Cube moves for $s$-embeddings and $\alpha$-realizations

Chelkak introduced $s$-embeddings as tilings by tangential quads which provide the right setting to study the Ising model with arbitrary coupling constants on arbitrary planar graphs. We prove the existence and uniqueness of a local transformation for $s$-embeddings called the cube move, which consists in flipping three quadrilaterals in such a way that the resulting tiling is also in the class of $s$-embeddings. In passing, we give a new and simpler formula for the change in coupling constants for the Ising star-triangle transformation which is conjugated to the cube move for $s$-embeddings. We introduce more generally the class of $\alpha$-embeddings as tilings of a portion of the plane by quadrilaterals such that the side lengths of each quadrilateral $ABCD$ satisfy the relation $AB^\alpha+CD^\alpha=AD^\alpha+BC^\alpha$, providing a common generalization for harmonic embeddings adapted to the study of resistor networks ($\alpha=2$) and for $s$-embeddings ($\alpha=1$). We investigate existence and uniqueness properties of the cube move for these $\alpha$-embeddings.


Introduction
The star-triangle transformation was first introduced by Kennelly in the context of resistor networks [16], as a local transformation which does not change the electrical properties of the network (such as the equivalent resistance) outside of the location where the transformation is performed.It consists in replacing a vertex of degree 3 by a triangle as on Figure 1, and the conductances after the transformation are given as some rational functions of the conductances before the transformation.It follows from the classical connection between resistor networks and random walks [15] that applying this star-triangle transformation to a graph with weights on the edges also preserves the probabilistic properties of the random walk on this graph.Conversely, a triangle can be made into a star, and the terminology "star-triangle transformation" is often used to refer to both of these operations.
Another probabilistic model known to possess a star-triangle transformation is the Ising model, a celebrated model of magnetization which samples a 1 arXiv:2003.08941v2[math.CO] 4 Oct 2021 random configuration of spins living at the vertices of a graph; this property appears in [26,28], see also Chapter 6 of [3].The probability distribution of this configuration depends on coupling constants attached to the edges of the graph.When these coupling constants are all positive (which is called the ferromagnetic regime), one can perform a local transformation of the graph as on Figure 1 without changing the correlations of spins outside of the location where the transformation is performed [26,28].Note however that the formulas relating the coupling constants before and after the transformation are not the same as those for the conductances in resistor networks.
Let G be a planar graph, that is, a graph that can be embedded in the plane -or equivalently in the sphere.We denote by G its diamond graph, whose vertex set is composed of the vertices and dual vertices of G and whose faces are all quadrilaterals, one for each edge of G (see Figure 2).We note that planar graphs are graphs that can be embedded in the plane but which do not come with a distinguished embedding.From now on we will assume that every planar graph G is 3-connected, which implies [29] that it possesses a unique embedding up to homeomorphisms of the sphere.In particular the faces of G are welldefined.Hence its diamond graph G is well-defined, is also 3-connected and has well-defined faces.
Figure 2: A piece of a planar graph G (black dots and solid lines), its dual vertices (white dots) and its diamond graph G (dotted lines).
To each of the two models described above (random walk and Ising model) is associated a class of graph embeddings.More precisely, if G denotes a planar graph carrying positive edge weights (conductances for the random walk, coupling constants for the Ising model), one embeds in the plane its diamond graph G .
The embeddings associated to random walks are called the Tutte embeddings or harmonic embeddings [27] and have the property that the faces of G are embedded as orthodiagonal quads, that is, their diagonals are perpendicular.The conductance of an edge of G corresponding to an orthodiagonal quad embedding of a face of G is given by a ratio of the lengths of the diagonals of the quad.The embeddings associated to the ferromagnetic Ising model were introduced by Chelkak [6,7] under the name of s-embeddings; in this case the faces of G are embedded as tangential quads, which are quads admitting a circle tangential to the four sides.The coupling constant of an edge e of G corresponding to a tangential quad embedding of a face of G is given by where tan 2 θ e = cotan δ + cotan β cotan α + cotan γ .
and α, β, γ, δ denote the half-angles of the tangential quad, as shown on Figure 3.
In the special case when the tangential quad is a rhombus, the angle θ e arises as the half-angle of a corner of the quad corresponding to a primal vertex.In the general case θ e does not seem to have a geometric realization as an angle.Existence and uniqueness questions for both harmonic embeddings and sembeddings may be asked either for finite weighted graphs or infinite weighted graphs periodic in two directions.Complete answers are known for harmonic embeddings in both cases and for s-embeddings in the periodic case, see [18,7] for a discussion.However, such questions are not relevant for our purposes, as we will start from one given embedding and discuss whether or not there exists another embedding related by a local transformation.
To avoid unnecessary complications related to boundary issues in the case of finite graphs, we will always assume that the edges involved in our star-triangle transformations are not boundary edges.Note that one could consider a broader framework including such boundary edges, provided that one replaces the notion of a dual graph by that of the augmented dual graph, which has several dual vertices associated with the outer face, one between each pair of consecutive boundary vertices, see e.g.[18].
A harmonic embedding or an s-embedding of the diamond graph G determines a drawing of G, since the vertices of G form a subset of the vertices of G .However a drawing of G does not uniquely determine a harmonic embedding or an s-embedding.
Such embeddings provide the appropriate geometric setting to observe conformally invariant objects such as Brownian motion or SLE processes when taking the scaling limits of random walks and Ising models on generic planar graphs [4,6,7].They generalize isoradial embeddings (embeddings of the faces of G as rhombi), for which specific techniques can lead to proofs of scaling limits (for instance in [17,8,9,12,5] and others), but which correspond to specific choices of weights on an already embedded graph.The star-triangle move for random walks (resp.the Ising model) translates into a geometric local move for harmonic embeddings (resp.for s-embeddings), whereby three orthodiagonal quads (resp.tangential quads) sharing a vertex as on the left-hand side picture of Figure 4 get erased and replaced by three other orthodiagonal quads (resp.tangential quads) with the same hexagonal outer boundary, as on the right-hand side picture of Figure 4. We call this geometric local move a cube move, phrase which was originally introduced in [19] only with a combinatorial meaning (and not a geometric one).We emphasize that cube moves for us are geometric local moves which apply to tilings of the plane by quads, while star-triangle transformations are combinatorial local moves which apply to graphs with edge weights.
The existence and uniqueness of the cube move for harmonic embeddings follows from a classical theorem of planar geometry called Steiner's theorem [21,18].In this article, we show the existence and uniqueness of the cube move for s-embeddings (see definitions in Section 2 and Theorem 3.1 for the exact statement).
Theorem 1.1.For any s-embedding with the combinatorics of the graph on the left-hand side of Figure 4, erase the central black point and keep all the other points fixed.Then there exists a unique central white point that gives an s-embedding with the combinatorics of the graph on the right-hand side of Figure 4.
Moreover it follows from the construction in Section 3 of the s-embedding after the cube move that the coupling constants associated with this new sembedding are the same as those obtained by applying the star-triangle transformation to the coupling constants associated with the s-embedding before the cube move.In other words, the star-triangle transformation of the Ising model only has a local effect on s-embeddings.This is illustrated by a commutative diagram in Figure 5.Our proof of existence relies only on [6,Proposition 4.7] by Chelkak, which connects a linear system on the graph (called propagation equations) to the construction of s-embeddings.We prove uniqueness via selfcontained geometric arguments (see Section 3).
In passing we prove new and simpler formulas for the transformation of coupling constants under the Ising star-triangle transformation: Theorem 1.2.Denote by J 2 , J 4 , J 6 (resp.J 1 , J 3 , J 5 ) the coupling constants before (resp.after) the star-triangle transformation for the Ising model, as on Figure 1.For every 1 ≤ i ≤ 6 let θ i be the unique angle in (0, π 2 ) such that where the indices are considered modulo 6.
Formulas (1.1) and (1.2) are considerably less involved than the classical transformation procedure which is described in Proposition 3.9.We stress that the relation between the coupling constants before and after the star-triangle transformation is the same as in [26,28,3]; what is new is the simpler expression for that relation.
In addition to the results on local transformations for s-embeddings and the Ising model presented above, the second main contribution of this article involves a new class of embeddings, called α-embeddings, which we introduce a one-parameter common generalization of harmonic embeddings and sembeddings.Given α ∈ R * , we call a quadrilateral ABCD drawn on the plane an α-quad if its side lengths satisfy AB α + CD α = AD α + BC α .The quadrilateral ABCD may be non-convex or even have its edges intersecting away from their endpoints.An α-embedding of a graph G is an embedding of its diamond graph G such that each face of G is drawn as an α-quad.This definition is firstly motivated by the introduction of a unified framework for properties of harmonic and s-embeddings, which correspond respectively to the cases α = 2 and α = 1.Secondly, the celebrated isoradial embeddings are α-embeddings for every α, since rhombi are α-quads for every α.More generally, kite embeddings are α-embeddings for every α, see Section 5. Thirdly, as we shall see, there is a whole range of parameters for which these quadrilaterals satisfy a version of a cube move.This property suggests the presence of an integrable system, and it would be remarkable to find a family of such systems indexed by a continuous parameter.
To state this version of the cube move, we introduce the weaker notion of an α-realization of G as a map from the vertices of G to the plane such that each face of G is mapped to an α-quad, with edges possibly intersecting.We also provide a definition of the above notions in the cases where α ∈ {−∞, 0, +∞}.In addition, we show that, for α > 1, the cube move is possible for α-realizations (although the solution may not be unique).See Theorem 4.7 for a precise statement.
Theorem 1.3.Let α > 1 be a real number.For any α-realization with the combinatorics of the graph on the left-hand side of Figure 4, erase the central black point and keep all the other points fixed.Then there exists a central white point that gives an α-realization with the combinatorics of the graph on the right-hand side of Figure 4.
One may wonder whether a probabilistic model can be associated to αembeddings beyond the cases where α is 1 or 2. In other words, is there a way to define the interaction constants of a probabilistic model from the local geometry of α-embeddings in such a way that the cube move for α-embeddings is conjugated to the star-triangle move for the probabilistic model?The FKpercolation model is a common generalization of the Ising model and of spanning trees (which have the same star-triangle transformation as random walks).This model also has a star-triangle transformation; however, it can be applied only when the weights satisfy some local constraint [17,11], whereas the startriangle transformation for the Ising model and spanning trees holds without any condition on the weights.We did not succeed in relating α-embeddings to FK-percolation for arbitrary weights that would satisfy the local constraint and do not expect such a general connection to hold true.Nevertheless there exists a subclass of weights among those satisfying the local constraint which can be associated to isoradial embeddings, that is, embeddings where the quadrilaterals are rhombi.In that case, the star-triangle transformation corresponds to a cube move for rhombus tilings, which exists and is unique.

Organization of the paper
We define in Section 2 our main object of interest, α-quads; we introduce αembeddings and α-realizations, before defining their cube move.Section 3 is devoted to the specific case α = 1, where we recall the connection with the Ising model and prove the existence and uniqueness of the cube move for properly embedded graphs.We also prove Theorem 1.2.In Section 4, we show that for α > 1, such a move is always possible under weaker conditions, which leads to the proof of Theorem 1.3.Finally in Section 5 we investigate some more basic geometric properties of α-quads and we propose a general framework to study a broader class of quadrilaterals.Appendix A contains a brief introduction to Jacobi elliptic functions, which are one of our main tools in the proofs of Section 3.

Definitions
This first section is devoted to the definition of the so-called α-quads -which are the main object of interest of this paper -of their embeddings and of their realizations.We introduce an operation on them called a cube move, before defining an important tool to prove this property, which we call construction curves.
In what follows, we denote by R * the set R \ {0} and by R the set R ∪ {−∞, +∞} of extended real numbers.For two points A, B in the plane, AB denotes the Euclidean distance between A and B, and we use a dot to write products of lengths such as AB.CD.

α-embeddings and realizations
Let us start by defining the notion of α-quads.
Note that the α-quads are not required to be convex nor even proper (see Definition 2.2), meaning that edges may intersect away from their endpoints.
One can immediately notice that α-quads (for α ∈ R) are simply quadrilaterals ABCD such that f α (AB, CD) = f α (AD, BC), where for x, y > 0 we set y).Such a notation calls for a generalization from f α to any homogeneous symmetric function f of two variables, which is done at the end of Section 5.
Of particular interest are the following three families of α-quads, which were already defined in a different context: • 1-quads correspond to tangential quads, i.e. quads such that there is a circle tangential to their four sides (see e.g.[13]) ; • 2-quads correspond to orthodiagonal quads, i.e. quads whose diagonals are perpendicular (see e.g.[14]) ; • 0-quads are known under the name of balanced quads [13] and contain a well-studied class of quads, the harmonic quads, which are defined as cyclic 0-quads (that is, 0-quads inscribed in a circle, see [2] for details).
Definition 2.2.For n ≥ 3, an n-tuple of distinct points A 1 , . . ., A n is said to be a proper polygon if the line segments do not intersect except possibly at their endpoints.In other words, the closed piecewise linear curve A 1 , A 2 , . . ., A n , A 1 is a Jordan curve.
A proper polygon is said to be positively (resp.negatively) oriented if the points on its boundary oriented counterclockwise (resp.clockwise) are A 1 , A 2 , . . ., A n in this order.
Let G be a planar graph, finite or infinite.Denote its dual graph by G * , whose vertices are faces of G and where we connect two dual vertices if the two corresponding faces share an edge.We construct the graph G as the bipartite graph whose black (resp.white) vertices are the vertices of G (resp.G * ) and where an edge connects a black vertex to a white vertex whenever the black vertex is on the boundary of the face associated with the white vertex.In particular, all the faces of G are quadrilaterals, see Figure 2.
The fact that G is embedded in the plane implies that the α-quads corresponding to internal faces are proper.In particular, for embedded graphs, edges cannot collapse to a point, two distinct edges cannot meet outside of their endpoints, faces have non-empty interiors and two distinct faces have disjoint interiors.This setting is the one preferred for the study of statistical mechanical models such as spanning trees and the Ising model.However, in the more general setting we consider, we need to allow drawings of graphs that are not necessarily embeddings.Definition 2.4.Let α ∈ R.An α-realization of G is defined to be any map from the vertices of G to the plane such that every face of G is mapped to an α-quad.
See Figure 7 for an example of a 4-embedding and a 4-realization of the same graph.
The combinatorics of a cube move for α-embeddings and αrealizations.
As mentioned in the introduction, there are two notable classes of examples of α-embeddings.The class of 1-embeddings of a planar graph corresponds to the class of s-embeddings defined by Chelkak in [6,7] (see also [24]), while the class of 2-embeddings corresponds to the class of harmonic embeddings [27,18].

The cube move: setting
We now define the property that we want to study on α-quads, which we call the flip property.This property states that it is possible to perform a cube move like that of Figure 6 locally on the realization or embedding, while keeping the global structure unchanged.Definition 2.5.For any α ∈ R, the set of all α-realizations is said to satisfy the flip property if, for any six distinct points in the plane A 1 , A 2 , . . ., A 6 such that A 1 , A 3 , A 5 (resp.A 2 , A 4 , A 6 ) are not aligned, the following are equivalent: • there exists a point A 0 such that A 0 , A 1 , . . ., A 6 is an α-realization of the graph on the left-hand side of Figure 6; • there exists a point A 7 such that A 1 , . . ., A 7 is an α-realization of the graph on the right-hand side of Figure 6.
The set of α-embeddings is said to satisfy the proper flip property if, in the equivalence of Definition 2.5, it is also required that the figures are αembeddings, and that each of the quadrilaterals is proper, with its boundary vertices oriented in the same order as in Figure 6.
In both cases, it is said to satisfy the unique (proper) flip property if it satisfies the (proper) flip property and if, in addition, when the points A 0 and A 7 exist they are unique.
Our ultimate goal is to understand for which values of α these properties are satisfied by the set of α-realizations or α-embeddings.In this direction, we notably prove Theorem 3.1 for 1-embeddings and Theorem 4.7 for α-realizations with α > 1, and conjecture a generalization to other values of α (see Conjecture 3.2).Note that it is already known that the set of 2-embeddings satisfies the unique proper flip property and that the set of 2-realizations satisfies the unique flip property.This can be proved using a theorem of Steiner, see [21,18].
There is a necessary condition on the outer hexagon A 1 . . .A 6 in order to be in a position to flip three quads: Proposition 2.6.Let A 1 , . . ., A 6 be six distinct points and let α ∈ R. Suppose that there is either a point A 0 producing an α-realization of the left-hand side of Figure 6 or a point A 7 producing an α-realization of the right-hand side of Figure 6.Then the side lengths of the hexagon A 1 . . .A 6 must satisfy Proof.When α = 0, formula (2.1) follows immediately from summing the three equations defining the three α-quads involving A 0 (resp.A 7 ).The case α = 0 works similarly.
The example on Figure 8 with α = 1 illustrates the fact that condition (2.1) is not sufficient to have an α-realization.Indeed, the existence of a point A 0 implies that condition (2.1) is satisfied, but there exists no point A 7 .We also point out that there is no analogue of Proposition 2.6 when α = ±∞.
Figure 8: A configuration for α = 1, with the corresponding construction curves.The green construction curves have an intersection point A 0 but the purple ones do not intersect.

Construction curves
In order to check the existence of the points A 0 and A 7 introduced in Definition 2.5, we will see them as intersection points of certain curves called construction curves.Let us first define them properly.Definition 2.7.Let A, B and C be three distinct points in the plane and let α ∈ R. The α-construction curve with foci A, B going through C is the set of points M such that ACBM is an α-quad.
Let A 1 , . . ., A 6 be the distinct vertices of a hexagon.The α-construction curves of the hexagon A 1 . . .A 6 are the six curves C i where for every 1 ≤ i ≤ 6, C i is defined to be the α-construction curve with foci A i−1 and A i+1 going through A i .Here indices are taken modulo 6. Remark 2.8.In order for the flip property to be satisfied when α ∈ R, it is actually enough to consider the intersection of two construction curves rather than three.Indeed, if α = 0 and if A 0 , A 1 , . . ., A 6 are seven points such that the quadrilaterals A 1 A 2 A 3 A 0 , A 3 A 4 A 5 A 0 and A 5 A 6 A 1 A 0 are all α-quads, then by Proposition 2.6 formula (2.1) is satisfied.Hence, if C 1 and C 3 have a common point A 7 , then combining the equation of the two α-quads A 2 A 3 A 4 A 7 and A 6 A 1 A 2 A 7 with formula (2.1) yields that A 7 also lies on C 5 .The case α = 0 works similarly, replacing formula (2.1) by formula (2.2).
In a few cases, the construction curves are well known.Fix A and B two points in the plane.If C is a point such that AC = BC, then clearly for any α the construction curve going through C is the perpendicular bisector of [AB], which we denote from now on by P (AB).Without loss of generality, we may assume in what follows that BC < AC.See Figure 9 for the general shape of construction curves depending on the value of α. • For α = 0, it is a circle going through C; • For α = 1, it is the branch closest to B of the hyperbola with foci A, B going through C; • For α = 2, it is the perpendicular to (AB) going through C.
The explicit description follows from a simple case handling, depending of the length realizing the minimum.The same goes for α = +∞.For α = 0, the curve is the set of points M such that the ratio M A M B is fixed, which is a circle by Apollonius's circle theorem.
For α = 1, the curve is the set of points M such that AM − BM is a fixed positive number, which is a branch of hyperbola with foci A, B.
For α = 2, as stated in Section 5, the quadrilateral ACBM is a 2-quad if and only if (AB) and (CM ) are perpendicular.

The cube move for α = 1
Our goal in this section is to prove the unique proper flip property for 1embeddings.
Theorem 3.1.The set of 1-embeddings satisfies the unique proper flip property.
Although our proof only works in this specific case, based on the observation of many numerical examples we expect the following more general result to hold: Conjecture 3.2.For any α ∈ [−∞, 1], for any six distinct points in the plane A 1 , A 2 , . . ., A 6 such that A 1 , A 3 , A 5 (resp.A 2 , A 4 , A 6 ) are not aligned, the following are equivalent: • there exists a unique point A 0 such that A 0 , A 1 , . . ., A 6 is an α-embedding of the graph on the left-hand side of Figure 6; • there exists a unique point A 7 such that A 1 , . . ., A 7 is an α-embedding of the graph on the right-hand side of Figure 6.
Notice that this is weaker than the unique proper flip property, as it is possible that several points A 0 exist and no point A 7 .Such a configuration is shown in the top-left picture of Figure 10.For the sake of completeness, let us also mention that the set of 1-realizations does not satisfy the flip property (when there is no "proper" requirement): there exist configurations such that there is a point A 0 yielding 1-realizations, but there is no point A 7 (see Figure 8).
The rest of this section is devoted to the proof of Theorem 3.1.

Uniqueness
In this first part, we prove the following proposition, which states the uniqueness of the point A 0 , if it exists.
Proposition 3.3.For any proper positively oriented hexagon A 1 , A 2 , . . ., A 6 , there exists at most one point A 0 such that A 0 , A 1 , . . ., A 6 is a proper 1-embedding of the left-hand side of Figure 6.
In order to prove it, we first need some information on the geometric properties of 1-quads.Notice that if three distinct points A, B, C are fixed, then the construction curve for 1-quads is the set of points D such that Hence, it is a hyperbola branch with foci A, C (with possible degenerate cases being the perpendicular bisector of [AC], and half-lines A + t(A − C) or C + t(C − A) for t ≥ 0).We put all these cases under the same name: Definition 3.4.Let A, C be two distinct points in the plane.For any λ ∈ R, the set of points D in the plane such that is called a generalised hyperbola branch with foci A and C.
The following lemma already implies that there are at most two admissible points A 0 , in the sense of Proposition 3.3.Although this result, which appears in [25,30], has a very classical flavor, we could not find any earlier reference.We give an alternative proof below.

Lemma 3.5. Assume that two generalised hyperbola branches have exactly one common focus, then they have at most two intersection points.
Let us start with some definitions.The focus of a hyperbola branch B which belongs to (resp.does not belong to) the convex hull of B is said to be the interior (resp.exterior) focus of B. For example, on Figure 11, A 1 is the interior focus of both branches while A 3 and A 5 are the exterior foci of these branches.Assume that B 1 is a hyperbola branch with foci A and C and B 2 is a hyperbola branch with foci A and E, with A, C, E distinct.Then any intersection point of B 1 and B 2 lies on a hyperbola branch B 3 with foci C and E (whose equation is obtained by subtracting the equations (3.1) for B 1 and B 2 ), and we can also suppose that B 3 is not a line or a half-line.Then there is at least one of the three points A, C or E which is the interior focus of one branch and the exterior focus of another branch.Without loss of generality, we assume that A is the interior focus of B 1 and the exterior focus of B 2 .We will show that these two branches intersect in at most two points.
Suppose that B 1 and B 2 intersect at three distinct points S, T, U .As these points belong to a non-degenerate hyperbola branch, they are not aligned.The lines (ST ), (T U ), (SU ) therefore delimit exactly seven open regions in the plane, three of which touch the triangle ST U only at one vertex.We call these three regions the corner chambers of S, T, U .Lemma 3.6.Let S, T and U be three distinct points on a hyperbola branch B. Then the exterior focus of B belongs to a corner chamber of S, T, U , while the interior focus does not.
Proof of Lemma 3.5.If one of the generalised branches is a line or a half-line, the result easily comes from the fact that a hyperbola and a line have at most two points of intersection.
Suppose now that it is not the case.The proof of Lemma 3.5 follows from Lemma 3.6, as A should be both in a corner chamber of S, T, U (because A is the exterior focus of B 2 ) and not in one (because A is the interior focus of B 1 ).
Proof of Lemma 3.6.For any two distinct points A, B on B, the line (AB) cuts the interior of the convex hull of B into a finite part and an infinite part.The half-plane delimited by (AB) that contains the finite part is called the exterior half-plane of A, B. It is easy to see that the exterior focus belongs to the exterior half-plane of A, B, for instance by noting that this property is invariant by affine transformations of the plane, and is straightforward to prove for the special branch {(x, y) ∈ (0, ∞) 2 | xy = 1}.
Suppose that T belongs to the exterior half-plane of S, U (which is equivalent to saying that S, T, U are met in that order when following the branch B).Then, applying the previous property to (S, T ) and (T, U ), we get that the exterior focus has to belong to the corner chamber that touches T .
We now consider the interior focus of B. By convexity of B, the corner chambers are disjoint from the interior of the convex hull of B, which by definition contains its interior focus.The result follows.
We can now prove Proposition 3.3.
Proof of Proposition 3.3.By Lemma 3.5, there exists at most two such points.Suppose that there are two, A 0 and A 0 .We claim that A 1 , A 3 , A 5 belong to a unique hyperbola branch with foci A 0 , A 0 .Indeed, since A 0 , A 0 are on the same hyperbola branch with foci A 1 , A 3 , we have which in turn means that A 1 , A 3 are on the same hyperbola branch with foci A 0 , A 0 .The same holds for A 5 .Therefore, by Lemma 3.6, one of the points A 0 , A 0 is in a corner chamber of the triangle A 1 A 3 A 5 , and the other is not.But notice that for any point M that does not belong to the lines (A 1 A 3 ), (A 1 A 5 ), (A 3 A 5 ), the cyclic order of the vectors is always the same when M belongs to the union of the three corner chambers, and the opposite when M is outside that union.However, the cyclic order of the vectors should be the same as the cyclic order of the vectors , since this order should be fixed by the proper embedding.Therefore it is impossible for both A 0 and A 0 to correspond to proper embeddings.

Existence
The second part of this section consists in proving that, if we start with a proper 1-embedding A 0 , A 1 , . . ., A 6 of the left-hand side of Figure 6, then there actually exists a point A 7 inducing a proper 1-embedding of the right-hand side.To show this, we transform the problem into a linear one by using the propagation equations and s-embeddings defined by Chelkak [6,7].We briefly explain this construction (we refer to Chelkak's original papers for more details), then give a few extra properties concerning the ordering of the vertices of the quads it gives, and finally apply it to our setting.

Ising model, propagation equations and s-embeddings
Let G := (V, E) be a finite planar graph, in which each edge e ∈ E carries a positive weight J e > 0. The weights (J e ) e∈E are called the coupling constants of the ferromagnetic Ising model on G, that is, every spin configuration σ ∈ {±1} V is assigned the Boltzmann weight and a spin configuration is randomly sampled with probability proportional to its Boltzmann weight.Note however that we will not refer to any statistical mechanical property of the Ising model thereafter; all the proofs are of purely geometric nature.
One checks that, for every e ∈ E there exists a unique θ e ∈ (0, π 2 ) such that

.2)
We also set Let G c be the weighted graph whose vertices are the corners of the faces of G, and whose edges are of two types: 1. those connecting two corners that correspond to the same vertex and to the same edge e of G; such edges carry the weight cos θ e ; 2. those connecting two corners that correspond to the same face and to the same edge e of G; such edges carry the weight sin θ e .
See Figure 12 for an illustration.There exists a double cover Υ × of G c that branches around every edge, vertex and face of G, graphically represented around an edge in Figure 13 (see also Figure 17), which inherits the edge weights of G c .Let V × be the vertices of Υ × .We say that a function X : V × → C satisfies the propagation equation if, for every v ∈ V × with neighbours v , v ∈ V × around an edge e like in Figure 13, It is easy to check that if X satisfies the propagation equation, its value is multiplied by −1 whenever we change sheets above a vertex of G c .If X is a solution to the propagation equation, then one can construct a function S : G → C, which is called the s-embedding associated to X, in the following way.
Fix the image S(u 0 ) of a base vertex u 0 of G in the plane.Then define S such that for every vertex u of G and every face f adjacent to it, denoting by c the corner between u and f , where X c is any of the two values of X above the corner c; both values produce the same constraint on S. See Figure 14 for an example.Notice that it is not clear a priori that S is well-defined, as one needs to check at least that conditions (3.4) are closed around an edge, as in Figure 14.We will rely on the following result of Chelkak [6] that asserts that the s-embedding S is welldefined, and identifies s-embeddings with our notion of 1-embedding.Proposition 3.7 ([6]).For any solution X of the propagation equation such that Re(X), Im(X) are two vectors independent over R, the associated s-embedding S is well-defined, and is such that every face of G is sent to a proper 1-quad in the complex plane.Conversely, for any 1-embedding T of G, for any edge e ∈ E let θ e be the unique angle in 0, π 2 such that, using the notation of Figure 3, Then T is an s-embedding associated to a solution of the propagation equations on G with parameters (θ e ) e∈E .
The plan of our proof of the existence part of the flip property is therefore to translate the initial configuration into a solution of the propagation equations, then show that one can apply a star-triangle transformation on this solution, and finally go back to embeddings.However, we need more information than what is contained in Proposition 3.7, as we want to keep track of the orientation of quadrilaterals.This is the aim of the following subsection.• (i) ⇒ (ii): Assume that (ii) is not verified, that is, Im(b) < 0. Then consider b, c, d, which is still a solution to the propagation equation as this equation has real coefficients.These values do satisfy (ii), hence also (iii) and thus (i) by the previous points.Note that they correspond to a quadrilateral that is the image by a reflection of our desired quad S(u)S(x)S(v)S(y), and therefore for our initial solution (i) does not hold.

Star-triangle transformation on propagation equations
This part consists in rephrasing Baxter's results on the star-triangle transformation of the Ising model (see Section 6.4 of [3]).It is slightly easier to present it in the triangle-star direction, i.e. from the right-hand side to the left-hand side of Figure 16, although the converse is also possible.
Let us suppose that the graph G contains a triangle, as on the right-hand side of Figure 16.We label its edges with the parameters θ i and define x i , J i as in (3.2) and (3.3).It is possible to transform the triangle into the star displayed on the left-hand side, while finding parameters such that both Ising models are coupled and agree everywhere except at A 7 .This is called the star-triangle transformation. .
Then the parameters θ 2 , θ 4 , θ 6 obtained by the star-triangle transformation are the unique angles in (0, π 2 ) such that where the labels of the angles are considered modulo 6.
Remark 3.10.This transformation may be expressed in different ways; in this remark, we recall a convenient elliptic parametrization due to Baxter [3], also used in [5].The previous definition of k naturally comes from the use of an elliptic modulus k ∈ iR ∪ [0, 1) such that k 2 + k 2 = 1; see Appendix A for a short introduction to elliptic functions.We can define elliptic angles τ 1 , . . ., τ 6 and θ 1 , . . ., θ 6 by where F (resp.K) is the incomplete (resp.complete) integral of the first kind, see Appendix A for definitions and details.The normalization is such that both θ and θ variables live in (0, π 2 ), while τ variables live in (0, K(k)).Then k can be seen as the only modulus such that the θ angles satisfy In these parameters, the star-triangle transformation simply reads [3,5] ∀i ∈ {1, 3, 5}, We emphasize that this simple formula requires the introduction of a local elliptic modulus, while Equations (1.1),(1.2) do not.
Proof.It is easy to see that any values of (y 1 , y 3 , y 5 ) uniquely characterize the solution of the propagation equation on the triangle graph.Thus, the set of possible vectors (w 1 , . . ., w 6 ) is a 3-dimensional subspace V .By setting (y 1 , y 3 , y 5 ) to be the elements of the canonical basis of C 3 and solving the propagation equation, we get a basis of V : Similarly, the set of values of (w 1 , . . ., w 6 ) for the star graph is a subspace V with basis Up to a cyclic shift, this is the same as (3.9), so it holds by the previous discussion.
As a byproduct of the previous proof, we obtain Theorem 1.2, which provides formulas expressing the change of θ parameters in the star-triangle transformation.These formulas are much simpler than the classical computation described in Proposition 3.9 and to the best of our knowledge, they are new.
Proof of Theorem 1.2.For i = 4, this follows from combining equations (3.8) and (3.9).Cyclic shifts of indices give the cases i = 2 and i = 6.
To obtain the other three values of i, we apply the Kramers-Wannier duality of the Ising model [22], which has the effect of transforming the variables θ i into π 2 − θ i .In this duality, the star graph becomes its dual, i.e. a triangle, and vice-versa.Hence the formula can be deduced from the previous one by changing sines into cosines and vice-versa.
We now have all the elements to prove the unique proper flip property of 1-embeddings.
Proof of Theorem 3.1.We start with A 0 , A 1 , . . ., A 6 , a proper embedding of the left-hand side of Figure 6.As uniqueness is a consequence of Proposition 3.3, we just have to prove that there exists a point A 7 such that A 1 , A 2 , . . ., A 7 is a proper embedding of the right-hand side.
By Proposition 3.7, there exists a solution (w 1 , w 2 , w 3 , w 4 , w 5 , w 6 , y 1 , y 3 , y 5 ) of the propagation equation as on the left-hand side of Figure 17 such that the points A 0 , A 1 , . . ., A 6 are the s-embedding of this solution.Hence, by Proposition 3.11, there exists (z 2 , z 4 , z 6 ) such that (w 1 , w 2 , w 3 , w 4 , w 5 , w 6 , z 2 , z 4 , z 6 ) is a solution to the propagation equation of the right-hand side of Figure 17.Let us consider its s-embedding.It has the same boundary as the initial onehence the points A 1 , . . ., A 6 are unchanged -and we have a new point A 7 .It remains to prove that the three new 1-quads are proper and oriented as on the right-hand side of Figure 6.Then w 1 /w 2 is not a real number.Indeed, if it were the case, the propagation equations would imply that the variables (w 1 , w 2 , w 3 , w 4 , w 5 , w 6 , z 2 , z 4 , z 6 ) all have the same argument, and the initial embedding would not be proper.Moreover the initial quad A 1 A 2 A 3 A 0 is proper and positively oriented.Hence we can apply Lemma 3.8 to obtain the ordering of the arguments of w 1 , w 2 , y 3 , y 1 .By doing the same for the other two initial quads, the order of the arguments of the variables w i and y i is that on the left of Figure 18.Let us prove that the new (proper) quad A 1 A 2 A 7 A 6 is positively oriented.By Lemma 3.8, it is enough to prove that Im(w 6 /w 1 ) > 0. If this is not the case, then we are in the situation of the second configuration of Figure 18, where all the arguments are included in a half-circle.Therefore we can get the order of the arguments of the w 2 i , as in the picture on the right of Figure 18.However, as in the proof of Lemma 3.8, the successive internal angles of the hexagon A 1 A 2 A 3 A 4 A 5 A 6 can be expressed as the successive oriented angles in direct order in this last figure, and in particular their sum is 2π.This contradicts the fact that the hexagon is non-crossed, as the sum should be 4π.By symmetry, the three new quads are properly oriented, which concludes the proof.Remark 3.12.The so-called exact bosonization operation provides a map from two independent Ising models on some planar graph G to a bipartite dimer model on a modified graph G [10].Under this correspondence, the Ising startriangle move corresponds to the composition of six local moves for the dimer model called urban renewals ( [20], Fig. 6).On the other hand, embeddings as centers of circle patterns are known to be adapted to bipartite dimer models for any fixed x M ≥ 0, lim y M →∞ F α (M ) = +∞.
if x M > 0 and y M > 0, for α < 2, ∂Fα ∂x (M ) < 0 and for α > 2, Proof.Firstly, all the claims on the regularity of F α are clear.Furthermore, the sign of this function on the quadrant (R + ) 2 is apparent as well, since M A ≥ M B with equality only if x M = 0.
Let us now deal with the asymptotic behaviour of F α .For α < 0, M B α goes to +∞ as M → B and therefore lim F α = −∞.On the other hand, for any α < 1, notice that by the mean value theorem, for any M there exists a where we used the triangular inequality.This goes to 0 as M B grows.In the case α > 1, let x M be fixed.For any large enough y M , by the mean value theorem, there exists a u ∈ For large y M , this is equivalent to αy α−1 M and so tends to +∞.We finally consider the partial derivatives of F α .For any M ∈ (R * + ) 2 , one can compute: and its sign is apparent.Regarding the second partial derivative of F α , we compute:

The space of α-quads and f -quadrilaterals
In this last section, we provide different geometric characterizations of α-quads.
It appears that one of them holds in the broader context of f -quadrilaterals (see Definition 5.5).The first characterization of α-quads is related to the existence in the quadrilateral of a so-called extremal pair.
Definition 5.1.A pair of opposite sides of a quadrilateral is called an extremal pair if these two sides achieve both the maximum and the minimum of the side lengths of the quadrilateral.
Proposition 5.2.A quadrilateral is an α-quad for some α ∈ R if and only if it has an extremal pair.
Proof.Let Q be a quadrilateral whose side lengths are denoted by 1 , 2 , 3 , 4 in cyclic order (starting from any of them).Assume that Q has no extremal pair.Then, without loss of generality, one may assume that 1 is the maximal length and 4 the minimal length, and they are distinct.Since ( 1 , 3 ) is not an extremal pair, 3 does not achieve the minimum.Thus, 3 > 4 and Q is not a −∞-quad.Similarly, 2 does not achieve the maximum; thus, 2 < 1 and Q is not a +∞-quad.By these inequalities, α and in any of these cases Q is not an α-quad).Furthermore, 1 3 > 2 4 and Q is not a 0-quad.Hence, Q is not an α-quad for any α ∈ R.
Conversely, suppose that Q has an extremal pair.We can assume that 1 is the maximal length, 3 the minimal length, and that 2 ≤ 4 (up to a possible mirror symmetry).If 2 = 3 or if 4 = 1 then Q is respectively a −∞-quad or a +∞-quad.Hence, we can assume that 3 < 2 ≤ 4 < 1 .Consider now the function g on R * defined as The function g can be extended to a continuous function on R by setting g(0) = ln 1 3 2 4 . By the previous inequalities, for α → −∞, g(α) ∼ α 3 α < 0 and for α → +∞, g(α) ∼ α 1 α > 0. Hence, by the intermediate value theorem, there exists an α ∈ R such that g(α) = 0 and Q is an α-quad.
We now characterize quadrilaterals that are α-quads for at least two distinct values of α.Definition 5.3.A quadrilateral ABCD is said to be a kite if its side lengths satisfy {AB, CD} = {BC, DA}.
Remark in particular that a kite is an α-quad for all values of α ∈ R.
Proposition 5.4.Let α = α ∈ R, and let Q be a quad which is an α-quad and an α -quad at the same time.Then Q is a kite.
This generalizes a result of [13] which claims that a quad which is both a 0-quad and a 1-quad is a kite.
Proof.Denote by 1 , 2 , 3 , 4 the side lengths of Q in cyclic order.Assume first that neither α nor α take the values −∞, ∞ or 0.Then, up to replacing i by α i , one may assume that α = 1.Write s = 1 + 3 and t = α 1 + α 3 .Since Q is a 1-quad, we have 4 = s − 2 .Since Q is also an α-quad, we have Assume now that α = 0 and α is finite.One may again assume that α = 1, in which case the sums and products of each pair {l 1 , l 3 } and {l 2 , l 4 } are equal, and Q is again a kite.If α ∈ {−∞, ∞} and α is finite non-zero, one may again assume that α = 1 and the conclusion follows easily.The case when α ∈ {−∞, ∞} and α = 0 is similar.Finally the case when {α, α } = {−∞, ∞} is also easy.
We finally provide another characterization, involving circumradii of the four triangles delimited by the sides and diagonals of the quad.This last characterization actually holds in a more generic setting, which we now introduce.In what follows, we say that a symmetric function of two variables f : (R + ) 2 → R is homogeneous if there exists u ∈ R such that for any λ, x, y > 0, we have f (λx, λy) = λ u f (x, y).Multiplying f by a non-zero scalar, or more generally post-composing it with a bijection, produces the same class of quads.As an example, for α ∈ R * , α-quads correspond to the function f α : (x, y) → x α + y α .By definition, 0quads, +∞-quads an −∞-quads are also f -quadrilaterals for some homogeneous function.
Proposition 5.6.Let ABCD be a quad.Let P denote the intersection point of its diagonals and suppose that P is distinct from A, B, C, D. We denote the respective circumradii of the triangles ABP, BCP, CDP and DAP by R 1 , R 2 , R 3 and R 4 .Let f : (R + ) 2 → R be a symmetric homogeneous function.The following are equivalent: This result was already known for orthodiagonal quads and tangentials quads [14] and our proof below is a straightforward extension of the proof of [14,Theorem 9].

Figure 3 :
Figure 3: Relation between an abstract weighted graph and an s-embedding.

Figure 4 :
Figure 4: The transformation on G induced by the star-triangle transformation on G: a cube move.

Figure 5 :
Figure 5: The star-triangle transformation for the Ising model is conjugated to a local geometric transformation called a cube move, via Chelkak's s-embeddings introduced in [6, 7].Notice that in the embedded graphs, only the central points P and P differ.

Figure 7 :
Figure 7: Here α = 4. Left: an α-embedding of the portion of the graph on the left-hand side of Figure 6.Right: an α-realization of the portion of the graph on the right-hand side of Figure 6.The quad A 4 A 5 A 6 A 7 on the right is not proper.

Figure 9 :
Figure 9: Construction curves with foci A and B going through C. The fact that they intersect only at C and at its symmetric with respect to (AB) is a consequence of Proposition 5.4.The asymptotic behavior of construction curves is described in Lemma 4.6.

Figure 10 : 7 . 2 . 9 .
Figure 10: Examples of construction curves for some hexagons.Top left: for α = 0, two possible points A 0 and no point A 7 .Top right: for α = 0, two points A 0 and two points A 7 .Pictures with the same number of solutions as the top two pictures can be obtained for α close to 0. Bottom: for α = 10, three points A 0 and one point A 7 .

Figure 11 :
Figure 11: A case of two branches intersecting at two points.The focus A 1 is the interior focus of both branches.

e sin θ e cos θ e Figure 12 :
Figure 12: The corner graph G c (dashed) around an edge e of G (solid).

Figure 13 :
Figure 13: The double cover Υ × around the edge e of G.

2 Figure 14 :Lemma 3 . 8 .Figure 15 := cos 2 θ
Figure14: An edge e ∈ E with vertices u, v and adjacent faces x, y, with a solution of its propagation equation, and the corresponding s-embedding, where the numbers a 2 , . . ., d 2 are the complex coordinates of the vectors bounding the quad.

Figure 18 :
Figure 18: The cyclic order of the arguments of the variables w i and y i .

Definition 5 . 5 .
Let f : (R + ) 2 → R be a non-constant homogeneous symmetric function of two variables.A quadrilateral ABCD is called an f -quadrilateral if: f (AB, CD) = f (BC, AD).

Proof.B 2 .B 2 ∈ 3 = h −α 2 + h −α 4 ,Question 5 . 8 .
Let us denote the centers of the circumcircles of ABP, BCP, CDP, DAP by respectively O1 , O 2 , O 3 , O 4 .Since AO 1 B is isoceles in O 1 , we have AB = 2R 1 sin AO 1Since P lies on the circle centered at O 1 and going through A and B, we have thatAO 1 AP B, π − AP B , hence AB = 2R 1 sin AP B.Similarly we haveBC = 2R 2 sin BP C CD = 2R 3 sin CP D DA = 2R 4 sin DP A Observing that AP B = CP D = π − BP C = π − DP A, we deduce that the quadruples (AB, BC, CD, DA) and (R 1 , R 2 , R 3 , R 4 ) are proportional.Since f is homogeneous, f (AB, CD) = f (BC, DA) if and only if f (R 1 , R 3 ) = f (R 2 , R 4 ).This concludes the proof.Remark 5.7.Let h 1 , h 2 , h 3 and h 4 be the altitudes through P in the triangles ABP , BCP , CDP and DAP .For α ∈ {1, 2}, ABCD is an α-quad if and only if h −α 1 + h −α see [14,Section 5].However this does not hold for general values of α.We end this section with an open question.Apart from f α , are there homogeneous functions f such that f -quadrilaterals satisfy one of the flip properties discussed in this paper?