, mq + np − pq). Therefore, one has:

p | m ⟺ a_{2} | a_{3} ⟺ R can be replaced by a point on the z_{2} axis,
q | n ⟺ a_{1} | a_{3} ⟺ R can be replaced by a point on the z_{1} axis.

(3.1)

3.2 Stability of the invariants under the deformations. 3.2.1 Proposition. Consider two isolated singularities with non-degenerate Newton principal parts whose Newton boundaries are equivalent in the sense of 3.1.3. Then the following invariants associated with these germs are the same:

(a) the Milnor number μ;

(b) the link K;

(c) more generally, the embedded topological type;

(d) the spectral numbers (in particular, the geometric genus); the equivariant Hodge numbers;

(e) the multiplicity.

Moreover, a deformation associated with M1_{+} or M2_{+} admits a weak simultaneous resolution.

Proof. First of all, (a) can be easily verified by direct computation (left to the reader) by Kouchnirenko’s formula (2.4). Item (b) can also be checked directly from Oka’s algorithm [Oka87] (§4.2 here), and (e) is also elementary. But there are also (more) conceptual short-cuts: The existence of a weak simultaneous resolution follows from a result of Oka [Oka89] (after we add some high degree monomials in the non-convenient case, and we notice that our moves are ‘negligible truncations’ in the sense of Oka), which implies (b) by a result of Laufer [Lau83]. For (d) one can use Varchenko’s result [Var82], which says that the spectrum is constant under a μ-constant deformation. Notice also that the geometric genus is the number of spectral numbers in the interval (0, 1]. Finally, a μconstant (f + tg)-type deformation (cf. 3.1.6) is topological trivial by a result of Parusiński [Par99] (proving (c)), and is equimultiple (e.g.) by Trotman [Tro80].

3.2.2 Remark. By similar proof as in [Ste85] (valid for the spectrum), one can show that the set of spectral pairs (equivalently, the equivariant Hodge numbers) of f are also determined by Γ, and are stable with respect to the ∼-deformation. Cf. also with [Dan79].

3.2.3 Corollary. Fix a Newton diagram Γ which satisfies (2.1). Then the following facts are equivalent:

(a) {(0, 1, 1), (1, 0, 1), (1, 1, 0)} ∩ Γ ≠ ∅;

(b) Γ is equivalent to a diagram which has no 2 dimensional faces;

(c) Γ is equivalent to the segment-diagram [(0, 1, 1) , (n, 0, 0)] for some n ⩾ 2; (d) f_{Γ}(z) := ∑_{p∈Γ} a_{p}z^{p} (with generic coefficients {a_{p}}_{p}) is an A_{n−1} singularity (the unique hyper-

surface cyclic quotient singularity with μ = n − 1) for some n ⩾ 2.

(e) The minimal dual resolution graph of {f_{Γ} = 0} is a string (with determinant n).

In fact, the integers n in (c), (d) and (e) are equal.

Proof. (b) ⟹ (c) follows from 2.1.4 and 3.1.5. The implication (c) ⟹ (b) is clear. For (a) ⟹ (c), using 3.1.5, it is enough to prove that if (0, 1, 1) ∈ Γ then Γ ∼ [(0, 1, 1) , (n, 0, 0)] for some n. If Γ intersects the z_{3} axis at some point (n, 0, 0), then [(0, 1, 1) , (n, 0, 0)] ⊂ Γ and one may use 3.1.4. Otherwise, one considers, like in 2.1.5, the projection (z_{1}, z_{2}, z_{3}) ↦ (z_{1}, z_{2}) restricted to Γ. By (2.1) there is at least one edge whose projection has the form [(a, 0), (0, 1)] (up to a permutation). Consider the edge-projection of this type which is closest to (1, 1), let its preimage in Γ be [(a, 0, c) , (0, 1, b)]. This choice guarantees that Γ contains the triangular face △ with vertices (a, 0, c), (0, 1, b), (1, 1, 0). Since △, as a diagram, satisfies (2.1), Γ ∼ △ by 3.1.4. By M2_{+} one can add to △ the vertex (ab + c, 0, 0), and apply again 3.1.4 to show that △ ∼ [(0, 1, 1) , (n, 0, 0)] with n = ab + c.

Next, notice that a_{1}z_{1}^{n}+a_{2}z_{2}z_{3} (a_{1}, a_{2} ≠ 0) defines an A_{n−1} singularity. Hence (c) ⟹ (d) follows from 3.2.1(b), since the A_{n−1} singularity is characterized by the fact that its link is the lens space L(n, n − 1). For (d) ⟹ (a) one uses that the quadratic part of the Taylor-expansion of an A_{n−1} singularity f_{Γ} (in any coordinate system) has rank at least two.

(d) ⟺ (e) follows from the fact that the A_{n−1} singularities are the only hypersurface singularities whose minimal resolution graphs are strings.

3.2.4. Not all discrete analytic invariants of the germs remain constant under the above equivalence relation. The following example was provided by J. F. de Borbadilla, A. Melle-Hernández and I. Luengo (private communication), in which Teissier’s invariant μ^{∗} jumps.

3.2.5 Example. Consider the deformation f_{t} = z_{3}^{3} + z_{2}^{4}z_{1} + z_{1}^{10} + tz_{2}^{3}z_{3}, which corresponds to Move 1, hence f_{1} and f_{0} are equivalent in the sense of 3.1.3. But the Milnor numbers of the generic hyperplane sections are not the same: μ^{(2)}(f_{1}) = 7, while μ^{(2)}(f_{0}) = 8. In particular, by [Lau87], this deformation does not admit a strong simultaneous resolution. Similar example was constructed by Briançon and Speder [BS75] (cf. also with [Oka89]); the main difference is that in the present case the stable link K(f_{1}) = K(f_{0}) is a rational homology sphere. Notice also that f_{0} is weighted homogeneous and deg(z_{2}^{3}z_{3}) > deg(f_{0}). (This example also shows that [Ném04, Question 13.12] has a negative answer: i.e. for a deformation which admits a weak simultaneous resolution the existence of a strong simultaneous resolution is not guaranteed, even if the stable link is a rational homology sphere.)

3.2.6 Example. The recent manuscript [ABLdBMH07, §4] provides a μ-constant deformation of singularities with non-degenerate Newton principal part and b_{1}(K) > 0 such that the homeomorphism of the tangent cone jumps, providing a counterexample to [Zar71, Conjecture B.]. In fact, a counterexample also exists among rational homology spheres, e.g. the deformation f_{t} = z_{1}^{3}z_{2} + z_{3}^{5} + z_{2}^{11} + tz_{1}^{2}z_{3}^{2} of type Move 1. Then the homeomorphism type of {z_{1}^{3}z_{2} + tz_{1}^{2}z_{3}^{2} = 0} is not constant.

3.3 Distinguished representatives 3.3.1. In this subsection we assume that all our Newton diagrams satisfy (2.1) and (2.5). It is preferable to have in each ∼-equivalence class a well-characterized and easily recognizable representative to work with. In its choice we are guided by the following principles (motivated by 4.2.5, which says that such a ‘minimal’ diagram reflects better the minimal resolution graph G_{min}(f) of the germ f):

(a) the representative should have a minimal number of faces;

(b) all the faces which cannot be eliminated by M1_{−} should be ‘minimized as much as possible’ by M2_{−}; (c) a representative may contain a trapezoid only if the trapezoid cannot be replaced by a triangle

in its class.

This motivates the following:

3.3.2 Definition. A diagram Γ is called M1-minimal if by the direct application of a move of type M1_{−} one cannot eliminate any of its faces.

Notice that at least one M1-minimal representative exists in any equivalence class.

Also, one can decide the M1-minimality of a diagram by analyzing the lattice points sitting on it, without any information about the other diagrams in its class. But, exactly for this reason, the above definition does not exclude the possibility that an M1-minimal diagram may have another diagram in its class with less faces. In fact, this may occur:

3.3.3 Example. (Cf. the proof of Corollary 3.2.3.) The diagram consisting of the unique triangular face with vertices (a, 0, c), (0, 1, b), (1, 1, 0) is M1-minimal, but it is equivalent to the segment [(1, 1, 0) , (0, 0, ab + c)].

The next lemma guarantees that this is the only pathological case when such a phenomenon may occur. Below #(Γ) denotes the number of faces of Γ.

3.3.4 Lemma. Fix a diagram Γ, which is not of type characterized by Corollary 3.2.3.

(a) Then Γ is M1-minimal if and only if for any Γ′ ∼ Γ one has #(Γ′) ⩾ #(Γ).

(b) If Γ and Γ′ are both M1-minimal and Γ ∼ Γ′, then they can be connected by a sequence of diagrams related to each other only by moves M2_{±}. In particular, the set of supporting planes of the faces of the two diagrams are the same.

Notice that the assumption is essential for part (b), too: see e.g. the segments of 3.1.5.

Proof. A sequence of diagrams Γ_{1}, Γ_{2}, …, Γ_{k} connects Γ_{1} and Γ_{k} if Γ_{i} and Γ_{i+1} (for 1 ⩽ i ⩽ k − 1)

are related by one of the moves M1_{±} or M2_{±}, denoted by Γ_{i} ⟶^{Mj_{±}} Γ_{i+1}. Our goal is to replace a given sequence of diagrams connecting Γ and Γ′ by another one which has the additional property that all M1_{−} moves appear first. For this, first we analyze how one can modify two consecutive moves in a sequence, where the second one is M1_{−}:

Fact. Γ_{i} ⟶^{Mj_{±}} Γ_{i+1} ⟶^{M1_{−}} Γ_{i+2} can be replaced either by moves Γ_{i} ⟶^{M1_{−}} Γ′_{i+1} ⟶^{Mj_{±}} Γ_{i+2}, or by a

single move of type Γ_{i} ⟶^{M1_{−}} Γ_{i+2}, or both moves can be eliminated, i.e. Γ_{i} = Γ_{i+2}.

Indeed, if the two moves operate on different faces of the diagram then they can be performed in the reverse order with the same effect. So we can assume that the two moves operate on the same face △.

If, additionally, the two moves have the same axis AB (see 3.1.3), then the moves eliminate one triangle from both sides of AB, hence Γ_{i} ∼ AB contradicting our assumption. Hence the two axes are different. This can occur only if △ is a triangle and the composition of the two moves is the removal of △, i.e. a move of type M1_{−}. This finishes the proof of the fact.

The easy consequence of the fact is that if Γ ∼ Γ′ then they can be connected by a sequence Γ_{i}, in which all the M1_{−} moves appear first (preceding the moves of other type). In particular, if Γ is M1-minimal then this sequence does not contain any M1_{−} moves, so the number of faces #(Γ_{i}) is non-decreasing along the sequence. This proves the non-trivial part of 3.3.4(a). If Γ′ is also M1-minimal then (applying the above also for the reverse sequence) #(Γ_{i}) must be constant, i.e. the sequence does not contain any move of type M1_{±}, finishing the proof of (b).

3.3.5 Definition (Canonical and minimal representatives). Fix the equivalence class of a diagram which does not satisfy 3.2.3, and consider all M1-minimal representatives. By 3.3.4(b) they are related to each other by moves M2_{±}. Clearly, this set has a unique maximal element with respect to M2_{±} (or equivalently, with respect to the inclusion). This diagram will be called the canonical representative of the class. It can be easily recognized: it is M1-minimal, and all its faces are as large as possible.

The canonical representative satisfies the principle (a) of 3.3.1, but not (b). For (b), we would need the unique minimal element with respect to M2_{±} of all M1-minimal representatives; but such an element, in general, does not exist. Nevertheless, we consider the set of minimal elements (diagrams which cannot be reduced by M2_{−}) of all M1-minimal representatives. We call these representatives minimal. By 3.3.4, these are those representatives which cannot be reduced by any move Mj_{−}.

3.3.6 Example.

(a) Fix a trapezoidal face △ of Γ with vertices as in 2.3.2. One can remove the vertex D if and only if either n = 1 or r_{2} + tq = 1. The vertex A can be removed if and only if r_{2} + q = 1. (There are analogous characterizations for B and C, too.) The case n = 1 is the ‘moving triangle’ situation 3.1.7. If r_{2} + q = 1, then there are (at least) two possibilities for the choice of the axis of M2_{−}, namely the segments [(0, 1, n), (r_{1} + tp, 0, 0)] and [(p, 0, n), (r_{1} + tp − p, 1, 0)]. One of them replaces the trapezoid by a triangle, while the other replaces it by a smaller trapezoid. Hence, in any situation, if a trapezoid can be decreased in some way, then it can be replaced by a triangle in the equivalence class of the diagram. Otherwise, it is called non-removable (this happens if n > 1, r_{1} + p > 1, r_{2} + q > 1).

(b) If above q = p = 1 and r_{1} = r_{2} = 0, then △ is the canonical representative of its class. One has four possible axes, and △ can be reduced to the trapezoid (0, 1, n), (1, 0, n), (t−1, 1, 0), (1, t−1, 0) or to the triangles (0, 1, n), (t, 0, 0), (1, t − 1, 0) or (1, 0, n), (t − 1, 1, 0), (0, t, 0). These are the minimal representatives.

3.3.7 Remark.

(a) Let △ be a triangular face of an M1-minimal representative Γ. Then in any minimal representative of Γ, which is obtained from Γ via moves M2_{−}, △ survives as a triangular face which is independent of the choice of the minimal representative. This happens, because the axes of all the moves M2_{−}, which can be applied to △, cannot intersect each other, hence all of them can be applied ‘simultaneously’ (a fact, which is not true in the case of removable trapezoids, see 3.3.6 above).

(b) Therefore, any class whose canonical representative has a non-removable trapezoid, or a central triangle or a central edge, admits a unique minimal representative

3.3.8 Discussion/Definition (d-minimal representatives). Fix a class. It may contain many minimal representatives; we will distinguish one of them, and we call it d-minimal (distinguishedminimal). If the class admits a unique minimal representative, then there is no ambiguity for the choice. This happens e.g. in all the situations 3.3.7(b).

For the sake of completeness, we allow diagrams satisfying 3.2.3. For such a class, the (d-)minimal representative is the segment [(0, 1, 1), (n, 0, 0)], for some n ⩾ 2, as given in 3.2.3(c).

Next, assume that a canonical representative contains a removable trapezoid (i.e. one replaceable by a triangle). Using the notations of 2.3.2, if n > 1, then again there is a unique minimal representative, unless we are in the situation of 3.3.6(b) (when there are two, but they correspond to each other by a permutation of coordinates). By definition, this is the d-minimal representative (in the last case it is well-defined up to the permutation of coordinates).

If n = 1, then we are in the situation of a moving triangle 3.1.7, and the class may contain many minimal representatives. (An even more annoying fact is that such a class may contain two equivalent diagrams such that one of them has a central triangle while the other has a central edge.) We will declare the position of the moving point R for the d-minimal representative as follows. Assume that p < q (for q < p interchange z_{1} and z_{2}). If R cannot be moved to any of the coordinate axis (cf. (3.1)), then take for R that possible lattice point which is closest to the z_{1} axis. If R can be moved to exactly one coordinate axis, then move it there. If R can be moved to both axes, then move to the z_{1} axis. (Since the determinants of QR and P R are p and q, respectively, by this choice of R, the determinant of the edge lying on the coordinate plane is larger. There is no deep motivation for this choice, except that we need one. In the ‘inverse’ algorithm the very same choice is built in.)

3.3.9 Corollary (Structure of d-minimal representatives). (Cf. also with 2.3.9.) The d-minimal representative of an equivalence class (which does not satisfy 3.2.3) sits in exactly one of the following three disjoint families of diagrams characterized by the existence of

(1) a non-removable trapezoid,

(2) a central triangle, or

(3) a central edge.

3.3.10 Notation. The three disjoint families listed in 3.3.9 will be denoted by ■, ▲, l. They can be divided further according to the number of hands. This number will appear as a subscript. E.g., ■_{3} denotes that family of classes of Newton boundaries whose d-minimal representative has a nonremovable trapezoid and 3 hands.

The first fruit of the minimality of a graph is the following arithmetical criterion:

3.3.11 Proposition. Fix a minimal representative Γ_{min} of a class which does not satisfy 3.2.3. Consider an edge of ∂Γ_{min} which is the intersection of the faces △ and ▽ of Γ_{min,+}, where the second one is non-compact. Then n_{△,▽} > 1.

Proof. We have to analyze two types of edges, cf. 2.1.5. First we discuss edges on a coordinate plane, say [(q_{1}, 0, q_{3}), (q′_{1}, 0, q′_{3})]. Take a triangle in △ which satisfies the criterions of 8.1.2(c). Then n_{△,▽} = n_{△,\overrightarrow{e_{2}}} = p_{2}. But, if p_{2} = 1, then this triangle can be eliminated by Mj_{−}.

Now we turn to the other type of edges, which have the form AB = [(a, 0, c) , (0, 1, b)] with a > 0. Take a third vertex C = (r, s, u) on △ such that △_{ABC} is empty. Then the identity (8.8) of 8.1.3 can be applied: n_{△,▽} = r + (s − 1)a. Assume that r + (s − 1)a = 1. If s = 0 then [(r, 0, u), (0, 1, b)] is an axis of a move Mj_{−}, hence △_{ABC} can be eliminated. If s = 1, then r = 1, hence by (2.5) u = 0, which contradicts our assumption about 3.2.3. The remaining case s ⩾ 2 imposes r = 0, s = 2, a = 1, with [(1, 0, c), (0, 2, u)] an axis of Mj_{−} which eliminates △_{ABC}.

3.3.12 Remark.

(a) By the above proof, when we eliminate triangles from a diagram by moves Mj_{−}, then, in fact, we eliminate those ‘mixed determinants’ (i.e. when a face is non-compact) with n_{△,▽} = 1. By 3.3.11, by repeated application of Mj_{−}, we can eliminate all such mixed determinants, provided that the class does not satisfies 3.2.3. (Otherwise this is not true: 3.3.3 shows a minimal triangle with a ‘mixed determinant’ 1.)

(b) The statement of 3.3.11 is also true for a class which satisfies 3.2.3 (where △ and ▽ are noncompact and contain [(0, 1, 1), (n, 0, 0,)]): n_{△,▽} = n > 1.

4. The dual resolution graph

4.1 Graph terminology 4.1.1. Recall that any resolution graph G(f) of {f = 0} is also a possible plumbing graph of the link K(f) of f. The link K(f) is a rational homology sphere if and only if G(f) is a tree, and the genera of all the vertices are 0. In such a case, G(f) has only one set of decorations: each vertex carries the self-intersection number of the corresponding irreducible exceptional divisor. In this subsection we recall the terminology of resolution graphs, and we present a construction which ‘simplifies’ a given graph. Its output will be called the orbifold diagram.

4.1.2. Let G be a decorated tree with vertices V and decorations {b_{v}}_{v∈V}. The entries of intersection matrix (I_{vw})_{v,w∈V} of G are I_{vv} = b_{v}, and for v ≠ w one sets I_{vw} = 1 if [vw] is an edge, and I_{vw} = 0 otherwise. We assume that I is negative definite (since the matrix of a dual resolution graph is so [Mum61]). By definition, det(G) := det(−I) is the determinant of the graph G.

A node of G is a vertex whose degree is at least 3. Let N be their collection. A chain is the path between two nodes excluding the endpoints, which does not contain any nodes. (We say that the chain connects the two nodes.) Similarly, a leg of G is a path between a degree 1 vertex and a node containing the degree 1 vertex but not the node, and containing no other nodes, either.

If r, s ∈ N are connected by a chain in G, then the determinant of this chain (i.e., the determinant of the corresponding subgraph) will be denoted by n_{rs}.

A star-shaped graph is a graph with a unique node. For any r ∈ N there is a unique maximal star-shaped subgraph G_{r} of G which contains r.

In general, a star-shaped graph is a plumbing graph of a Seifert 3-manifold. This has a natural S^{1}-action and orbifold structure. If the star-shaped graph G_{r} has normalized Seifert invariants, say, (α_{i}, ω_{i})_{i} (here, each pair is associated with one of the legs of the subgraph, α_{i} is the leg-determinant, 0 ⩽ ω_{i} < α_{i}, and we put the pair (1, 0) for legs with determinant one), and central vertex with decoration b_{r}, then the orbifold Euler number of G_{r} is e_{r} := b_{r} + ∑_{i} ω_{i}/α_{i}, see e.g. [Wag83] for details.

4.1.3. The orbifold diagram. Sometimes we do not need all the data of G, but only its shape and the determinants of some of its subgraphs. This information will be codified in a simpler graph-like diagram, the orbifold diagram associated with G, denoted by G^{o}.

G^{o} is constructed from G as follows. G^{o} has vertices, edges connecting two vertices, and half-free edges. A half-free edge is attached with one of its ends to a vertex, while its other end is free. The vertices of G^{o} are the nodes of G. The (ordinary) edges of G^{o} are the chains of G. The endpoints of an edge are the two nodes it connects as a chain. The half-free edges are the legs of G. The endpoint of a half-free edge is the node to which it is adjacent in G as a leg. Then we decorate G^{o}: we put on each edge the determinant of the corresponding chain or leg, and we label each node r with the orbifold Euler number e_{r} of the star-shaped subgraph G_{r}. (In the special case when G has no nodes then G^{o} is a ‘free’ edge decorated by det(G).)

The half-free edges of the orbifold diagram G^{o} will still be called legs.

The entries of the orbifold intersection matrix (I_{rs}^{o})_{r,s∈N} of G^{o}, by definition, are I_{rr}^{o} = e_{r}, and for r ≠ s one sets I_{rs}^{o} = 1/n_{rs} if [rs] is an edge of G^{o}, and I_{rs}^{o} = 0 otherwise. (Here we will not explain the ‘orbifold geometry’ behind this definition. Nevertheless, for a possible motivation, see §4.4.) Similarly as above, we set det(G^{o}) := det(−I^{o}).

4.1.4 Lemma. Fix a graph G as above with N ≠ ∅. Let Π be the product of the determinants of all the chains and legs of G. Then I^{o} is negative definite, and

det(G) = det(G^{o}) · Π.  (4.1)

Proof. The negative definiteness of I^{o} follows from (4.1) applied to some subgraphs. The equality (4.1) is elementary linear algebra, it follows (e.g.) by induction on #N. If #N = 1, then (4.1) is well-known, see e.g. [Neu83]. The induction runs as follows. Fix r, s ∈ N which are connected by a chain G_{rs}. The connected components of G \ ({r, s} ∪ G_{rs}) are {G_{i}}_{i}, the connected component of G \ {r} which contains s is G_{(s)}, and similarly one defines G_{(r)}. Then det(G) · det(G_{rs}) = det(G_{(s)}) · det(G_{(r)}) − ∏_{i} det(G_{i}).

4.1.5 Remark. The orbifold diagram has exactly the same shape as the splice diagram considered in [NW02], but it has different decorations. Nevertheless, by similar identities what we used in the proof of 4.1.4, one can show that the orbifold diagram contains the same amount of information as the splice diagram and det(G) altogether.

4.2 Oka’s algorithm for G(f). The case of minimal representatives. 4.2.1. Let f: (ℂ^{3}, 0) → (ℂ, 0) be a germ with isolated singularity and non-degenerate Newton principal part whose link K(f) is a rational homology sphere. In particular, its Newton boundary Γ(f) satisfies (2.1) and (2.5). In the first part of this subsection we recall the combinatorial algorithm of M. Oka [Oka87, Theorem 6.1], which provides a (possible, in general non-minimal) dual resolution graph G(f) of the surface singularity ({f = 0}, 0) from Γ(f).

In order to emphasize the dependence of the output upon Γ(f), we write G(Γ(f)).

4.2.2. Notations. Recall that Γ(f) is the union of compact faces of Γ_{+} := Γ_{+}(supp(f)), which can be recovered from Γ(f) as Γ_{+}({vertices of Γ(f)}). Hence, they contain the same amount of information. Similarly as above, F denotes the collection of all faces of Γ_{+}, and F_{c} denotes the set of all compact faces of Γ_{+}. For any △ ∈ F_{c}, we write F_{△} for the collection of all faces of Γ_{+} adjacent to △. Other notations are from 2.1.7.

4.2.3. The algorithm. The graph G(Γ(f)) is a subgraph of a larger graph G̃(Γ(f)), whose construction is the following. To start with, we consider F as a set of vertices (we will call them face vertices). Then, if △, ▽ ∈ F are two adjacent faces, then we connect them by t_{△,▽} copies of the following chain.

If n_{△,▽} > 1 then let 0 < c_{△,▽} < n_{△,▽} be the unique integer for which

\overrightarrow{c}_{△,▽} := (\overrightarrow{a}_{▽} + c_{△,▽} \overrightarrow{a}_{△})/ n_{△,▽}  (4.2)

is an integral vector. Let us write n_{△,▽} / c_{△,▽} as a continued fraction:

\frac{n_{△,▽}}{c_{△,▽}} = b_{1} − \frac{1}{b_{2} − \frac{1}{⋯ − \frac{1}{b_{k}}}},  (4.3)

where each b_{i} ⩾ 2. Then the chain with the corresponding self-intersection numbers is

[figure: △; ▽; −b_{1}; −b_{2}; −b_{k}; ⋯]

Figure 2. Chain between two face vertices

The left ends of all the t_{△,▽} copies of the chain (marked by △) are identified with the face vertex corresponding to △, and similarly for the right ends marked by ▽.

If n_{△,▽} = 1 then the chain consists of an edge connecting the vertices △ and ▽ (we put t_{△,▽} of them). Also, in this case we set c_{△,▽} := 0 and \overrightarrow{c}_{△,▽} := \overrightarrow{a}_{▽}.

Next, we compute the decoration b_{△} of any face vertex △ ∈ F_{c} by the equation:

b_{△} \overrightarrow{a}_{△} + ∑_{▽∈F_{△}} t_{△,▽} \overrightarrow{c}_{△,▽} = \overrightarrow{0} .  (4.4)

What we get in this way is the graph G̃(Γ(f)). Notice that the face vertices corresponding to noncompact faces are not decorated. If we delete all these vertices (and all the edges adjacent to them) we get the dual resolution graph G(Γ(f)).

Notice that G̃(Γ(f)) has no legs, but some of the chains of G̃(Γ(f)) become legs of G(Γ(f)) when we delete the vertices corresponding to F \ F_{c}. Regardless whether a chain in Figure 2 transforms into a chain or a leg of G(Γ(f)), it keeps its determinant n_{△,▽}.

4.2.4 Remark. If one starts with another Newton diagram, say Γ′(f), obtained from Γ(f) via Moves 1 or 2, then the graph G(Γ′(f)) can be obtained from G(Γ(f)) by some blow-ups, in accordance with 3.2.1(b). Hence, in general, G(Γ(f)) is not a good minimal resolution graph. Recall that a dual resolution graph G(f) with all genera vanishing is good minimal if all its (−1)-vertices are nodes. Each normal surface singularity admits a unique good minimal resolution.

4.2.5 Proposition. If the Newton diagram Γ_{min}(f) is a minimal representative of its class then the output G(Γ_{min}(f)) of Oka’s algorithm is the good minimal resolution graph. In fact, G(Γ_{min}(f)) reflects the shape of the diagram Γ_{min}(f) (preserving the corresponding adjacency relations):

(a) the nodes of G(Γ_{min}(f)) correspond bijectively to the faces of Γ_{min}(f);

(b) the chains and legs of G(Γ_{min}(f)) correspond bijectively to the edges of Γ_{min}(f) not lying in ∂Γ_{min}(f) and the edges lying on ∂Γ_{min}(f), respectively. (In the case of 3.2.3 we understand by this that Γ = ∂Γ is a segment, and G(Γ_{min}(f)) is a string.)

Proof. The chains in Figure 2 contain no (−1)-vertex, any face has at least three edges, and all the leg-determinants are greater than 1 by 3.3.11 and 3.3.12, hence the statement follows.

The legs corresponding to different primitive segments of the same edge form a leg group.

In the next subsection we make a more direct connection between the normal vectors of faces, the coordinates of vertices of Γ_{min}(f), and the determinants of legs in G(Γ_{min}(f)).

4.3 Leg-determinants in G(Γ_{min}(f)) We fix a minimal Newton diagram Γ_{min} = Γ_{min}(f) which does not satisfy 3.2.3, and let △ ∈ F_{c} be one of its faces. Let us consider the legs in G(Γ_{min}) adjacent to △. By 4.2.5 they correspond to the primitive segments lying on the edges of ∂Γ_{min} ∩ △. The next proposition summarizes the divisibility properties of the determinants of these legs. We will refer to such a leg-determinant as the determinant D(α) of the corresponding edge α. (The coordinate choices are accidental, they can be permuted arbitrarily.) As usual, we write \overrightarrow{a}_{△} = (a_{1}, a_{2}, a_{3}).

4.3.1 Proposition.

I. Any edge α of ∂Γ_{min} ∩ △ satisfies the next divisibility properties: 1. If α ⊂ {z_{3} = 0}, then D(α) = gcd (a_{1}, a_{2}) and gcd (D(α), a_{3}) = 1.

2. If α = [(p, 0, a) , (0, 1, b)] crosses the z_{3} axis, then D(α) = a_{3}, and D(α) does not divide any of {a_{1}, a_{2}}, unless the edge can be ‘moved’ to a coordinate plane, see 3.1.7. If this happens, then D(α) divides the corresponding two coordinates as in part (I1), and either of the following cases holds.

(a) The edge belongs to a ‘moving triangle’ so that it can be moved to a coordinate plane by moving the moving vertex to a coordinate axis, see (3.1).

(b) (0, 2, c) is on △, hence △ can be extended (in the class of Γ_{min}) with a new vertex (0, 0, 2b − c), which lengthens the edge β = [(0, 1, b) , (0, 2, c)] by an extra primitive segment. We interpret this as moving α to this extra segment whose determinant is D(β).

II. The determinants belonging to different edges of △

1. differ, except in the case of (I2b), where D(α) = D(β);

2. are pairwise relative prime except the two cases below:

(a) an edge lying on a coordinate plane is adjacent to a crossing edge: the determinant of the former one divides the determinant of the latter one;

(b) an edge α lying on a coordinate plane is adjacent to two crossing edges: then D(α) is the greatest common divisor of the determinants of the crossing edges.

Proof. (I1) is clear by (2.2) since \overrightarrow{a}_{△} is primitive. (8.7) implies the first part of (I2). Since \overrightarrow{a}_{△} is orthogonal to α, i.e. ⟨\overrightarrow{a}_{△} , (p, 0, a)⟩ = ⟨\overrightarrow{a}_{△} , (0, 1, b)⟩, one has

a_{2} = pa_{1} + (a − b)a_{3}.  (4.5)

Since \overrightarrow{a}_{△} is primitive a_{3} ∤ a_{1}, and if a_{3} | a_{2} then D(α) = a_{3} | p, too. This will be used later.

Let P be a lattice point of △, such that the triangle ▽ (a part of the face △) formed by α and P is empty. First notice that P cannot be on the z_{1}z_{3} plane, since then ▽ would be removable. If P lies on the z_{2}z_{3} plane and has the form (0, q, c), then by (8.8), D(α) = (q − 1)p. If D(α) | p then q = 2. This is the case (I2b).

Assume now that P lies on the z_{1}z_{2} plane, P = (r_{1}, r_{2}, 0) with r_{1} and r_{2} positive. Then, again by (8.8), D(α) = r_{1} + p(r_{2} − 1). Thus, if D(α) | p then r_{2} = 1 and r_{1} | p. But then the triangle ▽ is movable (as in 3.1.7), and the vertex (p, 0, a) can be moved to the point (0, 0, a + bp/r_{1}) lying on the third coordinate axis. This is the case (I2a).

For part (II), assume that the leg-determinants belonging to two different edges are not relative prime. If one of the edges lie on a coordinate plane, then (2.2) and (8.7) show that we are in the situation (II2a) or (II2b). Otherwise, if one of the edges crosses, say, the z_{1} axis, and the other edge crosses the z_{2} axis, then their endpoints sitting on the z_{1}z_{2} plane do not coincide, and hence case (II2b) holds. Indeed, assume that the two endpoints do coincide. This common point cannot be (1, 1, 0) by our assumption, cf. 3.2.3. Otherwise, by a relation similar to (4.5), the third coordinate of \overrightarrow{a}_{△} is an integral linear combination of the first two ones (which are the determinants), contradicting the fact that \overrightarrow{a}_{△} is primitive.

Finally, if two determinants are equal, then by part (II2) the corresponding edges must be as in (II2a). But then, by (8.6) and (8.8), we have q = 2 which leads to (I2b).

4.3.2 Corollary (Non-removable trapezoids). The leg groups of a non-removable trapezoid have different determinants. Hence, the collection of chains and legs adjacent to the vertex corresponding to the trapezoid can be separated in 4 distinguishable groups.

Since a vertex corresponding to a triangular face has at most 3 such groups of distinguishable legs and chains, the vertex of a non-removable trapezoid can be recognized in the resolution graph.

Proof of 4.3.2. Assume the contrary. Then by 4.3.1, we are in the situation of (I2b) with the points (p, 0, a), (0, 1, b) and (0, 2, c) on the trapezoid (modulo a permutation of the coordinates). Then this face can be extended by the vertex (0, 0, 2b − c). This extended face is a trapezoid, too. But by 3.3.6(a), this is a removable trapezoid.

4.4 The orbifold diagram 4.4.1. We fix a minimal representative Γ_{min}(f) as in 4.2.5. §4.2 provides a good minimal resolution graph G(Γ_{min}(f)) from the Newton diagram Γ_{min}(f). On the other hand, to any graph G, the general procedure 4.1.3 associates a diagram G^{o}. In the present situation this will be denoted by G^{o}(Γ_{min}(f)). Although, by the very construction of G^{o}, we (apparently) throw away some information, we prefer to use G^{o}(Γ_{min}(f)) since it reflects more faithfully the Newton diagram. For the convenience of the reader, in short, we sketch how one can draw G^{o}(Γ_{min}(f)) directly from Γ_{min}(f).

Similarly as in §4.2, first we construct a decorated graph G̃^{o}. Its vertices are the elements of F, i.e. all the faces of Γ_{min,+}. If △, ▽ ∈ F are adjacent in Γ_{min,+}, then we connect them by t_{△,▽} edges in G̃^{o}, and we label each of these edges with the number n_{△,▽}. Finally, we label each △ ∈ F_{c} with the orbifold Euler number of the maximal star-shaped subgraph containing △, which is

e_{△} := b_{△} + ∑_{▽∈F_{△}} t_{△,▽} \frac{c_{△,▽}}{n_{△,▽}} .  (4.6)

In this way we get the labelled graph G̃^{o}. If we remove the vertices {v_{△} : △ ∈ F \ F_{c}} (but we keep the edges—i.e. the new legs—adjacent to them), we get the diagram G^{o}(Γ_{min}(f)). For any △ ∈ F_{c}, we call e_{△} the orbifold Euler number of △.

4.4.2. The point is that (4.4) can be transformed via the orbifold Euler numbers into some (more natural) identities which only involve the normal vectors of the faces.

4.4.3 Proposition. Fix the representative Γ_{min}(f). Then for any △ ∈ F_{c} one has:

e_{△} \overrightarrow{a}_{△} + ∑_{▽∈F_{△}} \frac{t_{△,▽}}{n_{△,▽}} \overrightarrow{a}_{▽} = \overrightarrow{0} .  (4.7)

Proof. Use (4.2), (4.4) and (4.6).

Obviously, if one wishes to recover the equation of a face of Γ_{min}(f), one needs its normal vector \overrightarrow{a}_{△}, and its face value, i.e. the value of \overrightarrow{a}_{△} on any of the face’s point:

m_{△} := ⟨\overrightarrow{a}_{△} , P ⟩ , P ∈ △.  (4.8)

It turns out that these numbers {m_{△}}_{△∈F} also satisfy a similar equation:

4.4.4 Proposition. Fix Γ_{min}(f) as above. Then for any △ ∈ F_{c} one has:

e_{△} m_{△} + ∑_{▽∈F_{△}} \frac{t_{△,▽}}{n_{△,▽}} m_{▽} = − g(△) .  (4.9)

Proof. Denote the vertices of △ by P_{0}, …, P_{k} in this order, and set P_{k+1} := P_{0}. Assume that P_{i}P_{i+1} is the common edge of △ and ▽. Then, by (8.4), one has

g(△_{P_{0}P_{i}P_{i+1}}) = \frac{t_{△,▽}}{n_{△,▽}} ⟨\overrightarrow{a}_{▽} , P_{0} − P_{i}⟩ = \frac{t_{△,▽}}{n_{△,▽}} ⟨\overrightarrow{a}_{▽} , P_{0}⟩ − \frac{t_{△,▽}}{n_{△,▽}} m_{▽} ,  (4.10)

with g(△_{P_{0}P_{0}P_{1}}) = g(△_{P_{0}P_{k}P_{k+1}}) = 0. Then, by (4.7) and (4.10)

− e_{△} m_{△} = − e_{△} ⟨\overrightarrow{a}_{△} , P_{0}⟩ =

∑ \frac{t_{△,▽}}{n_{△,▽}} ⟨\overrightarrow{a}_{▽} , P_{0}⟩ = ∑ \frac{t_{△,▽}}{n_{△,▽}} m_{▽} + ∑ g(△_{P_{0}P_{i}P_{i+1}}) .

Then use the additivity of the combinatorial area.

4.4.5 Corollary. For any subset \overline{F}_{c} ⊂ F_{c} consider the system of equations (4.7) for all △ ∈ \overline{F}_{c} in unknowns \{\overrightarrow{a}_{△} : △ ∈ \overline{F}_{c}\} (or in one of their fixed coordinates). Then this system is non-degenerate. The same is true for equations (4.9) instead of (4.7).

Proof. The matrix of the system (for \overline{F}_{c} = F_{c}) coincides with the matrix I^{o} of the orbifold diagram G^{o}(Γ_{min}(f)) (cf. 4.1.3), which is negative definite by 4.1.4.

4.4.6 Remark. If one wishes to solve the above equations, one needs the values for non-compact faces. If such a face is supported by a coordinate plane, then its normal vector is a coordinate vector, and its face value is 0. Otherwise, if it has an edge of type [(a, 0, c) , (0, 1, b)] (a > 0), then its normal vector is (1, a, 0) and its face value is a.

The next lemma connects the face value of a central triangle with entries of normal vectors:

4.4.7 Lemma. Let △ be an empty central triangular face with three adjacent faces ▽_{i}. Write n_{i} := n_{△,▽_{i}}, \overrightarrow{a}_{▽_{i}} = (a_{1}^{(i)}, a_{2}^{(i)}, a_{3}^{(i)}) (1 ⩽ i ⩽ 3); and \overrightarrow{a}_{△} = (a_{1}, a_{2}, a_{3}). Then

− m_{△} = a_{1}a_{2}a_{3} · (e_{△} + ∑_{i=1}^{3} \frac{a_{i}^{(i)}}{n_{i}a_{i}} ) .  (4.11)

Proof. Let (0, p_{2}, p_{3}) be a vertex of △. Then m_{△} = p_{2}a_{2} + p_{3}a_{3}. By (8.3) (and by a sign check) a_{1}^{(2)}a_{2} − a_{2}^{(2)}a_{1} = n_{2}p_{3} and a_{1}^{(3)}a_{3} − a_{3}^{(3)}a_{1} = n_{3}p_{2}. Use these identities and (4.7).

4.4.8 Remark.

(a) Of course, all the results proved for G(Γ_{min}(f)) can be transformed into properties of G^{o} = G^{o}(Γ_{min}(f)). E.g., 4.2.5 reads as follows. The diagram G^{o} reflects the shape and adjacency relations of Γ_{min}(f): the vertices of G^{o} correspond to the faces of Γ_{min}(f). The edges of G^{o} connecting vertices correspond to edges of Γ_{min}(f) not lying in ∂Γ_{min}(f), and the legs of G^{o} correspond to the primitive segments lying on the edges of ∂Γ_{min}(f). By 3.3.11, all the legdecorations are greater than 1, and they satisfy the divisibility properties of 4.3.1. Moreover, by the very definition, the intersection orbifold matrix I^{o} (cf. 4.1.3) can be read from G^{o} and also the combinatorial areas {g(△)}_{△∈F_{c}}, needed in (4.9). Indeed, g(△) + 2 equals the degree of the corresponding vertex in G^{o} (cf. (2.3) and (2.5)).

(b) One may ask: how easily can Γ_{min}(f) be recognized from G^{o}? Well, rather hardly! Already the types (1–3) of 3.3.9 are hardly recognizable. The ‘easy’ cases include ■_{∗} (cf. 3.3.10) recognized via 4.3.2, or ▲_{3} (G^{o} has a vertex adjacent with three other vertices). In these cases, one also recognizes the vertices corresponding to hands (vertices adjacent with one vertex), or to central faces. Also, if G^{o} has one vertex, then it corresponds to a one-faced diagram. But all the other families cannot be easily separated. E.g., it is hard to separate the case of a central triangle with two non-degenerate arms from the case of a central edge. In these cases, it is not easy at all to find the hands or central triangles. Another difficulty arises as follows. Consider an arm (with many triangles) crossing, say, the z_{3} axis. There are two types of triangles in it, depending on whether the non-crossing edge is on the z_{1}z_{3} or z_{2}z_{3} plane. These types are invisible from the shape of G^{o} (and will be determined using technical arithmetical properties of the decorations).

5. Starting the inverse algorithm

5.1 The main result 5.1.1. We consider analytic germs f: (ℂ^{3}, 0) → (ℂ, 0) with isolated singularity at 0, with nondegenerate Newton principal part, and with rational homology sphere link K(f). At a combinatorial level, this means that we consider all the Newton boundaries Γ with (2.1) and (2.5). Oka’s algorithm §4.2 provides a resolution graph G(Γ) for each such Γ. Such a graph, in general, is not good minimal. But 4.2.5 guarantees that if some graph G can be obtained by this procedure, then also the good minimal resolution graph G_{min} associated with G (obtained from G by repeated blow downs of (−1)- vertices of degree less than 3) can be obtained by running the algorithm for a minimal representative Γ_{min} of Γ.

Recall that all the resolution graphs which are equivalent modulo blowing up/down (−1)-vertices can be regarded as the plumbing graphs of the same plumbed 3-manifold, the link K(f). By [Neu81], this class of graphs, and the unique good minimal one, can be recovered from the oriented topological type of the link K(f).

Recall also that to any graph G one can associate the orbifold diagram G^{o}.

Our next result, which also implies Theorem 1.0.1 from the introduction, says that Oka’s algorithm is, basically, injective:

5.1.2 Theorem. The d-minimal representative Γ_{min} (up to a permutation of coordinates) can be uniquely recovered from the orbifold diagram G^{o} associated with G(Γ_{min}).

5.1.3 Corollary. We consider germs as in 5.1.1. Then one has:

(i) The orbifold diagram associated with the good minimal resolution contains the same information as G_{min} itself.

(ii) If the links of two germs f_{0} and f_{1} are homeomorphic then there exist germs {g_{i}}_{i=0}^{k} (as in 5.1.1) and a coordinate-permutation σ so that g_{0} = f_{0}, g_{k} = f_{1} ∘ σ, and g_{i} + t(g_{i+1} − g_{i}) (0 ⩽ i < k) is a μ-constant deformation corresponding to one of the moves Mj_{±}.

5.1.4. Outline of the algorithm. The inverse algorithm which recovers the d-minimal representative from G^{o} is rather long. It distinguishes 3 cases depending on the number N of nodes:

N=0 The minimal resolution graph has no nodes. This is the simplest case solved in 3.2.3: Let n denote the determinant (equivalently, G^{o} is a ‘free’ edge with decoration n). The representative is the diagram of z_{1}^{n} + z_{2}z_{3}. N=1 The minimal resolution graph is star-shaped (equivalently, G^{o} has only one vertex). This case corresponds to (deformations of) isolated weighted homogeneous germs. Then [MN05] proves that from the resolution graph one can recover the supporting plane π of the unique face of the (representative) Newton boundary (or the weights). Next we provide an even shorter argument. By [Pin77], the Poincaré series of the graded algebra of the germ f is recovered from G(f). But this is a rational function of type (t^{m} − 1)/((t^{a_{1}} − 1)(t^{a_{2}} − 1)(t^{a_{3}} − 1)) codifying the equation ∑ a_{i}z_{i} = m of π, cf. [Wag83]. Putting all the possible lattice points on π, we get the canonical representative of Γ.

A long combinatorial case by case verification recovers π from G^{o}, too, which the patient reader may rediscover using the classification in Appendix 8.2. N>1 The orbifold diagram has at least two vertices. This is the subject of the remaining sections, an outline of it is given here. The procedure involves three main technical steps:

(1) arm preprocessing which provides partial information about the arms and about the face(s) behind the shoulders (§5.2);

(2) determination of the center (§§ 5.4, 6 and 7);

(3) arm postprocessing which calculates the arms completely (§5.3).

In fact, (1) for an arm runs only if we know the position of the corresponding hand, otherwise it should be preceded by a hand-search step.

In the next two subsections, we are going to discuss the easier arm preprocessing and postprocessing, respectively. They are uniform no matter how the diagram looks like.

On the other hand, we devote more sections to determine the center, since it distinguishes many cases depending on how the center looks like and uses a separate algorithm in every case.

5.2 Arm preprocessing 5.2.1 Definition. Let us consider a non-degenerate arm of a d-minimal Newton diagram in the direction, say, of the z_{3} axis. Its basic data consists of the following:

(1) the correspondence κ between the triangles and edges of the arm and the corresponding vertices and decorated edge groups of G^{o}, respectively;

(2) the first and second coordinates of the vertices of the triangles of the arm;

(3) the third coordinates of the normal vectors of all the triangles of the arm and also of the (compact or non-compact) face of Γ_{+} opposite the shoulder;

(4) the face values of the non-compact faces adjacent to the triangles of the arm associating these numbers to the corresponding ‘half-free’ edges of G^{o}.

The basic data is an invariant of the arm, i.e. it is independent of the parts of Γ outside the arm, and also does not depend on the choice of coordinates; explicit coordinates are used in the definition only for simplicity of language. In particular, in the language of (1)–(4) above, it is only well-defined up to a permutation of the first two coordinates. Nevertheless, this permutation is global: if we exchange the coordinates in one triangle, then we have to exchange in all of them. (In (4) the face values are independent of the permutation of the first two coordinates. In fact, they are 0 excepting maybe one leg of the hand.)

It is convenient to distribute the basic data among the triangles: The basic data of a triangle △ of an arm (in the direction of the z_{3} axis) consists of the first and second coordinates of its vertices and the third coordinate of its normal vector, and also the correspondence κ between the edges of △ in Γ and edge groups of κ(△) in G^{o}. The basic data of a triangle is part of the basic data of the containing arm; the choice of coordinates agree with the choice of coordinates for the arm. (If a triangle is contained in several arms then it has a separate basic data for each of the containing arms.)

5.2.2. The aim of arm preprocessing. Assume that we identify in G^{o} the vertex corresponding to the hand of an arm in the d-minimal representative. The aim of arm preprocessing is to determine from G^{o} the basic data of this arm. We will compute the basic data of the triangles of the arm one after the other beginning at the hand. Meanwhile, we will also recognize when we reach the shoulder of the arm, and we will compute the third coordinate of the normal vector of the next face as required by 5.2.1(3).

5.2.3. We start with the basic data of the hand △. At this stage we are free to make any choice of coordinates: we assume that the arm is in the direction of z_{3} axis; and if △ has any edge with interior lattice points (say, t − 1 of them) then this edge sits on the z_{2}z_{3} plane. Let κ(Δ) be the corresponding vertex in G^{o}. The next paragraph collects some facts about decorations of the legs adjacent to κ(Δ). By 3.3.11, all of them are greater than 1.

5.2.4. If △ intersects the z_{3} axis, then there are two types of leg-decorations (cf. 4.2.5): t legs decorated with n_{△,\overrightarrow{e_{1}}} and one leg with n_{△,\overrightarrow{e_{2}}}. Notice that by 4.3.1(II) one has gcd (n_{△,\overrightarrow{e_{1}}} , n_{△,\overrightarrow{e_{2}}}) = 1, and by (8.6) the vertices of △ have the form (0, 0, ∗), (0, t n_{△,\overrightarrow{e_{2}}} , ∗), and (n_{△,\overrightarrow{e_{1}}} , 0, ∗). This last fact together with (8.5) implies a_{3} = n_{△,\overrightarrow{e_{1}}} · n_{△,\overrightarrow{e_{2}}}.

Otherwise, if △ has an edge of type [(a, 0, c) , (0, 1, b)] (a > 0), set \overrightarrow{n} := (1, a, 0) as in 8.1.3. Then there are t legs with decoration n_{△,\overrightarrow{e_{1}}} and one leg with n_{△,\overrightarrow{n}}. By (8.7) we have n_{△,\overrightarrow{n}} = a_{3}, by (8.6) we have n_{△,\overrightarrow{e_{1}}} = a, and by (8.8) we have n_{△,\overrightarrow{e_{1}}} | n_{△,\overrightarrow{n}} (since r = 0). Again by (8.8), the vertices of △ have the form (0, 1, ∗), (n_{△,\overrightarrow{e_{1}}} , 0, ∗) and (0, 1 + t n_{△,\overrightarrow{n}} / n_{△,\overrightarrow{e_{1}}} , ∗). Notice that n_{△,\overrightarrow{n}} = n_{△,\overrightarrow{e_{1}}} may happen only in the case 4.3.1(I2b).

For the corresponding face values, see 4.4.6.

5.2.5. Algorithm: the basic data of a hand. Let N be the set of all the decorations of the legs adjacent to κ(Δ). One may have the following situations:

(a) N = {n_{1}, n_{2}} with gcd (n_{1}, n_{2}) = 1. One of them, say n_{2}, decorates exactly one leg; the other one, n_{1}, decorates several legs, say t of them. (If t = 1 then the construction is symmetric.) Then (up to a permutation of the coordinates, cf. 5.2.1) the vertices of the hand have the form (0, 0, ∗), (0, tn_{2}, ∗) and (n_{1}, 0, ∗); and the third coordinate of \overrightarrow{a}_{△} is n_{1}n_{2}. All the face values are 0.

(b) N = {n_{1}, n_{2}} with n_{1} | n_{2}. Then the number of legs decorated by n_{1} will be denoted by t, and (automatically) n_{2} decorates one leg. The face value of legs with n_{1}-decoration is 0, but the face value of the unique n_{2}-decorated leg is n_{1}. The vertices of the hand have the form (0, 1, ∗), (n_{1}, 0, ∗) and (0, 1 + tn_{2}/n_{1}, ∗) and the third coordinate of \overrightarrow{a}_{△} is n_{2}.

(c) N = {n}. Then let t + 1 be the total number of legs. We set n_{1} = n_{2} = n. Then the basic data of the hand is given by the same formulas as in (b). We separate one leg (with face value n), the others form another group (with face value 0).

5.2.6. Arm continuation. Assume that we have computed from G^{o} the basic data of the triangles △_{1}, …, △_{k} (belonging to an arm in the direction of z_{3}, where △_{1} is the hand, and △_{i} is adjacent to △_{i+1}) in such a way that the coordinate-ambiguities are compatible (i.e., if we fixed coordinates for △_{1}, then for all the other △_{i} we respect the same choice). Our aim is to determine the part of the basic data corresponding to the next face, i.e. the third coordinate of its normal vector, and whether it belongs to the arm. If yes then we also compute its basic data.

We write △ := △_{k} and set κ(△) for the corresponding vertex of G^{o}. By the inductive step, we have already computed the correspondence κ of all the edges of △ with the edges adjacent to κ(△). Let ▽ be the next face of Γ_{+}, adjacent to △, and set γ := ▽ ∩ △. The face ▽ is compact if and only if κ(γ) connects two vertices of G^{o}, one of them is obviously κ(△). If this is the case, we set κ(▽) for the other end.

In any situation, we need the third coordinate a_{3} of the normal vector \overrightarrow{a}_{▽}. This can be computed from G^{o} and the basic data of the triangles △_{i} using (4.7).

Next, if ▽ is compact, we wish to decide whether it belongs to the arm. The face ▽ is a non-removable trapezoid if and only if κ(▽) admits four distinguishable groups of adjacent edges (cf. 4.3.2). In this case, clearly, ▽ does not belong to the arm. The same is true if κ(▽) has no legs (which happens if and only if ▽ is a central triangle). Therefore, assume that ▽ is a triangle with at least one adjacent leg.

5.2.7 Lemma. The face ▽ belongs to the arm if and only if κ(▽) has an adjacent leg whose decoration divides a_{3}.

Proof. Write γ = AB, and let C be the third vertex of ▽. If C is on the z_{2}z_{3} or z_{1}z_{3} planes, then the leg associated with BC or AC divides a_{3} by 4.3.1(I1). If C is not sitting on one of these two planes, then it can have a leg only if at least one of AC and BC is a crossing edge. But, by 4.3.1(I2), such an edge determinant divides a_{3} only if ▽ is a moving triangle whose moving vertex C can be moved to a coordinate axis. But this would contradict the definition of the d-minimal representatives in 3.3.8.

If ▽ does not belong to the arm, then we stop (having all the basic data of the arm).

Next, assume that ▽ belongs to the arm. Then we have to identify edges of ▽ with edges adjacent to κ(▽) in G^{o}, and to determine the first two coordinates of C.

First we identify the leg-decoration n_{▽}, adjacent to κ(▽), which corresponds to the edge α of ▽ which lies on a coordinate plane. By 4.3.1(I1), it divides a_{3}. We claim that n_{▽} is the largest leg-decoration adjacent to κ(▽) which divides a_{3}. Indeed, we have to check only the case when κ(▽) has two leg-decorations N = {n_{1}, n_{2}} (the determinants of AC and BC, one edge sitting on a coordinate plane, the other being a crossing edge), both dividing a_{3}. Then, by 4.3.1, ▽ is a moving triangle such that C can be moved to both coordinate axes, and by the construction of the d-minimal representative (cf. 3.3.8), the determinant of the edge which lies on the coordinate plane is the larger one.

Now, we fix an edge of △ (whose determinant will be denoted by n_{△}) which lies on a coordinate plane (which is either z_{1}z_{3} or z_{2}z_{3} determined clearly by the basic data of △). Denote this plane by π. Then, by 8.1.4, α lies on π if and only if n_{△} = n_{▽} | n_{△,▽}.

This is valid for C, too, hence this clarifies whether C is on the z_{1}z_{3} or z_{2}z_{3} plane. Finally, we have to compute the first two coordinates of C. One of them is 0 (depending whether π is the z_{1}z_{3} or z_{2}z_{3} plane), the other can be determined using (8.5).

Then we add ▽ to the triangles {△_{i}}_{i} and repeat the ‘arm continuation’ process by induction.

5.3 Arm postprocessing 5.3.1. The arm postprocessing step assumes the knowledge of two sets of data: the first one is the basic data coming from arm preprocessing, the second one is some knowledge about the face on the other side of the shoulder (which is usually the center but not always). More precisely, let ▽ be the face in Γ_{+} containing the shoulder of the arm, but not contained in the arm. (It is non-compact if and only if the edge κ(▽) in G^{o} is a leg.)

I. The first set of data: the basic data of an arm in the direction of the z_{3} axis modulo the ambiguity of a permutation of the first two coordinates. Recall that the basic data of the arm also includes the knowledge of ⟨\overrightarrow{a}_{▽} , \overrightarrow{e_{3}}⟩.

II. The second set of data: consists of all the coordinates of \overrightarrow{a}_{▽} and of the shoulder of the arm in some choice of coordinates z_{1} and z_{2}. The coordinate z_{3} is the same as in the first set.

Hence, by assumption, we have a ‘half-compatibility’ connecting the two choices of coordinates in the two sets of data: the third coordinate z_{3} in the basic data (I) and for the pair (\overrightarrow{a}_{▽} , shoulder) in (II) are matched. But, a priori, we do not know how to identify the other (i.e. the first two) coordinates in the two sets of data.

The aim of arm postprocessing is (using (I), (II) and G^{o}) to compute all the coordinates of the vertices of the arm in a unified choice of coordinates for the sets of data.

5.3.2. Unifying the first two coordinates. Let z_{1}, z_{2}, z_{3} be the coordinates in which we describe the second set of data (II). Assume that in these coordinates the end points of the shoulder are A = (0, p_{2}, p_{3}) and B = (q_{1}, 0, q_{3}). (Notice that the data (I) recognizes the first two coordinates up to their permutation of A and B. Hence, if p_{2} ≠ q_{1} then these information already unifies the coordinates. But, in general, we have to do more.)

Let △ be the last triangle of the arm (i.e. △ ∩ ▽ = AB). Fix an edge α of Δ in Γ whose edgegroup κ(α) in G^{o} contains, say, t legs. We will determine whether α is on the z_{1}z_{3} or z_{2}z_{3} plane: this will orient all the basic data (I) in accordance with {z_{i}}_{i}.

First we compute (in the coordinates \{z_{i}\}_{i}) the normal vector \overrightarrow{a}_{△}. For this, notice that n_{△,▽} · \overrightarrow{AB} = \overrightarrow{a}_{△} ×\overrightarrow{a}_{▽} . This follows from 8.1.2(a) up to a sign; the sign is a consequence of the right-hand rule for vector products. Since \overrightarrow{AB}, \overrightarrow{a}_{▽} and n_{△,▽} are known, this identifies \overrightarrow{a}_{△} up to a summand of a scalar multiple of \overrightarrow{a}_{▽}. Since ⟨\overrightarrow{a}_{△} , \overrightarrow{e_{3}}⟩ is also known (from (I)) and ⟨\overrightarrow{a}_{▽} , \overrightarrow{e_{3}}⟩ is not 0, these facts determine \overrightarrow{a}_{△} completely.

Now, we determine whether α lies on z_{1}z_{3} or z_{2}z_{3} plane. Recall that from the basic data (I), we know the set of the first two coordinates of α: one of them is 0, the other one is, say, δ(α) > 0. E.g., if α = [(q_{1}, 0, q_{3}), (q′_{1}, 0, q′_{3})], then δ(α) = q_{1} − q′_{1}, and δ(α)/t is a positive integer, known from the basic data (I). Then (8.5) and (2.2), for i ∈ {1, 2}, reads as:

α ⊂ z_{i}z_{3} plane ⟹ a_{3} = gcd (a_{3}, a_{i}) · δ(α)/t.

Since \overrightarrow{a}_{△} is primitive, and the above greatest common divisor is, in fact, a leg-determinant, hence it is greater than 1 by 3.3.11, the right-hand side of the above identity cannot be true for both i = 1, 2 simultaneously. This fact determines which coordinate plane contains α.

5.3.3. The complete determination of the arm. Now, using 5.3.2, we can write all the basic data (I) in the coordinates z_{1}, z_{2}, z_{3} of (II). Notice that the basic data (I) determines completely all the normal vectors and all the face values associated with the non-compact faces adjacent to the arm (cf. 4.4.6). Moreover, from (II) we know the normal vector and the face value of ▽. Hence the affine equations of all the triangles in the arm follow from the systems 4.4.5 (where \overline{F}_{c} is the index set of triangles of the arm).

5.4 The complete inverse algorithm for ▲_{3}. We end this section by clarification of case ▲_{3}. First notice that this family can be identified using the diagram G^{o}: it has a (unique) vertex v with three adjacent vertices and without any legs. In this subsection we assume that G^{o} has this property.

The vertex v corresponds to the central triangle △. The other vertices can be grouped in three, each group consisting of a string of adjacent vertices corresponding to the three arms of the diagram. The hands correspond to vertices with exactly one adjacent vertex.

We mark the three vertices corresponding to the hands (or directions of the arms) with the three coordinates. Here we are free to make any marking (up to a permutation of the coordinates). We fix one. Once this choice is made, let us denote the coordinates of △ by (0, p_{2}, p_{3}), (q_{1}, 0, q_{3}) and (r_{1}, r_{2}, 0). At this stage these entries are unknowns.

Now, we preprocess the arms. E.g., for the arm in the direction of z_{3} we obtain the basic data of that arm up to a permutation of the first two coordinates. In particular, we obtain

(a) the first two coordinates of the shoulder [(0, p_{2}, p_{3}), (q_{1}, 0, q_{3})] up to a permutation, hence the set S = {q_{1}, p_{2}};

(b) ⟨\overrightarrow{a}_{△} , \overrightarrow{e_{3}}⟩;

(c) the face values of the legs of this arm.

Summing up for all three arms, we get

(A) the pairs of coordinates {r_{2}, q_{3}}, {q_{1}, p_{2}}, {p_{3}, r_{1}}; (B) \overrightarrow{a}_{△}; (C) the face values of all the legs and, hence, the face value m_{△} of △, too, from the system (4.9)

via 4.4.5.

5.4.1 Lemma. Let (0, p_{2}, p_{3}), (q_{1}, 0, q_{3}) and (r_{1}, r_{2}, 0) be the vertices of an empty triangle △ (with p_{2}, p_{3}, q_{1}, q_{3}, r_{1}, r_{2} > 0). Then these coordinates are uniquely determined by:

(i) the sets {r_{2}, q_{3}}, {q_{1}, p_{2}}, {p_{3}, r_{1}}; (ii) the normal vector (a_{1}, a_{2}, a_{3}) = \overrightarrow{a}_{△} of △; (iii) the face value m_{△} of △.

Coming back to our original situation, 5.4.1 determines the central triangle △. Then we postprocess the arms to calculate all the missing data about Γ.

Proof of 5.4.1.. We have a ℤ_{2}-ambiguity for each set of (i) and we wish to select the correct choice from the 2^{3} possibilities. For this, first assume that we are able to decide which element of the set {r_{2}, q_{3}} is r_{2} and which one is q_{3}. Then we claim that the other two ambiguities disappear. Indeed, the face value identities written for the vertices of △

a_{1}q_{1} + a_{3}q_{3} = a_{2}p_{2} + a_{3}p_{3} = a_{1}r_{1} + a_{2}r_{2} = m_{△} ,  (5.1)

(where a_{k} > 0 for all k) and (i) provide △. Hence, we have at most two choices: either the correct one for every arm or the wrong one for every arm (i.e. when we interchange r_{2} with q_{3} and q_{1} with p_{2} and p_{3} with r_{1}). We claim that the wrong choice can be ruled out. Indeed, assume that both choices of system of integers satisfy the formula (8.2) for \overrightarrow{a}_{△} and (5.1). Notice that without loss of generality, we may assume that r_{1} is the smallest among r_{1}, p_{2} and q_{3}. Write (a part of) (8.2) and (5.1) for both choices:

p_{2}q_{3} + r_{2}p_{3} − r_{2}q_{3} = q_{1}r_{2} + q_{3}r_{1} − q_{3}r_{2} = a_{1},  (5.2)
q_{3}r_{1} + p_{3}q_{1} − p_{3}r_{1} = r_{2}p_{3} + r_{1}p_{2} − r_{1}p_{3} = a_{2},  (5.3)
a_{2}p_{2} + a_{3}p_{3} = a_{1}r_{1} + a_{2}r_{2} = m_{△} = a_{2}q_{1} + a_{3}r_{1} = a_{1}p_{3} + a_{2}q_{3}.  (5.4)

Then (5.2) implies that 0 ⩽ (p_{2} − r_{1})/r_{2} = (q_{1} − p_{3})/q_{3}, hence q_{1} ⩾ p_{3} too. Thus:

a_{2} = q_{3}r_{1} + p_{3}q_{1} − p_{3}r_{1} ⩾ r_{1}^{2} + p_{3}^{2} − r_{1}p_{3} = (r_{1} − p_{3})^{2} + r_{1}p_{3} > |r_{1} − p_{3}|.

On the other hand, from (5.4) expressing a_{1} and a_{3} yields: a_{1} = a_{2}(q_{3} − r_{2})/(r_{1} − p_{3}) and a_{3} = a_{2}(p_{2} − q_{1})/(r_{1} − p_{3}). But this contradicts to the fact that \overrightarrow{a}_{△} is primitive:

gcd (a_{1}, a_{2}, a_{3}) = \frac{a_{2}}{|r_{1} − p_{3}|} gcd (q_{3} − r_{2}, r_{1} − p_{3}, p_{2} − q_{1}) > 1.

6. The inverse algorithm for the families ■_{∗}

6.1 The start 6.1.1. By 4.3.2, the family ■ can be identified from G^{o}: it has a (unique) vertex v with four different types of edges. In this section we assume that G^{o} has this property.

The vertex v corresponds to a non-removable trapezoid △. This vertex always has at least one leg group (corresponding to the bottom edge).

6.1.2 Lemma. The diagram has at least one non-degenerate arm.

Proof. Assume that the top edge is the shoulder of a degenerate arm. Then, write the coordinates of the vertices △ as in 2.3.2. Then (up to a permutation of the first two coordinates) q = 1, and by 3.3.6, r_{2} > 0 and n > 1. For r_{2} = 1, the trapezoid △ can be enlarged and is removable. Hence Γ has a non-degenerate arm in the direction of z_{1}.

The hands can also be identified in G^{o}: they are those vertices (different from v) which have one adjacent vertex. The next algorithm splits according to their number.

Notice also, that by 8.1.1, and with the notation of 2.3.2, the normal vector of △ is

\overrightarrow{a}_{△} = (nq, np, r_{1}q + r_{2}p + (t − 1)pq) .  (6.1)

6.2 The case of three non-degenerate arms: ■_{3}. This case has many similarities with 5.4; but, in fact, it is simpler since the legs of v help in the procedure. Let the decoration of the unique leg group of v be d.

We start by preprocessing the three arms. This provides the coordinates of \overrightarrow{a}_{△} (up to a permutation). Since we already identified the vertex of the central face, we know when we arrive to the shoulder. Nevertheless, at this step, we see a difference between the side-arms and the top-arm. Consider e.g. a side arm and the ‘hidden’ triangle (as part of △) formed by the shoulder and the base edge. With this triangle the arm continuation procedure 5.2.6 is not obstructed, in other words, (by 5.2.7) d divides the corresponding coordinate of \overrightarrow{a}_{△}. For the top-arm this is not the case.

Therefore, d divides exactly two coordinates of \overrightarrow{a}_{△}. We attach the coordinate z_{3} to the arm for which this divisibility does not hold (in this way its shoulder will be the top edge and the bottom edge will sit on the z_{1}z_{2} plane). The coordinates z_{1} and z_{2} (chosen arbitrarily) will be attached to the other two strings of vertices.

Since the face values of the legs of v are 0, and all the other face values associated with legs have been determined during arm preprocessing, (4.9) and 4.4.5 provide the face value of △. In particular, we get the equation of the affine plane supporting △. Since the top edge is primitive (and parallel to the z_{1}z_{2} plane), this is enough for its identification. In particular, with the notations of 2.3.2, we get n, p and q (in fact, n = d by (6.1)). During preprocessing of the arm in the direction z_{1}, we have obtained the set {n, r_{2}}, but n is already identified, hence we obtain r_{2}, too. Similarly, we get r_{1}. Thus we know all the vertices of △, hence the algorithm finishes by postprocessing the arms.

6.3 The case of two non-degenerate arms: ■_{2}. 6.3.1. We have two different leg groups (one of them attached to the bottom), and two non-degenerate arms. First we have to determine whether the shoulders of the non-degenerate arms are the side edges, or one of them is the top edge. This can be decided by the following divisibility property. Its proof uses (6.1) and the fact that \overrightarrow{a}_{△} is primitive (the details are left to the reader):

6.3.2 Lemma. Consider a trapezoid △ with coordinates as in 2.3.2. Assume that it has two nondegenerate arms. Then (up to permutation of the first two coordinates) there are two possibilities:

Case 1. q = 1, n > 1, r_{1} > 1 and r_{2} > 1, the direction of the non-degenerate arms are the z_{1} and z_{2}

axes, and \overrightarrow{a}_{△} = (a_{1}, a_{2}, a_{3}) = (n, np, ∗), hence a_{1} | a_{2}. Case 2. r_{1} = 0 or r_{1} = 1, and n > 1, p > 1, q > 1, the direction of the non-degenerate arms are

the z_{1} and the z_{3} axes, but the coordinates (a_{1}, a_{2}, a_{3}) of \overrightarrow{a}_{△} do not satisfy any divisibility

relation: a_{1} ∤ a_{3}, a_{3} ∤ a_{1}.

Thus the algorithm starts by preprocessing the two non-degenerate arms to get two coordinates of \overrightarrow{a}_{△}. If one of them divides the other then we are in Case 1 above, otherwise we are in Case 2. Next we treat each case independently.

6.3.3. Case 1. Side edges as shoulders of non-degenerate arms. Preprocessing the arms has provided (say) the first two coordinates of the normal vector \overrightarrow{a}_{△}. We name the coordinate axes so that the smallest of the first two coordinates of \overrightarrow{a}_{△} is the first coordinate. Then we compute n as the first coordinate and p as the fraction of the first two coordinates. Since q = 1, at this point, we know all coordinates of the top edge. Then we end this case by the same argument as in 6.2. Preprocessing of the arm in the direction z_{1} has provided the set {n, r_{2}}. Since n is already identified, we obtain r_{2}. Similarly, we get r_{1}, too. Finally, t + 1 is the number of legs of △. Knowing all the vertices of △, we finish by postprocessing the arms.

6.3.4. Case 2. Top edge as shoulder of a non-degenerate arm. Here, one may proceed in the spirit of the other cases 6.2 and 6.3.3, but one may use the following observation as well. We may think about this situation as the degeneration of ▲_{3} (cf. 5.4). Indeed, consider the trapezoid △ as in 2.3.2 and cut it into two triangles along [(r_{1} + tp, r_{2}, 0) , (0, q, n)]. Let △_{1} denote the lower triangle (whose vertices are (0, q, n), (r_{1} + tp, r_{2}, 0) and (r_{1}, r_{2} + tq, 0)), and let △_{2} denote the upper triangle. We may consider △_{1} as a ‘virtual’ hand with two different leg groups, and △_{2} as a central triangle with two ‘genuine’ and one ‘virtual’ arms. The degeneration consists of the fact that △_{1} and △_{2} are in the same plane. Nevertheless, we can apply the same argument. The basic data of the ‘virtual hand’, similarly as in 5.2.5, together with the basic data of the ‘genuine’ arms provide all the data necessary to apply 5.4.1 for the empty triangle △_{2}. Therefore, we obtain △_{2} (up to a permutation of the coordinates). Postprocessing the two arms and completing △_{2} to a trapezoid (in its supporting plane) ends the procedure.

6.4 The case of one non-degenerate arm: ■_{1}. The central vertex is attached to one non-degenerate arm and three leg groups. We denote the set of decorations of these legs by D, which contains three different elements (cf. 4.3.2). Preprocessing of the arm provides a coordinate of \overrightarrow{a}_{△}, denoted by A, and two coordinates of the shoulder (see e.g. 5.4(a)) forming the set S. The above discussions (and/or Appendix 8.1) determine these three objects A, D, S in terms of the integers used in 2.3.2 for the trapezoid △. Basically, (up to a permutation of the first two coordinates) there are two possibilities (depending on whether the shoulder of the arm is a side or top edge). (For the coordinates (a_{1}, a_{2}, a_{3}) of \overrightarrow{a}_{△} see (6.1), which shows that gcd (a_{3}, n) = 1. Moreover, if r_{1} = 0 then p | a_{3}, and if r_{2} = 0 then q | a_{3}.)

I. q = 1, r_{2} > 1, p > 1, n > 1, and r_{1} = 0 or r_{1} = 1, and the (minimal) diagram has an arm in the direction of the z_{1} axis. Then, A = a_{1} = n, S = {n, r_{2}}, and D = {n, a_{3}, n^{r_{1}}p}. If r_{1} = 0 then gcd (n, p) = 1.

II. p > 1, q > 1, but r_{1} and r_{2} are 0 or 1, and the arm is in the direction of the z_{3} axis. Then A = a_{3}, S = {p, q}, and D = {n, n^{r_{1}}p, n^{r_{2}}q}, where gcd (p, q) = 1.

The first case satisfies A ∈ S ∩ D. If A ∈ S ∩ D happens in case II then r_{1} = 0 and r_{2} = t = 1 (or vice versa), and △ is a parallelogram with two sides on coordinate planes. Hence, by a permutation of the z_{1} and z_{3} axes one arrives to the situation I.

Analyzing the above data, one derives the next algorithm to recover △ from G^{o}. It has two cases.

1. If A ∈ S ∩ D then set n = A and r_{2} is the other element of S. The arm is in the direction of z_{1}. Two subcases may occur:

(a) If there exists D ∈ D \ {n} with n | D, then set r_{1} = 1 and p = D/n;

(b) If D \ {n} = {D_{1}, D_{2}} with D_{1} | D_{2}, then set r_{1} = 0 and p = D_{1}.

2. If A ∉ S ∩ D, then we obtain n as the smallest element of the set D \ {all divisors of A}. The set S has two (relative prime) elements; we declare them (arbitrarily) p and q (hence we will get △ up to a permutation of the first two coordinates). r_{1} and r_{2} are determined by the fact that D = {n, n^{r_{1}}p, n^{r_{2}}q}. The arm is in the direction of the z_{3} axis.

In this way, we recover △ and the position of the arm in both cases, hence the algorithm ends by postprocessing the arm.

7. The inverse algorithm for the remaining cases ▲_{1}, ▲_{2}, l_{1}, l_{2}.

7.1 Find a hand! 7.1.1. Next we assume that G^{o} has at least two vertices, each vertex has at most two adjacent vertices, and for each vertex the number of adjacent vertices and leg groups together is three. An end vertex has one adjacent vertex. The diagram G^{0} has two end vertices. The minimal subgraph generated by vertices and edges connecting them is a string.

Clearly, the Newton diagram has at least one hand. Compared with the previous cases, now it is much harder to recognize the vertices of G^{o} corresponding to hands (and/or centers). A hand always corresponds to an end vertex, but end vertices may also correspond to central triangles (e.g. the case of moving triangle), or to the last triangle (adjacent to the shoulder) of an arm (e.g. the diagram of z_{3}^{a}z_{2} + z_{1}^{b} + z_{1}^{c}z_{2}^{d} + z_{2}^{e}).

For any end vertex v, consider its two leg groups. Let t_{i} and n_{i} denote the number of legs and the decoration of the leg groups for i = 1, 2. Set r(v) := n_{1}t_{2} + n_{2}t_{1}.

By 4.3.1(II2), if the two decorations of an end vertex v are not relative prime then v is a hand. We call such an end vertex an easily recognizable hand (ER-hand for short).

7.1.2 Lemma. Assume that neither of the end vertices v_{1} and v_{2} of G^{o} is an ER-hand. We mark one of them as follows. If r(v_{1}) < r(v_{2}) then v_{1} is marked. If r(v_{1}) = r(v_{2}) then the one with greater orbifold Euler number is marked. If even their orbifold Euler numbers are equal, then G^{o} has only two vertices (namely, v_{1} and v_{2}) and it has an isomorphism permuting these two vertices. Then we mark arbitrarily one of the vertices.

All in all, the marked vertex is always a hand (in the last case up to this isomorphism).

7.1.3 Example. The symmetric case occurs if the Newton diagram has only four vertices (0, 0, 2), (p, 0, 1), (0, q, 1) and (r_{1}, r_{2}, 0) (satisfying r_{1}q + r_{2}p > 2pq). Then G^{o} has two vertices, each having two legs decorated by p and q. One can check that even the resolution graph is symmetric. This is surprising since the Newton diagram is not symmetric at all: either face is a hand, the other is a moving (central) triangle. Nevertheless, the algorithm recovers the asymmetric Newton diagram from a symmetric orbifold diagram! (Up to permutation of coordinates, this is the only possibility for the symmetric case, see the proof below.)

Proof of 7.1.2.. Fix a non-degenerate arm in the direction of the z_{3} axis with hand v_{1}. Then the sum of the first two coordinates of the crossing edges of this arm strictly increases from the hand to the shoulder. For the first segment (closest to v_{1}) it is r(v_{1}), cf. 5.2.4. Assume that v_{2} corresponds to the triangle △_{P QR}, with P = (0, p_{2}, p_{3}), Q = (q_{1}, 0, q_{3}) and R = (r_{1}, r_{2}, 0). We may assume that p_{3} > 0 and q_{3} > 0 (otherwise v_{2} is a hand and we have nothing to prove). Thus, it is enough to show q_{1} ⩽ det(P Q) (and its analogue). Since by (8.8) this determinant is a_{2} (of △_{P QR}), and a_{2} = q_{3}r_{1} + p_{3}(q_{1} − r_{1}) by (8.2), we need q_{1} ⩽ q_{3}r_{1} + p_{3}(q_{1} − r_{1}). By (2.1) at least one of p_{3} and r_{1} is 1, hence the inequality follows. Moreover, r(v_{1}) = r(v_{2}) if and only if v_{1} and v_{2} are the only vertices of G^{o}, and p_{3} = q_{3} = 1; hence the Newton diagram is given by (0, 0, c), (0, p_{2}, 1), (q_{1}, 0, 1), (r_{1}, r_{2}, 0). For this, using 4.4, we get e_{v_{1}} ⩾ e_{v_{2}}, with equality if and only if c = 2. For c = 2 the graph is symmetric.

7.1.4. Start of the algorithm. We fix an end vertex v_{1} which correspond to a hand. We denote the other end vertex by v_{2} (we may not know yet whether it is a hand). The algorithm starts with preprocessing the arm with hand v_{1}. Depending on the outcome, we continue by 7.1.5, §7.2 or §7.3.

7.1.5. The arm contains all vertices. We assume that the arm of v_{1} contains all the vertices of G^{o}. We fix the coordinates in such a way that the arm is in the direction of z_{3}. Then the shoulder has the form [(r_{1}, 0, 0), (0, p_{2}, p_{3})] with p_{3} = 0 or p_{3} = 1, and r_{1} > 1. Let \overrightarrow{a} be the normal vector (of the non-compact face) beyond the shoulder. With this choice, preprocessing the arm has provided the set \{r_{1}, p_{2}\} and the third coordinate a_{3} of \overrightarrow{a}. Notice that if p_{3} = 0 then \overrightarrow{a} = \overrightarrow{e_{3}}, otherwise \overrightarrow{a} = (1, 0, r_{1}). Hence, if a_{3} = 1 then p_{3} = 0, but if a_{3} > 1 then p_{3} = 1 and r_{1} = a_{3}. In the p_{3} = 1 case we get the integers r_{1} and p_{2}, but in the case p_{3} = 0, the integers r_{1} and p_{2} behave symmetrically, so we distinguish them arbitrarily. The algorithm finishes by postprocessing the arm.

Notice that this algorithm covers not only the family l_{1}, but also some part of l_{2}. The remaining classes of l_{2} will be discussed in 7.3.2 (in accordance with this paragraph).

7.2 The case ▲_{1}. 7.2.1. We assume that the arm of v_{1} contains all vertices but one, which is not an ER-hand. Assume that the arm is in the direction z_{3}, and let [(q_{1}, 0, q_{3}), (0, p_{2}, p_{3})] be its shoulder with p_{2} ⩾ 2, q_{1} ⩾ 2. Since v_{2} is not an ER-vertex, p_{3} > 0 and q_{3} > 0 (cf. 5.2.4). If the third vertex of the face associated with v_{2} is (r_{1}, r_{2}, 0), then r_{i} > 0 (i = 1, 2) since otherwise v_{2} would be in the arm of v_{1}. Therefore, v_{2} corresponds to a central triangle with only crossing edges. Moreover, (2.1) guarantees that 1 ∈ {r_{1}, p_{3}} ∩ {r_{2}, q_{3}}. Let (a_{1}, a_{2}, a_{3}) be the normal vector of the face of v_{2}.

Let us collect some facts about such a Newton diagram in order to be able to find the right algorithm. Since r_{1} = r_{2} = 1 is not possible (see 5.1.4(N=0)), we may assume that p_{3} = 1. (This introduces a choice of the coordinates z_{1} and z_{2}, and at this moment it is not clear how this choice fits with any property of G^{o}; this will be explained later.)

We distinguish two cases. The first case is q_{3} = 1, then v_{2} is a moving triangle, hence a_{1} = p_{2} and a_{2} = q_{1}. The second case is q_{3} > 1, which we analyze in the rest of this paragraph. Since 1 ∈ {r_{2}, q_{3}}, we get r_{2} = 1. By (8.2) one has

(a_{1}, a_{2}, a_{3}) = (p_{2}q_{3} − q_{3} + 1, q_{3}r_{1} + q_{1} − r_{1}, r_{1}p_{2} + q_{1} − q_{1}p_{2}).  (7.1)

From this and q_{3} ⩾ 2 one gets a_{1} ⩾ 2p_{2} − 1 > p_{2} − 1 and a_{2} ⩾ r_{1} + q_{1} > r_{1}, q_{1}. In particular, a_{1} + a_{2} > p_{2} + q_{1} and hence {a_{1}, a_{2}} ≠ {p_{1}, q_{2}}. The face value computed via the two vertices (r_{1}, 1, 0) and (0, p_{2}, 1) gives r_{1}a_{1} + a_{2} = p_{2}a_{2} + a_{3}. Therefore, the integers r_{1}, p_{2}, q_{1}, a_{1}, a_{2} satisfy:

\begin{matrix}a_{3} = r_{1}a_{1} − (p_{2} − 1)a_{2} \\ 0 < r_{1} < a_{2}, \\ 0 < p_{2} − 1 < a_{1}. \end{matrix}  (7.2)

7.2.2. The algorithm. The two decorations of the legs of v_{2} are D = {a_{1}, a_{2}} (cf. (8.7)), where gcd (a_{1}, a_{2}) = 1 by 4.3.1(II). Preprocessing the arm (with hand v_{1}) has produced a_{3} and the set S = {p_{2}, q_{1}} (we cannot distinguish the two coordinates yet). We shall compute the coordinates of v_{2} below, and then postprocess the arm to determine the rest of the Newton diagram.

We distinguish two cases for computing v_{2}. First case: D = S. Let the two elements of this set be a_{1} = p_{2} and a_{2} = q_{1} (here is a choice between the z_{1} and z_{2} coordinates). We select the d-minimal (as explained in 3.3.8) solution (r_{1}, r_{2}) of positive integers of the equation r_{1}p_{2} + r_{2}q_{1} − q_{1}p_{2} = a_{3}. (The only reason for selecting the d-minimal solution is to obtain the d-minimal representative.) Then the vertices of v_{2} are (q_{1}, 0, 1), (0, p_{2}, 1), (r_{1}, r_{2}, 0).

Second case: D ≠ S. We choose the unique 6-tuple (r_{1}, p_{2}, q_{1}, q_{3}, a_{1}, a_{2}) of positive integers with S = {a_{1}, a_{2}}, D = {p_{2}, q_{1}} satisfying both (7.1) and (7.2). (Uniqueness will be proved in the next paragraph.) Then the vertices of v_{2} (up to a permutation of the first two coordinates) are (q_{1}, 0, q_{3}), (0, p_{2}, 1) and (r_{1}, 1, 0).

7.2.3. Uniqueness of the 6-tuple. Notice that once the choice between a_{1} and a_{2} is made, then (7.2) determines uniquely r_{1} and p_{2}. Then one gets q_{1} form D and also q_{3} = (a_{1} − 1)/(p_{2} − 1).

Assume for contradiction that by interchanging a_{1} and a_{2} we get another set of solutions r̃_{1}, p̃_{2} and so on. Then, by (7.2), r̃_{1} = a_{1} − p_{2} + 1 and p̃_{2} = a_{2} − r_{1}. Since p̃_{2} ∈ {p_{2}, q_{1}}, there are two cases.

If p̃_{2} = q_{1} then substituting this in the expression of p̃_{2} and using (7.1) for a_{2} produces (q_{3}−2)r_{1} = −1, whose left hand side is non-negative, a contradiction.

On the other hand, if p̃_{2} = p_{2} then from the expression of a_{3} in (7.1) (used for both sets of solutions) we obtain r̃_{1} = r_{1}. Thus, again from (7.1), we obtain a_{1} = a_{2} contradicting gcd (a_{1}, a_{2}) = 1.

7.3 Two non-degenerate arms. 7.3.1. Assume that there are either at least two vertices which are not in the arm of v_{1}, or there is only one such vertex, namely, v_{2}. In the latter case, we also assume that v_{2} is an ER-hand since the other case is treated in §7.2. Anyway, v_{2} is also a hand, so we preprocess its arm, too. We face two cases: either the two arms (of v_{1} and v_{2}) cover all the vertices of G^{o} (this fact characterizes the family l_{2}), or the arms contain all the vertices but one, which should be a central vertex/face (this is the family ▲_{2}).

7.3.2. The case l_{2}. If the arm of v_{2} contains all the vertices then we are in the situation of 7.1.5, and we are done. Assume that this is not the case. Fix the coordinates z_{i} so that the arm of v_{i} is in the direction of z_{i} (i = 1, 2). We select (arbitrarily) a common edge α = [(p, q, 0), (0, 0, c)] of the two arms, and let △_{i} be the face adjacent to it in the direction z_{i}. In particular, △_{i} lies in the arm of v_{i}. Let \overrightarrow{a}^{(i)} be the normal vector of △_{i}. We seek the coordinates of these vectors and the edge α.

By preprocessing the arms, we have obtained the sets {c, p} and {c, q}, and the first two coordinates of both \overrightarrow{a}^{(i)}. By (8.3) one has \overrightarrow{a}^{(1)} × \overrightarrow{a}^{(2)} = (−p, −q, c), hence c = a_{1}^{(1)}a_{2}^{(2)} − a_{2}^{(1)}a_{1}^{(2)}. Hence we recover α. Moreover, by face value computation, a_{3}^{(i)}c = pa_{1}^{(i)} + qa_{2}^{(i)}, hence we get the normal vectors as well. The algorithm finishes with postprocessing the arms.

7.3.3. The case ▲_{2}. Similarly as above, fix the coordinates z_{i} so that the arm of v_{i} is in the direction of z_{i} (i = 1, 2). We wish to determine the central triangle △ using 5.4.1, whose notations we will use. Preprocessing the two non-degenerate arms, we have determined the sets {r_{2}, q_{3}} and {p_{3}, r_{1}}, and the first two coordinates of \overrightarrow{a}_{△}. The third coordinate of \overrightarrow{a}_{△} is the decoration of the leg adjacent to the vertex corresponding to △, hence \overrightarrow{a}_{△} is known from G^{o}. For 1 ⩽ i ⩽ 3 denote by ▽_{i} the face of Γ_{+} adjacent to △ in the direction of the axis z_{i}. Then, by the notations of 4.4.7, we already know the coordinates a_{1}^{(1)} and a_{2}^{(2)} from preprocessing the arms. Furthermore, ▽_{3} is a non-compact face with a_{3}^{(3)} = 0 by 8.1.3. Therefore, 4.11 gives the face value m_{△}. This, via the equations (4.7) and 4.4.5 provide all the face values, in particular the face value of ▽_{3} too. This is p_{2}q_{1}. Since either p_{2} or q_{1} is 1, we get the set {p_{2}, q_{1}} as well. Hence, 5.4.1 determines △ (up to a permutation of coordinates). Then postprocessing the arms recovers the Newton diagram.

8. Appendix

8.1 Some arithmetical properties of Newton boundaries. 8.1.1 Lemma. Let △ be a triangle whose vertices are lattice points. Let \overrightarrow{a} and \overrightarrow{b} be the vectors of two of its sides. Then

\overrightarrow{a} × \overrightarrow{b} = ± g(△) \overrightarrow{a}_{△} .  (8.1)

In particular, if △ is an empty triangle with vertices (0, p_{2}, p_{3}), (q_{1}, 0, q_{3}) and (r_{1}, r_{2}, 0), then

\overrightarrow{a}_{△} = (p_{2}q_{3} + r_{2}p_{3} − r_{2}q_{3}, q_{3}r_{1} + p_{3}q_{1} − p_{3}r_{1}, r_{1}p_{2} + q_{1}r_{2} − q_{1}p_{2}) .  (8.2)

Proof. By the additivity of g(△), we may assume that △ is empty. In that case \overrightarrow{a} and \overrightarrow{b} can be completed to a base (see e.g. [Oda88, p. 35]), hence \overrightarrow{a} × \overrightarrow{b} is primitive. The second part is a direct application. To verify the sign, note that the scalar product of both vectors in (8.2) with the vertices of the triangle are positive.

8.1.2 Lemma. Let △ and ▽ be two adjacent lattice polygons.

(a) Then the vector \overrightarrow{v} of their common edge is, up to a sign:

\overrightarrow{v} = ±\frac{t_{△,▽}}{n_{△,▽}} \overrightarrow{a}_{△} × \overrightarrow{a}_{▽} .  (8.3)

(b) Assume that △ and ▽ are adjacent faces of a Newton polytope, △ is a triangle, and let \overrightarrow{a} be a vector from a point from their common edge to the third vertex of △. Then

n_{△,▽} = \frac{t_{△,▽}}{g(△)} ⟨\overrightarrow{a} , \overrightarrow{a}_{▽}⟩ .  (8.4)

(c) Let △ be a triangle with vertices (0, p_{2}, p_{3}), (q_{1}, 0, q_{3}) and (q′_{1}, 0, q′_{3}) with q′_{1} < q_{1}, situated on a compact face of a Newton boundary. Assume that △ has no lattice points other than its vertices and possible internal lattice points on its side on the z_{1}z_{3} plane. Then the following expressions are equal and integers:

\frac{q_{1} − q′_{1}}{t_{△,\overrightarrow{e_{2}}}} = \frac{⟨\overrightarrow{a}_{△} , \overrightarrow{e_{3}}⟩}{n_{△,\overrightarrow{e_{2}}}} ∈ ℕ.  (8.5)

In fact,

n_{△,\overrightarrow{e_{2}}} = p_{2}.  (8.6)

Proof. (a) The vector ± \overrightarrow{v} is characterized by the fact that it is orthogonal to both normal vectors and it is t_{△,▽} times a primitive vector. The vector on the right-hand side of (8.3) has this property. For (b), since \overrightarrow{v} is orthogonal to \overrightarrow{a}_{▽}, Equations (8.3) and (8.1) give

± g(△) \frac{n_{△,▽}}{t_{△,▽}} \overrightarrow{v} = ± g(△) (\overrightarrow{a}_{△} × \overrightarrow{a}_{▽}) = (\overrightarrow{a} × \overrightarrow{v}) × \overrightarrow{a}_{▽} = ⟨\overrightarrow{a} , \overrightarrow{a}_{▽}⟩ \overrightarrow{v} .

This gives (8.4) up to a sign. Since scalar product of the normal vector of a face assigns its minimum on the face (when restricted to the Newton boundary), the scalar product in (8.4) is positive, and hence both sides of (8.4) are positive.

For (8.5), we apply (a) with \overrightarrow{a}_{▽} = \overrightarrow{e_{2}} and \overrightarrow{v} = (q_{1} − q′_{1}, 0, q_{3} − q′_{3}). First notice that \overrightarrow{v} is t_{△,\overrightarrow{e_{2}}} times a primitive vector, hence (q_{1} − q′_{1})/ t_{△,\overrightarrow{e_{2}}} = ⟨\overrightarrow{v} / t_{△,\overrightarrow{e_{2}}} , \overrightarrow{e_{1}}⟩ ∈ ℕ.

On the other hand, taking scalar product of (8.3) with \overrightarrow{e_{1}}, we obtain (8.5) up to a sign. Since both expression are positive in (8.5), the sign is correct. The last equality is a special case of (b) with ▽ the z_{1}z_{3} plane, because g(△) = t_{△,\overrightarrow{e_{2}}}.

Recall (cf. 2.1.5) that a non-compact face of Γ (with (2.1)) either lies on a coordinate plane, or it has an edge of type [(a, 0, c) , (0, 1, b)] and normal vector (1, a, 0) with a > 0.

8.1.3 Lemma. Let an edge AB = [(a, 0, c) , (0, 1, b)] lie on a compact face and on a non-compact one with normal vectors \overrightarrow{a} and \overrightarrow{n} := (1, a, 0), respectively (a > 0). Then

n_{\overrightarrow{n},\overrightarrow{a}} = ⟨\overrightarrow{a} , \overrightarrow{e_{3}}⟩ .  (8.7)

Assume that C = (r, s, u) is a third vertex of the compact face, such that the triangle △_{ABC} is empty. Then the determinant is also

n_{\overrightarrow{n},\overrightarrow{a}} = ⟨(r − a, s, u − c) , \overrightarrow{n}⟩ = r + (s − 1)a.  (8.8)

Proof. Let △ be the empty triangle on the non-compact face with vertices: (a, 0, c), (0, 1, b) and (0, 1, b + 1) and ▽ denote the compact face. Then (8.4) with \overrightarrow{a} = \overrightarrow{e_{3}} yields (8.7). The other equation is again an application of (8.4). However, this time ▽ is the non-compact face, and △ is the triangle with vertices (a, 0, c), (0, 1, b) and (r, s, u).

8.1.4 Lemma. Let △ and ▽ be two adjacent triangular faces of a Newton diagram whose vertices lie on the coordinate planes containing the z_{3} axis. Further, let us assume that △ has an edge on the z_{1}z_{3} plane, which contains all the lattice points of the triangle except the third vertex. Let its determinant n_{△,\overrightarrow{e_{2}}} be denoted by n_{△}. Similarly, we suppose that ▽ has an edge α either on the z_{1}z_{3} plane or on the z_{2}z_{3} plane containing all lattice points except the third vertex. Its determinant will be denoted by n_{▽}. Then

α ∈ z_{1}z_{3} plane  ⟺  n_{△} = n_{▽} | n_{△,▽} ,  (8.9)
α ∈ z_{2}z_{3} plane  ⟺  gcd (n_{△} , n_{▽} , n_{△,▽}) = 1.  (8.10)

Proof. Let \overrightarrow{v} be the vector of the common edge of the triangles. Let \overrightarrow{a} be the primitive vector parallel to the edge of △ lying on the z_{1}z_{3} plane. Finally, let \overrightarrow{c} be the primitive vector parallel to α. Now, (8.3) combined with (8.1) implies that n_{△,▽} equals the triple product \overrightarrow{a} \overrightarrow{v} \overrightarrow{c} (up to a sign).

If α lies on the z_{1}z_{3} plane, n_{△} = n_{▽} = ⟨\overrightarrow{v} , \overrightarrow{e_{2}}⟩ by (8.6). Since the second coordinates of \overrightarrow{a} and \overrightarrow{c} are 0, the number ⟨\overrightarrow{v} , \overrightarrow{e_{2}}⟩ divides the triple product. This proves the ⟹ part of (8.9).

If α lies on the z_{2}z_{3} plane, then ⟨\overrightarrow{a} , \overrightarrow{e_{2}}⟩ = ⟨\overrightarrow{c} , \overrightarrow{e_{1}}⟩ = 0. Therefore, n_{△,▽}, modulo the greatest common divisor d of n_{△} = ⟨\overrightarrow{v} , \overrightarrow{e_{2}}⟩ and n_{▽} = ⟨\overrightarrow{v} , \overrightarrow{e_{1}}⟩, is:

n_{△,▽} = \overrightarrow{a} \overrightarrow{v} \overrightarrow{c} ≡ − ⟨\overrightarrow{a} , \overrightarrow{e_{1}}⟩ · ⟨\overrightarrow{v} , \overrightarrow{e_{3}}⟩ · ⟨\overrightarrow{c} , \overrightarrow{e_{2}}⟩ (mod d).  (8.11)

The three terms of the right-hand side are relative prime to d because \overrightarrow{a}_{△} = ± \overrightarrow{a} × \overrightarrow{v}, \overrightarrow{a}_{▽} = ± \overrightarrow{c} × \overrightarrow{v}, and \overrightarrow{v} are primitive. Hence the ⟹ part of (8.10) follows. We end the proof by noticing that the right-hand sides of (8.9) and (8.10) are mutually exclusive.

8.2 The weighted homogeneous case (with one node). Below L = (d_{1}, k_{1}; …; d_{s}, k_{s}) means that the unique vertex of G^{o} has s leg-groups, the ith group has size k_{i} ⩾ 1 and decoration d_{i} > 1 (with d_{i} ≠ d_{j} for i ≠ j, and ∑_{i} k_{i} ⩾ 3). The number e is the orbifold Euler number. One has the following cases:

1. L = (d, k)

Equation: z_{1}^{d} + z_{2}^{k−1}z_{3} + z_{2}z_{3}^{k−1}.

2. L = (d, 2; D, 2)

Equation: z_{1}^{d}z_{3} + z_{2}^{2D} + z_{3}^{2}, equivalently z_{1}^{2d} + z_{2}^{D}z_{3} + z_{3}^{2}.

(The equations are ∼-equivalent.)

3. L = (d, k; D, 1), d | D Equation: z_{1}^{k}z_{2} + z_{1}z_{2}^{(k−1)D/d+1} + z_{3}^{d}.

4. L = (d, k; D, 1), gcd (d, D) = 1, −edD = 1 Equation: z_{1}^{d} + z_{2}^{(k−1)/D}z_{3} + z_{2}z_{3}^{k}.

5. L = (d, k; D, 1), gcd (d, D) = 1, −edD = k Equation: z_{1}^{d}z_{2} + z_{2}^{(D+1)(k−1)/k}z_{3} + z_{3}^{k}.

6. L = (a, 2; b, 2; c, 2)

Equation: z_{1}^{2a} + z_{2}^{2b} + z_{3}^{2c}.

7. L = (a, k; b, 1; c, 1), a | b, a | c Equation: z_{1}^{(bk)/a+1}z_{2} + z_{1}z_{2}^{(ck)/a+1} + z_{3}^{a}.

8. L = (a, k; b, 1; c, 1), b | c and k > 1 Equation: z_{1}^{a}z_{2} + z_{2}^{c/b+1} + z_{3}^{kb}.

9. L = (a, k; b, 1; c, 1), a | b, a ∤ c, −ebc = 1 Equation: z_{1}^{kc}z_{2} + z_{2}^{(bk)/a+1} + z_{3}^{a}.

10. L = (a, k; b, 1; c, 1), a | b, a ∤ c, A := −ebc > 1 Equation: z_{1}^{(kc−1)/A+1}z_{2} + z_{2}^{A} + z_{1}z_{3}^{a}.

11. L = (a, k; b, 1; c, 1), the numbers a, b, c does not divide each other, and −eabc = k^{2} Equation: z_{1}^{a} + z_{2}^{kc} + z_{3}^{kb}.

12. L = (a, k; b, 1; c, 1), the numbers a, b, c does not divide each other, k = 1 and A := −eabc > 1 Equation: z_{1}^{(A−b)/a}z_{2} + z_{2}^{(A−c)/b}z_{3} + z_{3}^{(A−a)/c}z_{1}, or z_{1}^{(A−b)/c}z_{2} + z_{2}^{(A−a)/b}z_{3} + z_{3}^{(A−c)/a}z_{1}. (Only one of the equations have integer exponents, and this one gives the right diagram.)

References

ABLdBMH07 Enrique Artal Bartolo, Ignacio Luengo, Javier Fernández de Bobadilla, and Alejandro Melle-

Hernández, Milnor number of weighted Iomdine singularities, manuscript, 2007.

AGZV88 V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. II, Monographs in Mathematics, vol. 83, Birkhäuser Boston Inc., Boston, MA, 1988. MR 966191 (89g:58024)

BS75 Joël Briançon and Jean-Paul Speder, La trivialité topologique n’implique pas les conditions de

Whitney, C. R. Acad. Sci., Paris Sér. A 280 (1975), no. 6, 365–367. MR 0425165 (54 #13122)

Dan79 V. I. Danilov, Newton polyhedra and vanishing cohomology, Funktsional. Anal. i Prilozhen.

13 (1979), no. 2, 32–47. MR 541636 (80h:14001)

HR89 H. Hauser and R. Randell, Report on the problem session, Singularities (Iowa City, IA, 1986), Contemp. Math., vol. 90, American Mathematical Society, Providence, RI, 1989, pp. 119–134. MR 1000597 (90c:32017)

Kou76 A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976),

no. 1, 1–31. MR 0419433 (54 #7454)

Lau83 Henry B. Laufer, Weak simultaneous resolution for deformations of Gorenstein surface singularities, Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol. 40, American Mathematical Society, Providence, R.I., 1983, pp. 1–29. MR 713236 (84k:32030)

Lau87 , Strong simultaneous resolution for surface singularities, Complex analytic singularities, Adv. Stud. Pure Math., vol. 8, North-Holland, Amsterdam, 1987, pp. 207–214. MR 894294 (88g:32024)

LVMHN05 I. Luengo-Velasco, A. Melle-Hernández, and A. Némethi, Links and analytic invariants of superisolated singularities, J. Algebraic Geom. 14 (2005), no. 3, 543–565. MR 2129010 (2005m:32057)

MN05 Robert Mendris and András Némethi, The link of {f(x, y)+z^{n} = 0} and Zariski’s conjecture,

Compos. Math. 141 (2005), no. 2, 502–524. MR 2134278 (2006c:32039)

MT80 Michel Merle and Bernard Teissier, Conditions d’adjonction, d’après Du Val, Séminaire sur les Singularités des Surfaces (Michel Demazure, Henry Charles Pinkham, and Bernard Teissier, eds.), Lecture Notes in Mathematics, vol. 777, Springer, Berlin, 1980, Held at the Centre de Mathématiques de l’École Polytechnique, Palaiseau, 1976–1977, pp. 229–247 (French). MR 579026 (82d:14021)

Mum61 David Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Études Sci. Publ. Math. (1961), no. 9, 5–22. MR 0153682 (27 #3643)

Ném04 András Némethi, Invariants of normal surface singularities, Real and complex singularities, Contemp. Math., vol. 354, American Mathematical Society, Providence, RI, 2004, pp. 161– 208. MR 2087811 (2005g:32040)

Neu81 Walter D. Neumann, A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981), no. 2, 299–344. MR 632532 (84a:32015)

Neu83 , Abelian covers of quasihomogeneous surface singularities, Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol. 40, American Mathematical Society, Providence, RI, 1983, pp. 233–243. MR 713252 (85g:32018)

NN02 András Némethi and Liviu I. Nicolaescu, Seiberg-Witten invariants and surface singularities, Geom. Topol. 6 (2002), 269–328 (electronic). MR 1914570 (2003i:14048)

NW02 Walter D. Neumann and Jonathan Wahl, Universal abelian covers of surface singularities, Trends in singularities, Trends Math., Birkhäuser, Basel, 2002, pp. 181–190. MR 1900786 (2003c:32028)

Oda88 Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988, An introduction to the theory of toric varieties, Translated from the Japanese. MR 922894 (88m:14038)

Oka79 M. Oka, On the bifurcation of the multiplicity and topology of the Newton boundary, J. Math.

Soc. Japan 31 (1979), no. 3, 435–450. MR 535090 (80h:32018)

Oka87 , On the resolution of the hypersurface singularities, Complex analytic singularities, Adv. Stud. Pure Math., vol. 8, North-Holland, Amsterdam, 1987, pp. 405–436. MR 894303 (88m:32023)

Oka89 , On the weak simultaneous resolution of a negligible truncation of the Newton boundary, Singularities (Iowa City, IA, 1986), Contemp. Math., vol. 90, American Mathematical Society, Providence, RI, 1989, pp. 199–210. MR 1000603 (90h:32030)

Par99 Adam Parusiński, Topological triviality of μ-constant deformations of type f(x) + tg(x), Bull.

London Math. Soc. 31 (1999), no. 6, 686–692. MR 1711027 (2001d:58048)

Pin77 H. Pinkham, Normal surface singularities with C^{∗} action, Math. Ann. 227 (1977), no. 2,

183–193. MR 0432636 (55 #5623)

Sai88 M. Saito, Exponents and Newton polyhedra of isolated hypersurface singularities, Math. Ann.

281 (1988), no. 3, 411–417. MR 954149 (89h:32027)

Ste77 J. H. M. Steenbrink, Mixed Hodge structure on the vanishing cohomology, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 525–563. MR 0485870 (58 #5670)

Ste85 , Semicontinuity of the singularity spectrum, Invent. Math. 79 (1985), no. 3, 557–565.

MR 782235 (86h:32033)

Tro80 David Trotman, Equisingularité et conditions de Whitney, Thèse d’Etat, Université de Paris-

Sud, Orsay, January 1980.

Var76 A. N. Varchenko, Zeta-function of monodromy and Newton’s diagram, Invent. Math. 37

(1976), no. 3, 253–262. MR 0424806 (54 #12764)

Var82 A.N. Varchenko, The complex exponent of a singularity does not change along strata μ=const.,

Funct. Anal. Appl. 16 (1982), 1–9 (English. Russian original).

VK85 A. N. Varchenko and A. G. Khovanskiĭ, Asymptotic behavior of integrals over vanishing cycles and the Newton polyhedron, Dokl. Akad. Nauk SSSR 283 (1985), no. 3, 521–525. MR 800892 (87e:32012)

Wag83 Philip Wagreich, The structure of quasihomogeneous singularities, Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol. 40, American Mathematical Society, Providence, RI, 1983, pp. 593–611. MR 713284 (85e:14051)

Wal99 C. T. C. Wall, Newton polytopes and non-degeneracy, J. Reine Angew. Math. 509 (1999),

1–19. MR 1679164 (2000g:32032)

Zar71 Oscar Zariski, Some open questions in the theory of singularities, Bull. Amer. Math. Soc. 77 (1971), 481–491. MR 0277533 (43 #3266)

Gábor Braun braung@renyi.hu Alfréd Rényi Institute of Mathematics, 1053 Budapest, Reáltanoda u. 13–15, Hungary

András Némethi nemethi@renyi.hu Alfréd Rényi Institute of Mathematics, 1053 Budapest, Reáltanoda u. 13–15, Hungary