From Compact Semi-Toric Systems To Hamiltonian S-1-Spaces

We show how any labeled convex polygon associated to a compact semi-toric system, as de fined by V (u) over tilde Ngoc, determines Karshon's labeled directed graph which classifies the underlying Hamiltonian S-1-space up to isomorphism. Then we characterize adaptable compact semi-toric systems, i.e. those whose underlying Hamiltonian S-1-action can be extended to an effective Hamiltonian T-2-action, as those which have at least one associated convex polygon which satisfies the Delzant condition.


1.
Introduction. This paper studies the relation between a certain family of completely integrable Hamiltonian systems on closed 4-dimensional symplectic manifolds and Hamiltonian S 1 -actions on these spaces. As such, it lies at the intersection of the theory of Hamiltonian torus actions on closed symplectic manifolds and the classification of completely integrable Hamiltonian systems. The former is a special case of Hamiltonian actions in symplectic and Poisson geometry, an area of mathematics which brings together algebraic geometry, Lie theory, Poisson geometry and differential topology amongst others. Of particular prominence for the purposes of this work is Karshon's monograph [12] on Hamiltonian circle actions on closed 4-dimensional symplectic manifolds, whose results have been extended to higher dimensions (cf. Karshon & Tolman [14,15,16]). The classification of completely integrable Hamiltonian systems is a driving question in Hamiltonian mechanics with The idea of the proof is to exploit the similarities between symplectic toric manifolds and compact semi-toric systems. For the former, Karshon [12] shows how to recover the labeled directed graph of the associated Hamiltonian S 1 -space (cf. Remark 3.2). Thus the aim is to mimic Karshon's ideas in the compact semi-toric case. However, compact semi-toric systems allow for focus-focus singular points which do not occur in the symplectic toric category; this difficulty is overcome by using (a) the so-called Eliasson-Miranda-Zung local normal form which gives control over the geometry of the system near such singularities (cf. Section 2.2.1) and (b) the connectedness of the fibers of Φ, an important fact proved by Vũ Ngo . c [26].
1. Following Pelayo & Vũ Ngo . c [21], the polygons shown in Figure 1.1 are two semi-toric polygons associated to the same compact semi-toric system (M, ω, Φ) on CP 2 , with one focus-focus point and the Taylor series invariant associated to the focus-focus critical point is taken to be 0 (cf. Pelayo & Vũ Ngo . c [20,21] for details). The graph below each polygon is the corresponding labeled graph for the underlying Hamiltonian S 1 -space, where the edge is a Z 2 -sphere, which corresponds to the "upper chain" of edges in each polygon (cf. Section 3.2.2), and the isolated vertex is the focus-focus point (cf. Section 3.1).  Semi-toric systems can be naturally divided in two families: those whose underlying Hamiltonian S 1 -action can be extended to an effective Hamiltonian T 2 -action (cf. Definitions 2. 16 and 3.11), and the rest. The former are called adaptable, while the latter are non-adaptable. Once the main result is proved, this article turns to obtaining a characterization of adaptable systems, stated below.  [23]). The other implication of Theorem 4.1 is obtained by giving a characterization of non-adaptable systems both near fibers containing focus-focus points and globally (cf. Proposition 4.10). Moreover, an explicit example of a non-adaptable system is constructed in Example 4.12, which, to the best of our knowledge, is the first of its kind.
Organization of the paper. After the introduction, Section 2 recalls the definition and properties of both Hamiltonian S 1 -spaces and compact semi-toric systems; many results are quoted with references where to find the proofs and more details. Section 3 states and proves the main result of the article. The proof of Theorem 3.1 is broken down into several steps. Section 4 studies adaptable and non-adaptable compact semi-toric systems, which are characterized by Theorem 4.1, Theorem 4.5, and Propositions 4.10 and 4.16.
Conventions. In the whole article, (M, ω) denotes a connected, closed symplectic manifold. Unless otherwise stated, group actions on manifolds are effective, i.e. there are no non-trivial elements of the group which act trivially on the whole space. The identification S 1 = R/2πZ is used throughout.
2. Hamiltonian S 1 -spaces and compact semi-toric systems. Let (M, ω) be a closed 2n-dimensional symplectic manifold. Since ω is non-degenerate, it induces where ω(X α , ·) = −α. Let C ∞ (M ) denote the vector space of smooth functions and let d be the exterior differential. The Hamiltonian vector field associated to F ∈ C ∞ (M ) is defined as X F = ω # (dF ). The Poisson bracket {·, ·} : C ∞ (M ) × C ∞ (M ) → C ∞ (M ) induced by ω is given by The triple (M, ω, Φ) is henceforth referred to as a Hamiltonian R k -space, and Φ is the moment map.
To see that Definition 2.1 yields an R k -action, let X F1 , . . . , X F k be the Hamiltonian vector fields associated to F 1 , . . . , F k , and denote by ϕ 1 t , . . . , ϕ k t the corresponding flows. These exist for all t ∈ R by compactness of M . Moreover, property (2.2) implies that they pairwise commute. Then the R k -action is given by . . . , t k ) · p := ϕ 1 t1 • · · · • ϕ k t k (p). Two families of Hamiltonian R k -spaces play an important role in this paper, namely • completely integrable Hamiltonian systems when k = n in Definition 2.1. • Hamiltonian T k -spaces if the flows of X F1 , . . . , X F k are periodic, and the induced torus action is effective. In both examples, the following additional condition holds: Henceforth (M, ω) is taken to be 4-dimensional, unless otherwise stated.
Example 2.4. Consider CP 2 with homogeneous complex coordinates [z 0 : z 1 : z 2 ] and the (standard) Fubini-Study symplectic form ω F S . The map J : is the moment map of the following effective Hamiltonian S 1 -action where λ ∈ S 1 . Thus the triple (CP 2 , ω F S , J) defines an object in Ham S 1 .
A source of interesting examples of Hamiltonian S 1 -spaces is provided by symplectic toric manifolds, which are defined below.
Remark 2.6. Given a symplectic toric manifold, there are several ways to obtain a Hamiltonian S 1 -space, corresponding to restricting the action to a subgroup S 1 ⊂ T 2 . Throughout this paper, the triple (M, ω, µ 1 ) is henceforth referred to as the Hamiltonian S 1 -space associated to (M, ω, µ = (µ 1 , µ 2 )). It is important to notice that not all Hamiltonian S 1 -spaces arise in this fashion (cf. Example 2.17).
Karshon's classification. The classification of Hamiltonian S 1 -spaces up to isomorphism has been carried out in Karshon [12], and is recalled below without proofs in order to introduce ideas and notation used in the rest of the paper.
Let (M, ω, J) be a Hamiltonian S 1 -space. For each subgroup G ⊂ S 1 , let M G be the set of points in M whose stabilizer is G. The connected components of M S 1 are symplectic submanifolds, hence either points or surfaces (since the action is effective); this follows from the following local normal form (presented below without proof, cf. Karshon [12,Cor. A.7] as a reference). Lemma 2.7 (Chaperon [4]). For each p ∈ M S 1 there exist neighborhoods U ⊂ M of p, U 0 ⊂ C 2 of (0, 0), and a symplectomorphism Ψ : [12], Lemma 2.2). The closure of each connected component of M Z k is a symplectic sphere on which S 1 /Z k acts with two fixed points, which are isolated fixed points in M S 1 .
Such submanifolds are called Z k -spheres, k being the isotropy weight, and the minimum (respectively maximum) of J on a Z k -sphere is called south (respectively north) pole.
The work in Karshon [12] provides an algorithm which associates a labeled directed graph Γ = (V, E) to (the isomorphism class of) (M, ω, J):  13. An important role in the proof of Theorem 2.11 is played by the so-called gradient spheres, whose definition is recalled below. Fix (M, ω, J) and let g be a compatible metric, i.e. an S 1 -invariant Riemannian metric such that the endomorphism J : TM → TM defined by g(u, v) = ω(u, J (v)) is an almost complex structure. Thus the gradient vector field of the moment map J satisfies grad(J) = −J (X J ).
By invariance of the metric, the flow generated by J (X J ) commutes with the circle action, thus obtaining an R × S 1 C × -action. The closure of each C × -orbit is a topological sphere, called a gradient sphere; as above, the minimum (respectively maximum) of J along one such sphere is called the south (respectively north) pole.
A gradient sphere is free if its stabilizer is trivial. A chain of gradient spheres is a sequence C 1 , . . . , C l of gradient spheres such that the south pole of C 1 is a minimum of J, the north pole of C i−1 coincides with the south pole of C i , for every i = 2, . . . , l, and the north pole of C l is a maximum for J. A chain of gradient spheres is trivial if it consists only of one free gradient sphere, and non trivial otherwise.
In Karshon [12] particular attention is given to the relation between Hamiltonian S 1 -spaces and symplectic toric manifolds. The latter have been classified in Delzant [5], where a special role is played by a family of convex polygons defined below.
• A convex polygon ∆ ⊂ R 2 is simple if there are exactly 2 edges meeting at each vertex.
• A simple polygon ∆ is rational if all edges have rational slope, i.e. they are subsets of straight lines of the form A simple, rational, convex polygon whose vertices are smooth is said to be Delzant.
Given a symplectic toric manifold (M, ω, µ), the image µ(M ) := ∆ is a Delzant polygon and, conversely, any Delzant polygon ∆ determines (up to T 2 -equivariant symplectomorphisms preserving the moment map) a symplectic toric manifold (cf. Delzant [5]). A natural question to ask is which Hamiltonian S 1 -spaces arise as those associated to symplectic toric manifolds (cf. Remark 2.6). To this end, Karshon [12] proves the following.  The following theorem of Karshon gives a sufficient condition for a Hamiltonian S 1 -space to be extendable. Theorem 2.18 (Karshon [12], Theorem 5.1). Let (M, ω, J) be a Hamiltonian S 1space whose fixed points are isolated. Then (M, ω, J) comes from a Kähler toric variety by restricting the action of the 2-torus to a sub-circle.

2.2.
Compact semi-toric systems. The aim of this section is to introduce the category of compact semi-toric systems, to provide some examples, and to describe how to associate a family of polygons to such a system, following Vũ Ngo . c [26].
2.2.1. Almost toric singularities. Let (M, ω, Φ = (J, H)) be a completely integrable Hamiltonian system. A point p ∈ M is singular or critical if Φ fails to be a submersion at p. In this case, the rank of p is defined to be rk D p Φ. Working with arbitrary types of singular points is beyond the scope of this paper. To this end, all singular points are henceforth assumed to be non-degenerate in the sense of Williamson [27], i.e. a generalization of the Morse-Bott condition in the symplectic category (cf. Zung [28] for a precise definition). This notion is generic and naturally extends to singular orbits of the R 2 -action, i.e. if an orbit O contains a singular nondegenerate point, then all points in O are non-degenerate. Moreover, the singular points considered here are not of hyperbolic type, as these are of an intrinsically different nature to the other ones (cf. Symington [24]). These are henceforth referred to as almost toric. This is no standard notation, introduced here for convenience. For the purposes of this paper, non-degeneracy amounts to controlling the local behavior of the action near compact singular orbits (cf. Eliasson [8], Miranda & Zung [19]). This can be made precise as follows and is henceforth referred to as the Eliasson-Miranda-Zung local normal form. The above assumptions imply that there are three types of singular orbits, two of rank 0 (i.e. fixed points) and one of rank 1 (i.e. a circle). Fixed points: Let (x, y, ξ, η) denote Darboux coordinates on (R 4 , ω 0 ). Elliptic-elliptic point: A point p ∈ M is said to be of elliptic-elliptic type if there exist open neighbourhoods U ⊂ (M, ω) of p, U 0 ⊂ (R 4 , ω 0 ) of 0 ∈ R 4 , a symplectomorphism Ψ : (U, ω) → (U 0 , ω 0 ) such that Ψ(p) = 0, and a local diffeomorphism ψ : R 2 → R 2 satisfying ψ(Φ(p)) = (0, 0), which make the following diagram commute where Φ ee = (q 1 , q 2 ) and q 1 = 1 2 (x 2 + ξ 2 ), q 2 = 1 2 (y 2 + η 2 ). Focus-focus point: A point p ∈ M is said to be of focus-focus type if there exist U, U 0 , Ψ, ψ as above making the diagram in equation (2.3) commute with respect to the map Φ ff = (q 1 , q 2 ), where q 1 = xη − yξ, q 2 = xξ + yη. Rank 1 orbits: Let (x, y, a, θ) denote Darboux coordinates on (R 2 × T * S 1 , ω 0 ).
Definition 2.19 (Compact semi-toric systems, Pelayo and Vũ Ngo . c [20]). The category ST is defined by • Objects: completely integrable Hamiltonian systems (M, ω, Φ = (J, H)) whose singular points are almost toric and such that (M, ω, J) is a Hamiltonian S 1space. These are henceforth called compact semi-toric systems.
Remark 2.20. In the original definition, semi-toric systems are defined on manifolds which are not necessarily compact, but J is asked to be proper (cf. Pelayo & Vũ Ngo . c [20,21]). Semi-toric systems on non-compact manifolds, while beyond the scope of the present paper, are of great interest in mathematical physics, i.e. in the study of the Jaynes-Cummings model from quantum optics, cf. Babelon & Doucot [2].
Intuitively, compact semi-toric systems lie at the intersection of completely integrable Hamiltonian systems and Hamiltonian S 1 -spaces, as formalized by the following remark.  Example 2.23.
The first examples of honest (i.e. with focus-focus points) compact semi-toric systems appeared in the study of coupled angular momenta carried out in Sadovskií & Zĥilinskií [23]. More generally, the methods of Pelayo & Vũ Ngo . c [21] allow to construct compact semi-toric systems by specifying some initial data. Semi-toric polygons. In analogy with the case of symplectic toric manifolds, it is possible to associate a family of simple, rational, convex polygons to (the isomorphism class of) a compact semi-toric system (cf. Vũ Ngo . c [26]). However, there are two differences: first, not all vertices need to be smooth, and, second, this family consists of more than one element. These polygons, called semi-toric, play an important role in the proof of the main result of this paper (cf. Theorem 3.1); as such, their construction is recalled below in some detail (cf. Vũ Ngo . c [26] for proofs).
Throughout this section, fix a compact semi-toric system (M, ω, Φ) with m f focus-focus critical points. The image B := Φ(M ) is called the curved polygon with marked interior points (often abbreviated to curved polygon) associated to (M, ω, Φ), where the marked interior points are critical values of Φ whose fiber contains focus-focus points (see Figure 2 Ngo . c [26]). The subset B \ {c 1 , c 2 , . . . , c m f } inherits the structure of a manifold with corners endowed with an integral affine structure, defined below for two dimensional manifolds.
The integral affine structure A on B is defined by taking the action coordinates given by the Liouville-Arnol'd theorem near regular values (cf. Duistermaat [6]), and by the Eliasson-Miranda-Zung local normal form of Section 2.2.1 near the boundary.
Remark 2.29. In integral affine coordinates the boundary ∂B ⊂ B is locally defined by hyperplanes whose normals have integer coefficients.
It is interesting to note that the integral affine structure A on B \{c 1 , c 2 , . . . , c m f } is not the one coming from the inclusion B ⊂ R 2 unless m f = 0 (cf. Zung [29]). In order to bypass this issue, Vũ Ngo . c [26] introduces vertical cuts on B in such a way that the resulting subset is a simply connected integral affine manifold with corners. These can be defined as follows. Let p 1 , . . . , p m f ∈ M be the focusfocus singular points of (M, ω, Φ) and order the corresponding focus-focus values For ε i = 1 (respectively −1) this is the closed vertical segment between c i and the upper (respectively lower) boundary of B.
denote the open vertical segments bẙ Each point s ∈l ε is labeled by the integer where j(c i ) is the multiplicity of the focus-focus critical value c i (cf. Remark 2.26). A choice of cuts ε determines a convex polygon P ε (these are henceforth called semi-toric) associated to (M, ω, Φ) in the following fashion.
Remark 2.31. In fact, any other semi-toric polygon associated to the same choice of cuts ε differs from P ε by composition with an element of  [26, Step 2, Theorem 3.8]). The adjective 'specific' refers to the fact that the first component J of Φ is chosen as an action coordinate (equivalently, the first standard coordinate x on R 2 is chosen as an integral affine coordinate on B), since it generates an effective Hamiltonian S 1 -action. Moreover, upon choosing an orientation on R 2 , f can always be chosen so that the top (respectively bottom) boundary of B is sent to the top (respectively bottom) boundary of P ε by changing the sign of its second component. Henceforth, whenever referring to the semi-toric polygon P ε associated to (M, ω, Φ) and ε ∈ {+1, −1} m f , it is understood that a choice of action variables (equivalently local integral affine coordinates on B) as above is fixed and that f is chosen to be orientation preserving (upon a choice of orientation on R 2 ), unless otherwise stated.
Remark 2.32. Let ε, ε be two choices of cuts for (M, ω, Φ) and denote the corresponding semi-toric polygons by P ε , P ε . Then there exists a continuous piecewise integral affine transformation τ such that P ε = τ (P ε ) with the property that τ preserves vertical lines, i.e. on each region on which it is defined by an integral affine transformation it is given by a restriction of an element in T. This can be used to give a geometric interpretation of the action of {+1, −1} m f on the space of semi-toric polygons associated to (M, ω, Φ) (cf. Vũ Ngo . c [26,Prop. 4.1]).
Fix a choice of cuts ε ∈ {+1, −1} m f and let P ε be the associated semi-toric polygon. The vertices of P ε fall into three categories, as described below.  Henceforth, P ε is also used to denote the labeled semi-toric polygon.
Remark 2.36. From the labeled semi-toric polygon it is possible to recover the total number of focus-focus points m f , since it is the sum of the multiplicities j(c i ) for all i. Let J min (respectively J max ) denote the minimum (respectively maximum) value taken by J. By construction of f , vertices lying on . Fix a vertex v of P ε strictly between the vertical lines between J min and J max . By Theorem 2.30, the edges incident to v have integral tangent vectors; denote their primitives with positive first component by u, w ∈ Z 2 . Throughout, u (respectively w) denotes the primitive tangent to the edge on the 'left' (respectively 'right') of v (the orientation is chosen so that J does not decrease going from left to right). As mentioned above, Z u, w does not need to be the standard Z 2 ⊂ R 2 , i.e. the vertex does not need to be smooth. However, using the results of Vũ Ngo . c [26], the lemma below proves some conditions on u, w, which generalise Pelayo & Vũ Ngo . c [21,Def. 4 n v , ε v being the degree and sign of v respectively.
which commutes with the Hamiltonian action defined by Φ. Thus, in this case, the result follows because the action is locally toric. If v is either hidden Delzant or fake, consider a semi-toric polygon P ε associated to the choice of cuts ε , which agrees with ε except that it changes the sign of the cuts going into f −1 (v). Let f : B → P ε denote the homeomorphism associated to ε as in Theorem 2.30.
If v was hidden Delzant for P ε , then v is Delzant for P ε , while if it was fake for P ε , it is not a vertex for P ε . The result in both cases follows from the fact that P ε = τ v (P ε ), where τ v is a piecewise integral affine transformation, which is the identity on the half-space to the left of the vertical line containing v and A v on the half-space to the right (cf.  In analogy with symplectic toric manifolds, P ε determines the derivative of the Duistermaat-Heckman function of (M, ω, J) as stated in the theorem below. This is an important fact which is used in the proof of the main result.
where α ± (x) denotes the slope of the top (respectively bottom) edge of P ε intersecting the vertical line through x. Otherwise, where e + (respectively e − ) is zero if there is no top (respectively bottom) vertex whose first coordinate is x, or e ± = − 1 a ± b ± , where a ± , b ± are the isotropy weights of the S 1 -action at the corresponding vertices, and j x is the number of focus-focus critical points of Φ lying in J −1 (x) ⊂ M .
3. The main theorem: From semi-toric polygons to labeled directed graphs. Recall that there is a functor F : ST → Ham S 1 , which sends a compact semi-toric system (M, ω, Φ = (J, H)) to its underlying Hamiltonian S 1 -space (M, ω, J) (cf. Remark 2.21). A natural question to ask is to describe how to recover the invariants of (the isomorphism class of) a Hamiltonian S 1 -space underlying (the isomorphism class of) a compact semi-toric system from the invariants of the latter. On the one hand, isomorphism classes of Hamiltonian S 1 -spaces are classified by their associated labeled directed graphs (cf. Section 2.1 and Karshon [12]); on the other, there is no theorem classifying isomorphism classes of semi-toric systems in full generality (cf. Pelayo & Vũ Ngo . c [20,21] for the classification of generic semitoric systems). However, all that is needed to recover the labeled directed graphs of Hamiltonian S 1 -spaces underlying compact semi-toric systems are the labeled associated semi-toric polygons introduced in Section 2.2: this is the content of the main theorem of this paper, stated below. Throughout this section, fix a compact semi-toric system (M, ω, Φ) along with a labeled semi-toric polygon P ε , and denote its underlying Hamiltonian S 1 -space by (M, ω, J) unless otherwise stated. This automatically sets the homeomorphism f : B = Φ(M ) → P ε given by Theorem 2.30. Recall that the labeled directed graph Γ associated to (M, ω, J) is determined by the vertex set V and its labeling (i.e. the connected components of the fixed point set M S 1 , and their topological and symplectic properties respectively), and the edge set E and its labeling (i.e. Z k -spheres). Recovering V and its labeling from P ε is the aim of Section 3.1, while Section 3.2 deals with 'seeing' Z k -spheres from P ε . The proof of Theorem 3.1 is then given in Section 3.3, which brings everything together.
The scheme of the proof of Theorem 3.1 mimics the relation between symplectic toric manifolds and their associated Hamiltonian S 1 -spaces, which is recalled below.
Remark 3.2 (Karshon [12], Section 2.2). The following table shows how to pass from a Delzant polygon ∆ associated to a symplectic toric manifold (M, ω, µ) to the labeled directed graph Γ of its associated Hamiltonian S 1 -space. This serves as a guide to follow the ideas of the forthcoming sections.

Edge set E:
An edge e ∆ of ∆ whose primitive tangent vector is of the form (k, b) ∈ Z 2 (for k ≥ 2) gives rise to an edge e Γ in Γ joining the vertices in V corresponding to the vertices of ∆ of e ∆ (note that these vertices can never be fat). Labeling of E: Each edge e Γ of Γ is labeled with the integer k ≥ 2, where (k, b) ∈ Z 2 is a primitive tangent vector to the corresponding edge e ∆ of ∆.
3.1. Fixed point set M S 1 : vertex set V and its labeling. This section describes how to construct and label the vertex set V of Γ from P ε . By definition, vertices of Γ correspond to connected components of M S 1 , the fixed point set of the S 1 -action whose moment map is J. Any point p ∈ M which is fixed by this S 1 -action satisfies dJ(p) = 0. Thus p is also a critical point of Φ. The next lemma, which is probably well-known, characterizes isolated fixed points of the S 1 -action in terms of critical points of Φ. However, since a suitable reference cannot be traced, a proof is included. Having dealt with isolated fixed points in M S 1 , consider the fixed surfaces, which, by the following proposition, are precisely as in the case of symplectic toric manifolds (cf. Remark 3.2). Proposition 3.4. Let Σ be a connected surface which is fixed by the S 1 -action. Then either Σ = J −1 (J min ) or Σ = J −1 (J max ), where J min (respectively J max ) is the minimum (respectively maximum) value of J on M . Moreover, Σ is a symplectic sphere.
Proof. By a standard argument which uses local normal forms (cf. Karshon [12, Cor. A.7]), Σ is a symplectic surface which is a local minimum or maximum for J. Since J is a moment map for an S 1 -action, by the Atiyah-Guillemin-Sternberg convexity theorem, a local minimum or maximum is global, and is connected (cf. Atiyah [1], Guillemin & Sternberg [11]). From this it follows that Σ = J −1 (J min ) or Σ = J −1 (J max ).
It remains to prove that Σ is a sphere. Without loss of generality suppose that Σ = J −1 (J min ). Then there exists Ngo . c [26,Prop. 2.12]). In other words, f • Φ(Σ) corresponds to a vertical edge of P ε and, near Σ, the second component of f • Φ is the moment map of an effective Hamiltonian S 1 -action, which can be restricted to Σ by the Eliasson-Miranda-Zung local normal form. Hence Σ is a symplectic surface with an effective Hamiltonian circle action, which implies that it is a sphere.
Remark 3.5. The arguments in the proof of Proposition 3.4 imply that f • Φ(Σ) ⊂ P ε is a vertical edge, and its normalized symplectic area is the length of the corresponding vertical edge, just as in the symplectic toric case. 3.2. Z k -spheres: Edge set E and its labeling. Recall that the edges in the labeled directed graph Γ associated to (M, ω, J) correspond to symplectic spheres in M which are stabilized by a finite subgroup Z k ⊂ S 1 with k ≥ 2; these are known as Z k -spheres. Fix one such symplectic sphere Σ; the action of S 1 on Σ has two fixed points (called the poles of the sphere), which are isolated fixed points in M S 1 (cf. Karshon [12]). An isolated fixed point in M S 1 is a pole of a Z k -sphere if and only if one of its isotropy weights equals k in absolute value (cf. Remark 2.8). By Lemma 3.3, the isolated fixed points in M S 1 are either focus-focus or elliptic-elliptic critical points satisfying property (V2). Thus, before trying to 'see' Z k -spheres from P ε , it is necessary to understand how to obtain the isotropy weights of isolated fixed points in M S 1 from P ε . This is the aim of the next subsection.
3.2.1. Isotropy weights from P ε . Lemma 3.3 proves that isolated fixed points in M S 1 satisfy either property (V1), i.e. they are of focus-focus type, or (V2), i.e. they are of elliptic-elliptic type with an extra condition. Note that the former do not arise when considering symplectic toric manifolds and, as such, need to be dealt differently. To this end, each of the two cases is discussed separately below.
Let p ∈ M S 1 be a focus-focus critical point for Φ. Recall that the Eliasson-
Definition 3.7 (Local system preserving actions). An S 1 -action on U is said to be local system preserving if for all λ ∈ S 1 and all p ∈ U , Φ(λ · p) = Φ(p).
Remark 3.8. Local system preserving actions play a crucial role in the topological and symplectic classification of completely integrable Hamiltonian systems developed in Zung [28].
In what follows it is shown that the isotropy weights at any focus-focus critical point are ±1. The proof is broken down into two steps: first it is shown that, near a focus-focus critical point, there is a unique (up to sign) local system preserving effective Hamiltonian S 1 -action, which in the above local normal form has moment map given by q 1 (cf. Zung [30,Theorem 1.2] and Proposition 3.9). Since for any compact semi-toric system (M, ω, Φ = (J, H)), J is the moment map of an effective local system preserving Hamiltonian S 1 -action, it follows that the isotropy weights of any two focus-focus points are equal, because they are both equal to the isotropy weights of the origin in the above local model. Therefore, these weights can be calculated in a single example, which is conveniently chosen to be an adaptable compact semi-toric system (cf. Definition 3.11 and Proposition 3.12). Proposition 3.9. There exists a unique (up to sign) local system preserving effective Hamiltonian S 1 -action defined in a neighbourhood of a focus-focus point.

SONJA HOHLOCH, SILVIA SABATINI AND DANIELE SEPE
Therefore, for i, j = 1, 2 and for all z ∈ U 0 \ {0}, The above equation implies that the functions F 1 , F 2 are basic, i.e. there exist smooth functions G 1 , G 2 : Consider the flow of X h , which is periodic with period 2π. Since the functions F j are basic, they are constant along orbits of X h , as they only depend on the values on the image of Φ 0 and the latter is constant on the orbits of X h . Using the fact that [X q1 , X q2 ] = 0, it therefore follows that the flow of X h is given by where w 1 = x + iξ, w 2 = y + iη, and F 1 , F 2 are smooth functions of z 1 , z 2 (in fact, of q 1 , q 2 ). Since ϕ 2π h = id, (3.3) implies that e (iF1+F2)2π = 1 e (iF1−F2)2π = 1, which implies that F 2 ≡ 0 and that 2πF 1 ∈ 2πZ. Since the S 1 -action is effective, it follows that |F 1 | ≡ 1. Thus, up to sign, h = q 1 on U 0 \ {0}; since both functions extend smoothly at 0, it follows that, up to sign, h = q 1 + const on U 0 , which completes the proof.
Since J is the moment map of a system preserving effective Hamiltonian S 1action near p, it follows that its isotropy weights at p equal those of the origin in R 4 with respect to the Hamiltonian S 1 -action whose moment map is q 1 in the above local normal form. In particular, all focus-focus critical points have the same isotropy weights for the S 1 -action; these are known to be {+1, −1} (cf. Zung [30,Theorem 1.2]). Remark 3.10. In order to calculate the isotropy weights of an S 1 -action at a fixed point, an S 1 -invariant almost complex structure J has to be fixed. Observe that the (integrable!) almost complex structure used in the proof of Proposition 3.9 is not invariant under the S 1 -action and, as such, cannot be used to compute the weights.
Below a different proof of the fact that focus-focus critical points have isotropy weights {+1, −1} is given; it uses the close relation between a special class of compact semi-toric systems and symplectic toric manifolds.   H)) denote a symplectic toric manifold whose Hamiltonian T 2 -action extends the one defined by J (this exists by Definition 2.16), and let ∆ be the associated Delzant polygon. With this notation in hand, the following proposition, already proved in Zung [30], can be deduced by exploiting the properties of adaptable compact semi-toric systems. Proof. Any isolated fixed point in M S 1 of focus-focus type for Φ corresponds to a vertex of ∆ (see Figure 3.1 below). In light of Proposition 3.9 and the subsequent discussion, it suffices to consider an adaptable compact semi-toric system which has only one focus-focus point, e.g. the system considered in Sadovskií & Zĥilinskií [23] or that one constructed in Example 3.17. Fix such a system and let x ∈ R be such that J −1 (x) contains the only focusfocus point. The Eliasson-Miranda-Zung local normal form implies that J min < x < J max , where, as above, J min (respectively J max ) denote the minimum (respectively maximum) value of J. Let ρ denote the Duistermaat-Heckman function associated to the S 1 -action (cf. Definition 2.39). Theorem 2.40 gives that a + b + , and a + , b + are the isotropy weights at p of the S 1 -action defined by J (note that p is an elliptic-elliptic point for (M, ω, µ = (J,H))). Observe that the left hand sides of (3.4) and (3.5) are equal, as they depend on the S 1 -action defined by J. Therefore, a + b + = −1, which implies that {a + , b + } = {+1, −1} as required.
Having found that the isotropy weights at focus-focus critical points are equal to 1 in absolute value, the following corollary is immediate.  Remark 3.14. Given (M, ω, µ = (µ 1 , µ 2 )) is a symplectic toric manifold, its Delzant polygon ∆ can be used to calculate the isotropy weights of the isolated fixed points of the Hamiltonian S 1 -action of the associated Hamiltonian S 1 -space (M, ω, µ 1 ) as follows. By Remark 3.2, such a point maps to a vertex v ∆ of ∆ not incident to a vertical edge. Let u, w be the primitive integral tangent vectors to the edges incident to v ∆ which come out of it. Then the isotropy weights of the chosen S 1 -action at the corresponding isolated fixed point are given by taking the first coordinates of u, w.
Let v be a Delzant or hidden Delzant vertex of P ε satisfying condition (V2) and choose primitive integral tangent vectors u, w to the edges incident to v as in Remark 3.14. The next proposition proves that the isotropy weights of the corresponding isolated fixed point in M S 1 can be calculated as in the case of symplectic toric manifolds described by Remark 3.14 above.
Proposition 3.15. With the notation as above, the isotropy weights of the isolated fixed point corresponding to v are given by the first coordinates of u and w.
Proof. If v is a Delzant vertex, then locally f • Φ defines a Hamiltonian T 2 -action which extends the S 1 -action whose moment map is J. Thus in this case the result follows, as the action is locally toric and the observations made in Remark 3.14 hold. On the other hand, if v is hidden Delzant for P ε , then there exists a different choice of cuts ε such that v is a Delzant vertex for P ε (ε agrees with ε except that there are no cuts going into f −1 (v), cf. proof of Lemma 2.38). If u, w are the primitive integral tangent vectors to the edges of P ε chosen as in Remark 3.14, then u, A v w are the corresponding ones for P ε , where A v = 3.2.2. Z k -spheres. Corollary 3.13 and Proposition 3.15 allow to find poles of Z kspheres; what this section is concerned with is to show that, in analogy with the case of symplectic toric manifolds (cf. Remark 3.2), these are mapped to the boundary of P ε under f • Φ. Recall that B = Φ(M ) is a curved polygon with curved edges (cf. Section 2.2.2).
Proof. Let p, q ∈ Σ be fixed by the S 1 -action. By the local normal form of Lemma 2.7, p and q can be chosen so that one of the isotropy weights of p (respectively q) is k (respectively −k). Corollary 3.13 implies that Φ(p), Φ(q) are vertices of B. Suppose thatB∩Φ(Σ) = ∅ and consider s ∈B∩Φ(Σ). Suppose that s is a focus-focus critical value, then Σ ∩ Φ −1 (s) does not contain focus-focus critical points and, therefore, consists only of regular points. However by Proposition 3.9, the Hamiltonian S 1action whose moment map is J is the unique local system preserving effective such, and hence its action is free on regular points lying on Φ −1 (s). This leads to a contradiction, as points on Σ are stabilized by Z k ⊂ S 1 . Therefore, s is not a focusfocus critical value, which implies that s is regular. Since r ∈ Σ ∩ Φ −1 (s), then the isotropy of the S 1 -action at r is Z k ; however, since the S 1 -action commutes with the Hamiltonian R 2 -action induced by the compact semi-toric system, it follows that all points on Φ −1 (s) are fixed by Z k . The Liouville-Arnol'd theorem implies that, locally near Φ −1 (s), there exists a free Hamiltonian T 2 -action; since the Hamiltonian S 1 -action defined by J is effective and commutes with the Hamiltonian R 2 -action defined by the compact semi-toric system, the free local Hamiltonian T 2 -action can be chosen to extend the one defined by J. This leads to a contradiction, which implies thatB ∩ Φ(Σ) = ∅. Thus Φ(Σ) ⊂ ∂B and Σ consists of elliptic-elliptic and elliptic-regular critical points for Φ.
It remains to prove that Φ(Σ) consists of the whole of a curved edge. This is a consequence of the Eliasson-Miranda-Zung local normal form, as outlined below. Suppose that s ∈ Φ(Σ) is not a vertex of B and let e denote the curved edge containing s . Since focus-focus critical values are isolated, there exists an open neighbourhood W ⊂ B of e such that the Hamiltonian action defined by Φ on (Φ −1 (W ), ω) descends to a Hamiltonian T 2 -action, i.e. there exists a diffeomorphism f : y)), i.e. fix the Hamiltonian vector field of J| Φ −1 (W ) to be an infinitesimal generator of the Hamiltonian T 2 -action. The Eliasson-Miranda-Zung local normal form implies that f (e) is a straight line with integral tangent vector (cf. Remark 2.29). Given the above choices, it follows that a primitive tangent vector u forf (e) is of the form (k, b), for some b = 0 (cf. Remark 3.14). It is standard to check that (f •Φ) −1 (f (e)) is a Z k -sphere (cf. Karshon [12]); since ((f • Φ) −1 (f (e))) ∩ Σ is not empty and contains a point that is not a pole of Σ, it follows that Σ = (f • Φ) −1 (f (e)) = Φ −1 (e), which completes the proof.
Unlike the case of symplectic toric manifolds, it is not necessarily true that a Z ksphere Σ of the underlying Hamiltonian S 1 -space (M, ω, J) of a compact semi-toric system (M, ω, Φ) is the preimage of an edge in P ε . This is because some of the cuts may break the curved edge in B whose preimage under Φ equals Σ, thereby introducing fake vertices (cf. Definition 2.33). This is illustrated by the following example.
Example 3.17. Following Pelayo & Vũ Ngo . c [21], the polygons shown in Figure  3.2 are two semi-toric polygons associated to a compact semi-toric system (M, ω, Φ), where m f = 1 and the Taylor series invariant associated to the focus-focus critical point is taken to be 0 (cf. Pelayo & Vũ Ngo . c [20,25] for details). The top edge, going from (0, 0) to (2,1), of the polygon in Figure 3.2 (b) corresponds to a Z 2sphere; however, the same Z 2 -sphere is the preimage of the union of the top edges of the polygon in In general, a Z k -sphere is the preimage of a chain of consecutive edges e 1 , . . . , e N joining two vertices v, v in P ε , whose 'initial' (respectively 'final') vertex v ∈ e 1 (respectively v ∈ e N ) is Delzant or hidden Delzant, has one of its isotropy weights equals to k (respectively −k), and whose other vertices are all fake. Note that the isotropy weights at v and v can be calculated using Proposition 3.15. The adjectives 'initial' and 'final' refer to direction of increasing first coordinate, which corresponds to the flow of the negative gradient of J with respect to a compatible metric (cf. Remark 2.13).
3.3. Proof of Theorem 3.1. Sections 3.1 and 3.2 allow to describe an algorithm to construct the labeled directed graph Γ of the Hamiltonian S 1 -space (M, ω, J) underlying a compact semi-toric system (M, ω, Φ) from the labeled semi-toric polygon P ε . This is explained in the proof of the main theorem below.
Proof of Theorem 3.1. As in Remark 3.2, all that is needed is how to construct the vertices and the edges of Γ and their labeling.

Edge set E:
Suppose that v Γ ∈ V has an isotropy weight equal to k ≥ 2 (note that Corollary 3.13 implies that v Γ does not correspond to a focus-focus critical point). By Proposition 3.15 this happens if and only if the corresponding vertex of v Pε of P ε has an 'outgoing' edge e 1 (in the direction of increasing J) whose primitive tangent vector is of the form (k, b), for some b ∈ Z. Construct a chain C of consecutive edges e 1 , e 2 , . . . , e N by moving along e 1 in the direction of increasing J until a Delzant or hidden Delzant vertex v Pε is reached (this process terminates). Let v Γ ∈ V denote the corresponding vertex. Note that by Proposition 3.16,

Labeling of E:
Each edge e Γ of Γ is labeled with the integer k ≥ 2, where (k, b) ∈ Z 2 is a primitive tangent vector to the edge e 1 in the corresponding chain C of edges e 1 , . . . , e N of P ε .
Example 3.18. The compact semi-toric system whose associated semi-toric polygons are shown in Figure 3.2 is defined on (CP 2 , ω F S ), where ω F S is the standard Fubini-Study symplectic form. In fact, the underlying Hamiltonian S 1 -space is described by Example 2.4.

4.
Adaptable and non-adaptable compact semi-toric systems. As remarked above and in the literature, compact semi-toric systems share many properties with symplectic toric manifolds (cf. Remark 2.24, the proof of Theorem 3.1, and Pelayo & Vũ Ngo . c [20,21], Vũ Ngo . c [26]). In light of the classification of symplectic toric manifolds carried out in Delzant [5], it is natural to ask whether compact semi-toric systems admit a semi-toric polygon which is Delzant in the sense of Definition 2.14.
Note that property (5) of Theorem 2.30 implies that a semi-toric polygon P ε may fail to be Delzant only if some vertices are not smooth (cf. Definition 2.14).
Recall that (M, ω, Φ) is adaptable if and only if its underlying Hamiltonian S 1space (M, ω, J) is extendable, which in turn means that the S 1 -action can be extended to an effective Hamiltonian T 2 -action on (M, ω) (cf. Definitions 2.16 and 3.11, and Theorem 2.15). The aim of this section is to prove the following result. In fact, Corollary 4.9 follows from Theorem 4.5, which proves a stronger property of adaptable compact semi-toric systems: Let (M, ω, Φ) be adaptable, denote by (M, ω, J) its underlying Hamiltonian S 1 -space, and let (M, ω, µ) be a symplectic toric manifold whose associated Hamiltonian S 1 -space (in the sense of Remark 2.6) is (M, ω, J). Then there exists a choice of cuts ε such that P ε = ∆, where ∆ is the Delzant polygon classifying (M, ω, µ). In other words, the family of semitoric polygons associated to (M, ω, Φ) contains all the Delzant polytopes classifying symplectic toric manifolds whose associated Hamiltonian S 1 -space is (M, ω, J).
Henceforth, fix a compact semi-toric system (M, ω, Φ). Let P ε be a semi-toric polygon associated to (M, ω, Φ). The following lemma gives a necessary and sufficient condition for vertices of P ε to be smooth.
and ε v and n v are the sign and the degree of v respectively. Let (u w) denote the matrix whose columns are u, w. Then, if u = (u 1 , u 2 ) T and w = (w 1 , w 2 ) T , a simple calculation shows that Since ε v n v u 1 w 1 = 0, it follows that, if v is smooth, then the signs of det(u A v w) and det(u w) are opposite. Recall that u, A v w are the left (respectively right) primitive tangent vectors to the edges incident to v in a distinct semi-toric polygon P ε whose choice of cuts agrees with ε except that there are no cuts into v (so that v is a Delzant vertex for P ε ). Moreover, P ε = τ v (P ε ), where τ v is a piecewise integral affine transformation which is the identity on the left of vertical line through v and A v on the right (cf. Vũ Ngo . c [26,Prop. 4.1]). This transformation preserves convexity of the semi-toric polygon and, thus, the sign of det(u w) agrees with that of det(u A v w). If v is smooth as a vertex of P ε , this leads to a contradiction. It remains to check that fake vertices are smooth if and only if they satisfy property (b). Suppose v is fake and let u, w ∈ Z 2 be as above. Lemma 2.38 gives that det(u A v w) = 0; thus equation (4.1) implies that The vertex v is smooth if and only if |det(u w)| = 1, which is equivalent to n V , u 1 , w 1 = 1 (since they are all positive integers). The first component of u, w is equal to 1 if and only if v does not lie on a chain of edges corresponding to a Z k -sphere (cf. proof of Theorem 3.1). This completes the proof.

4.1.
Adaptable compact semi-toric systems. By condition (E2) in Theorem 2.15, (M, ω, Φ) is adaptable if and only if every non-extremal level set of J contains at most 2 non-free orbits and all fixed surfaces are spheres. By Proposition 3.4, the latter is always satisfied. The only possibilities for non-free orbits of the S 1 -action which are not extremal are: • elliptic-elliptic points not lying on a symplectic sphere fixed by the S 1 -action, • focus-focus points, • points lying on a Z k -sphere but not in M S 1 (in light of Proposition 3.16, these lie on elliptic-regular orbits stabilized by a subgroup Z k ⊂ S 1 ). Note that in all the above cases, the non-free orbits of the S 1 -action consist of critical points of Φ. As before, let J min , J max denote the minimum and maximum values of J respectively, and let x ∈ ]J min , J max [. If J −1 (J(x)) does not contain a focus-focus point, then it contains at most two non-free orbits (cf. Vũ Ngo . c [26,Theorem 3.4]). Thus, in order to see whether (M, ω, Φ) is adaptable or not, it suffices to check that each J −1 (J(x)) containing at least one focus-focus point does not contain more than two non-free orbits of the S 1 -action. This happens if and only if J −1 (J(x)) contains either (A1) exactly one rank 0 critical point of Φ, which is of focus-focus type, and at most one elliptic-regular orbit lying on a Z k -sphere, or (A2) exactly two rank 0 critical points of Φ, which can be either both of focus-focus type, or one of focus-focus type and the other of elliptic-elliptic type, and no point lying on a Z k -sphere.
Remark 4.3. The proof of Theorem 3.1 implies that any semi-toric polygon P ε associated to (M, ω, Φ) can be used to check that the system is adaptable, i.e. to check that conditions (A1) and (A2) hold.
Until the end of this section, assume that (M, ω, Φ) is adaptable with m f focusfocus critical points, and denote by (M, ω, J) its underlying Hamiltonian S 1 -space. By definition, (M, ω, J) is extendable, which, by condition (E3) in Theorem 2.15, is equivalent to the existence of an S 1 -invariant metric with at most two non-trivial chains of gradient spheres (cf. Remark 2.13). In fact, the method employed in Karshon [12,Prop. 5.16] to construct a symplectic toric manifold (M, ω, µ) whose associated Hamiltonian S 1 -space is (M, ω, J) is completely determined by (K1) a choice of an S 1 -invariant metric as above, (K2) a suitable toric extension of the S 1 -action near the minimum of J.
Here 'suitable' indicates that the moment map associated to the toric extension near the minimum has components (J,H) as in Theorem 2.15. This construction consists of building the two sequences of directed edges e 1 , . . . , e r and e 1 , . . . , e r starting at the minimum of J and ending at the maximum defining (the boundary of) a Delzant polygon; in particular for each i > 1, the tangent to e i (respectively e i ) is completely determined by the S 1 -weights of its 'initial vertex' (in the direction of J increasing), the direction of e i−1 (respectively e i−1 ) and the convexity of the Delzant polygon. Analogously, a semi-toric polygon associated to (M, ω, Φ) is completely determined by (V1) a choice of cuts ε ∈ {+1, −1} m f , (V2) a choice of suitable local action-angle coordinates around the minimum of J.
Here 'suitable' implies that the angles are defined using the Hamiltonian vector fields X J , X f (2) (J,H) for some smooth function f (2) as in Theorem 2.30. Equivalently, (V1) and (V2) determine the homeomorphism f : Φ(M ) ⊂ R 2 → R 2 onto its image P ε uniquely.
It is clear that (K2) and (V2) are entirely analogous, since a choice of suitable local action-angle coordinates gives a suitable toric extension of the S 1 -action defined by J. The relation between (K1) and (V1) is explored in the theorem below. Fix choices for (K1) and (K2), so as to obtain a symplectic toric manifold (M, ω, µ) classified by the Delzant polygon ∆.  Before proceeding to the proof of theorem 4.5, note that different choices of metrics satisfying (K1) and different choices of toric extensions near the minimum of J, as in (K2), give rise to all symplectic toric manifolds whose associated Hamiltonian S 1 -spaces are all isomorphic to (M, ω, J). Thus an immediate consequence of Theorem 4.5 is the following corollary.   In order to obtain all Delzant polygons in Corollary 4.6, the homeomorphism f : B ⊂ R 2 → P ε ⊂ R 2 may have to be chosen to be orientation-reversing (once an orientation in R 2 is fixed), as illustrated by Example 4.7.
The main idea behind the proof of Theorem 4.5 is that the chosen metric determines the cuts.
thus illustrating the relation between (K1) and (V1). The proof itself proceeds by induction on the number of vertical lines on which the cuts lie. Given the standard coordinates x, y on R 2 , let x 1 ≤ x 2 ≤ . . . ≤ x m f denote the x-coordinates of the focus-focus critical values, and let J min (respectively J max ) denote the minimum (respectively maximum) value of J. Note that J min < x 1 and that x m f < J max .
Proof of Theorem 4.5. Recall that choices of (K1) and (K2) are fixed, so as to obtain a Delzant polygon ∆ classifying a symplectic toric manifold (M, ω, µ). Note that focus-focus critical points for (M, ω, Φ) map to vertices of ∆ which are not extremal with respect to J. Moreover, the x coordinate of the vertices of ∆ is given by the value of J at the corresponding critical point of Φ; in particular, if c j = (x j , y j ) is a focus-focus critical value, for j = 1, . . . , m f , then the corresponding vertex of ∆ has coordinates (x j , y j ). In analogy with the definition of slices given above, define for i = 0, . . . , N the ∆-slices by Note that there is a one-to-one correspondence between slices S 0 , . . . , S N and ∆slices S ∆ 0 , . . . , S ∆ N , which preserves the labeling as illustrated by the figure below.
The proof proceeds by induction on the number of slices of the curved polygon. If there is only one slice then there is nothing to prove, since the system is of toric type in the sense of Vũ Ngo . c [26,Def. 2.1 and Cor. 3.5]. Suppose there are slices S 0 , . . . , S N for N ≥ 1 of the curved polygon. Firstly, observe that the homeomorphism f 0 := f | S0 can be defined in such a way that f 0 (S 0 ) = S ∆ 0 ; this follows from the fact that the action of T (cf. Remark 2.31) on S ∆ 0 gives all suitable toric extensions of the S 1 -action defined by J near the minimum and f 0 is one such. This is nothing but restating the fact that choices (K2) and (V2) are equivalent. Therefore the following may be assumed. Inductive hypothesis: A choice of cuts has been made so that f is defined on k−1 i=0S i ∪ S k and that, for all i = 0, . . . , k, f (S i ) = S ∆ i . The idea behind the construction of f is to define it slice by slice, at each stage making a choice of cuts so as to obtain a unique extension to the next slice (cf.
x 2 x 2 x 1 H Figure 4.1. Slices and ∆-slices vertices in ∆ (up to sign). Note that the choice of cuts is going to be such that there will be a one-to-one correspondence between vertices of f k i=0S i ∪ S k+1 and of ∆ along the vertical line {(x, y) | x = x k+1 }. Since the compact semi-toric system is adaptable, condition (E2) of Theorem 2.15 implies that J −1 (x k+1 ) contains at most two non-free orbits of the S 1 -action, one of which must be of focus-focus type for Φ by definition of x k+1 . There are two distinct cases to consider, depending on conditions (A1) and (A2) described above.
Case (A1). exactly one critical point of rank 0 for Φ in J −1 (x k+1 ). There is only one vertex v of ∆ whose first coordinate equals x k+1 ; this follows from the description of the isolated fixed points of M S 1 (cf. Lemma 3.3 and Theorem 2.15). Moreover, since this critical point is of focus-focus type (by definition of x k+1 ), its isotropy weights for the S 1 -action are +1, −1 (cf. Proposition 3.12). By construction of ∆, the primitive tangent with positive first coordinate u ∈ Z 2 (respectively w ∈ Z 2 ) to the left (respectively right) edge incident to v is of the form (1, u 2 ) T (respectively (1, w 2 ) T ). Choose the (only!) cut so that v is a vertex of the image of the extension of f defined on k i=0S i ∪ S k+1 . This can be achieved as follows. By assumption, there is (part of!) a curved edge in Φ(M ) ∩ S k which, under f , maps to the edge incident to v on the left (in the direction of increasing J). Directing the cut towards this edge is the required choice; extend f to S k+1 using the method of Theorem 2.30. Observe that, by assumption and by the fact that f is continuous,