

CUBO A Mathematical Journal
Vol.18, No¯ 01, (69–88). December 2016

Parametrised databases of surfaces based on Teichmüller
theory

Armando Rodado Amaris1, Gina Lusares2

1 Departamento de Ciencias Exactas,

Universidad de los Lagos,

Av. Fuchslocher 1305, Osorno, Chile

2 Departamento de Estadistica,

Universidad de Valparaiso,

Blanco 951, Valparaiso, Chile

armando.rodado@ulagos.cl, gina.lusares@postgrado.uv.cl

ABSTRACT

We propose a new framework to build databases of surfaces with rich mathematical

structure. Our approach is based on ideas that come from Teichmüller and moduli

space of closed Riemann surfaces theory, and the problem of finding a canonical and

explicit cell decomposition of these spaces. Databases built using our approach will have

a graphical underlying structure, which can be built from a single graph by contraction

and expansion moves.

RESUMEN

Proponemos un nuevo marco teórico para construir bases de datos de superficies con

rica estructura matemática. Nuestro enfoque está basado en ideas que vienen de teoŕıa

de espacios de Teichmüller y espacios módulares de superficies de Riemann cerradas,

y el problema de encontrar una descomposición celular canónica y expĺıcita de estos

espacios. Las bases de datos construidas usando nuestro enfoque tendrán una estructura

gráfica subyacente, la que se puede construir a partir de un solo grafo por movimientos

de expansión y contracción.

Keywords and Phrases: Database of Surfaces Design Teichmüller space Moduli Space of Rie-

mann surfaces Canonical cell decomposition of Riemann surfaces Teichmüller surfaces descriptor.

2010 AMS Mathematics Subject Classification:


70 Armando Rodado Amaris, Gina Lusares CUBO
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1 Introduction

The idea of surface deformation in real life is common and increasingly important for sciences and

industry. For example, it is known that in the automotive industry around 90% of the time a new

surface design start with the smooth modification of a prototype surface [1]. A process that mirrors

the exploration of the possible paths in the Teichmüller or moduli space of Riemann surfaces. This

suggests the need of encouraging the application of these theories, which already have been applied

for instance to neurosciences [24, 25], but it is still basically unexploited.

A systematic study of surfaces, its parametrisation and classification are tasks of real importance

in applied sciences and industrial applications. Formal studies on spaces of surfaces can be traced

back to Riemann and to the work of Teichmüller [7, 11, 18]. Nowadays moduli and Teichmüller

spaces theories are deep and active branches of mathematics with many connections to many other

fields.

Hyperbolic geometry [16, 19] plays an important role in the understanding of Riemann surfaces

because by the classical uniformization problem each Riemann surfaces S of genus g ≥ 2 can be

represented as a quotient of H/G, where H is the hyperbolic upper half plane and G is a discrete

group of Mobiüs transformations keeping H invariant.

The quest to find a canonical and explicit cell decomposition of the moduli space of closed Rie-

mann surfaces lead to combinatorial structures and circle pattern systems. Indeed, a combinatorial

analysis shows the existence of a family of graphs that contains all possible graphs corresponding to

the 1-skeleton of the Voronoi cells, determined by the characteristic set W of S; and circle pattern

systems of equations come from the study of realization problems of graphs. In addition, circle

patterns systems lead to a polytope Pg complex which can be viewed as a parametrisation of Tg
for the genus two case [2], and more generally for the hyperelliptic locus of the Teichmüller and

Moduli spaces of closed Riemann surfaces for genus g ≥ 2.

The development of applications of moduli spaces of Riemann surfaces theory can be facilitated

by assigning a computable combinatorial structure to each surface S on the moduli space, Mg,

or the Teichmüller space, Tg, where g ≥ 2. Combinatorial structures can be based on W, a

characteristic set of points on S, which usually is the set of Weierstrass points on S, because it

carries a lot of information about a Riemann surfaces as shown on [2, 12, 10, 14, 15]. However, for

specific application to choose a different set could be sensible.

Since a surface S embedded in a 3-dimensional space has a conformal structure and, if it has a

negative Euler characteristic, it can be provided of a hyperbolic metric. [24, 25], We can define a

new surface descriptor from the embedding of S on the Poincare disk by following the next steps:

(1) compute the uniformization metric of the surface S

(2) get a conformal model S̄ of S defined as a quotient of the Poincaré disc D by a suitable

Fuchsian group


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Surface databases based on Teichmüller theory 71

(3) compute the set W of Weierstrass points of S̄

(4) assign the Voronoi diagram, A(S), determined by W to S

(5) assign a circle intersection angle θi to each edge i of A(S)

(6) define the descriptor Dθ(S) for the surface S by Dθ(S) = (A(S), Θ = (θi)i).

For any genus Dθ(S) depends only on the Teichmüller class of S up to labelling, and circle

pattern coordinates -intersection angles- can be translated to Teichmüller coordinates [2]. The

technical procedures that are needed to build Dθ(S) for the first and second steps are well known

[24, 25].

The location of the Weierstrass points for the genus two case are described on [10, 14, 15]. To

find the locations of the Weierstrass points of a Riemann surfaces is in general challenging, for this

reason we mainly study the descriptor Dθ(S) for genus 2 surfaces. Our ideas show a pathway for

the implementation of the general case.

Dθ(S) for genus 2 surfaces can be simplified and associated to a linear system, whose solutions

are on a polytopes complex. This will allow us to approach surfaces descriptions on a marked

polytope complex- each polytope marked by a graph. We propose the polytope complex that arise

as a natural structure to support database of surfaces.

In the sequel, we will describe the theory of graphs associated to Riemann surfaces based on

Weierstrass points, the theory of linear systems associated to graphs, explain the construction of

a polytope complex based on the previous ideas, and describe our proposal for databases designs

based on Teichmüller theory.

2 Graphs associated to Riemann Surfaces

For each Riemann surface S with its hyperbolic metric, the 1-skeleton of the Voronoi cell decom-

position determined by the Weierstrass points of S is a unique graph, Â(S), which is a subset of S

and depends only of the class of S in the Teichmüller space, Tg, and in the moduli space Mg, for

any genus g [2].

Determining which graphs are associated to Riemann surfaces based on Weierstrass points in

general is a difficult problem. We restrict ourselves to the hyperelliptic case, which allow us to

find all graphs associated to Riemann surfaces computationally by considering a similar problem

on the 2-dimensional sphere. Indeed, the hyperelliptic involution, τ, of S induces an action on S

such that S/τ is the two dimensional hyperbolic sphere. S/τ has exactly 2g + 2 cone points which

has measure π. Then, τ projects Â(S) into a graph on S/τ that we denote by A(S). By a standard

lifting procedure [2] we can recover S from the marked sphere S/τ. This shows the existence of

a one to one correspondence between set of hyperelliptic surfaces of a given genus and a set of

graphs, assigning S to A(S).


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Figure 1: The Weierstrass point on a hyperelliptic surface S of genus g are the 2g+2 fixed points
of its unique hyperelliptic involution. Here, we represent the genus 2 case.

Definition 2.1. The graph, A(S), associated to a hyperelliptic Riemann surface (S, τ) of genus g

is the image on S/τ under τ of the Voronoi graph determined by the Weierstrass fixed points of S.

Proposition 2.1. If S is a hyperelliptic hyperbolic Riemann surface of genus g, its associated

graph A(S) satisfies the following properties:

(1) G is connected

(2) G does not have monogons

(3) G divides S2 into 2g + 2 regions

(4) All vertices of G have valence ≥ 3.

The above properties of A(S) motivates the following definition of the family of CE(g) graphs.

Definition 2.2. A CE(g) graph is a connected graph G with vertices of valence greater than two

that can be embedded in the sphere S2, determining 2g + 2 regions and having no monogons. If the

valence of each vertex of G is 3, then we say that the graph is generic.

2.1 Generic graphs: the genus two case

To find all the possible generic graphs in the genus two case, we take advantage of the fact that

all generic graphs are connected by Whitehead moves and do not have any monogons. We find

20 generic graphs G1, G2, . . . G20 which are subset of the two dimensional sphere, 17 of which are

non-isomorphic, see Figure 3.

Counting the number of sides on each of the faces of these graphs, we always get six numbers

which are arrange in a non-increasing order. This gives a natural labelling to each generic graph,


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Surface databases based on Teichmüller theory 73

Figure 2: A Whitehead move on the red edge on graph G1, contracts the edge to a point, as
shown on the middle graph, followed by an edge expansion as shown on the right graph. The result
is the new graph which is represented on the right.

called the face labelling, which is unique except for the generic graphs G11 and G12. However, this

exception is not really important because G11 is not associated to a Riemann surface of genus two.

On Figure 4, the combinatorics of genus two generic graphs is represented by a graph whose

nodes are all possible generic graphs. We join graph Gi with Gj by an edge if there is a Whitehead

move transforming Gi into Gj.

2.2 Stratification of CE(g)

An immediate consequence of the Euler characteristic formula of the sphere is that any generic

graph that belong to CE(g) has 4g vertices, 6g edges and by definition 2g + 2 faces. Then, the

family CE(g) of graphs is stratified in 4g − 1 levels. Indeed, since the number of faces in all CE(g)

graphs is 2g + 2, and each vertex is of valence not smaller than 3 then 3v ≤ 2e. Hence, by the

Euler characteristic formula of the sphere v ≤ 4g.

The maximum number of vertices that a CE(g) graph can have is 4g and the minimum number

is 2, because no monogons are allowed. Thus, CE(g) has 4g−1 levels, if we define the k-th stratum

of CE(g) as the set of all graphs in CE(g) that have exactly 4g − (k − 1) vertices.

The members of the first stratum of CE(g) are all cubic graphs and is easy to verify that graphs

at this level has 2g + 2 faces and 6g edges. This stratum of CE(g) also has great importance since

any graph of CE(g) in a higher strata can be obtained by contraction moves starting from a graph

of the first strata.

The fact that cubic graphs are connected by a sequence of Whitehead moves [3], allow us to view

CE(g) as generated by any of its cubic graphs by sequences of Whitehead moves and contraction


74 Armando Rodado Amaris, Gina Lusares CUBO
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Figure 3: Genus two generic graphs and their face labelling-l[f1f2f3f4f5f6].

moves. In particular, any graph in CE(2) can be connected to the graphs which are illustrated on

Figure 3.

For another equivalent point of view, any stratum of CE(g) can be obtained inductively by

contracting each graph of the previous strata by a single contraction move on one of its edges in

such way that no monogon is created. Furthermore, if we define a generalised Whitehead move on

a graph as the graph obtained by contracting one edge and expanding a new edge in such way that


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Surface databases based on Teichmüller theory 75

no monogons are created, we have that two graphs at a fixed strata are connected by a sequence

of generalised Whitehead moves.

Finally, to complete a view of how graphs are connected on CE(g), its most contracted graph can

generate all graphs by a sequence of contraction and expansions moves. Then, CE(g) is analogous

to a universe that can be generated by transforming any of its graphs by contraction or expansion

moves from a single graph.

3 Generic graphs on the Teichmüller space

Informally, the Teichmülller of Riemann surfaces of genus g, Tg is a space whose elements are classes

of marked surfaces, and paths on Tg can be viewed as deformation of a surface. This intuitive idea

suggests that Tg or an equivalent space could be an ideal space to model the deformation of real

surfaces. For a background on Teichmülller theory see [18].

Next, we list a few facts about Teichmülller theory,

(1) Let S be a fixed Riemann surface of genus g. A marked surface is pair (R, [f]), where R is

a Riemann surface, [f] is the homotopy class of a homeomorphism f : S → R. Two marked

surfaces (S, [f]) and (S′, [f′]) are equivalent if there is a conformal map g : R → R′ such that

[g ◦ f] = [f′].

(2) The Teichmüller space is the set of marked classes.

(3) The Teichmüller space Tg,p has a natural topology, which makes it homeomorphic to an open

set of R6g−6+2p, where g and p are the genus and the number of punctures of each surface

in a class of Tg,p. On this space, we will only consider the case when p = 0.

(4) The Teichmüller space Tg,p can be parametrised by Fenchel-Nielsen coordinates [18]

In addition to the above, if G is a cubic graph associated to S, then G can be associated to an

open subset of Tg, with its Teichmüller metric. The Fenchel-Nielsen coordinates of Tg allow us to

see it as a 6(g − 1) real dimensional manifold, and informally we could imagine a deformation of S

by slightly changing its Fenchel-Nielsen coordinates which cannot change the graph G because all

its vertices are trivalent and then G is stable under small perturbations.

The following properties which are proved in [2] establish the relationship between generic graphs

and the Teichmüller space.

Proposition 3.1. If τ is a hyperelliptic involution of a hyperelliptic Riemann surface R and h :

R → R′ is an isometry, then R′ is hyperelliptic and its hyperelliptic involution is τ′ = h ◦ τ ◦ h−1.

Proposition 3.2. The associated graphs corresponding to two equivalent marked surfaces (R, [f]),

(R′, [f′]) are equal.


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Denote by ÕG, the set of all points in the Teichmüller space with associated graph G, where G

is a graph embedded in S2.

Proposition 3.3. If G is a cubic graph, then ÕG is an open set of the Teichmüller topology of Tg.

4 Linear systems associated to graphs

4.1 Delaunay realization problem

If a graph G on the 2-sphere is the associated graph to a hyperelliptic surface S of genus g, then

G is the boundary of a cell decomposition of the 2-sphere which is S/τ, with 2g + 2 faces, each one

containing an interior point with cone angles equal to π. By lifting S/τ to its two fold branched

covering space S can be recovered.

The lifting of S/τ is standard. However, for a given graph G, the problem of finding a hyperbolic

metric on the sphere with 2g + 2 π-cone angles whose associated graph is G is not trivial. We call

this problem the Delaunay Realization Problem (DRP).

A solution to the DRP defined by G can be identified with a unique hyperbolic surface S and

also with a collection of circles with a set of intersection angles. This connects the realizability

problem that we have described with the theory of circle patterns, which can be traced back to

Koebe’s work (1936), E.M. Andreev’s work (1970) and Thurston [13], and has had great impact in

many fields including conformal mapping, complex analysis, Teichmüller theory, brain mapping,

random walks, tilings, minimal surfaces and integrable systems, numerical analysis, metric spaces

and more [21].

4.2 Circle patterns and quotients

Given a closed Riemann surface S, and a cell decomposition {Ci}i∈I of S, which might have cone

singularities at a vertex or at the center of a cell, we say that a circle pattern is a configuration

of disks {Di}i∈I, where the boundary of each Di contains all the vertices of Ci and no vertex of

the cell decomposition is in the interior of any Di. In a circle pattern, for each edge e of the cell

decomposition, two circles have as intersection the extremes of e.

Let Ŝ be a hyperelliptic Riemann surface with metric d̃ and hyperelliptic involution τ. The

quotient space S = Ŝ/τ is also a metric space with the quotient metric d. We are interested

in Delaunay circle patterns where the vertices of the circle pattern are the fixed points of the

hyperelliptic involution. We will project a circle pattern on Ŝ to a circle pattern of the quotient

S/τ.

Proposition 4.1. A circle pattern of a cell decomposition of a hyperelliptic Riemann surface S of

genus g is projected to a circle pattern of the sphere by the hyperelliptic involution of S.


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Surface databases based on Teichmüller theory 77

To study circle patterns, several variational approaches have been introduced and several circle

packing results were proved using different functionals [23].

Figure 4: Above, we show the 10 realizable generic graphs with their groups of rigid symmetries.
Note that Whitehead moves on red edges are prohibited.


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4.3 Delaunay circle patterns, Voronoi cells and duality

A set of points F = {P1, P2, . . . , Pn} on a Riemann surface S determines a Voronoi decomposition

V of S: this is a cell decomposition determined from the points Pi ∈ F by taking the sets of points

closest to Pi, for each i. The open 2-cells are sets of form

Vi := {p ∈ S : d(p, Pi) < d(p, Pj) ∀j ∈ Ii}

where Ii = {1, 2, . . . , n} − {i}. Thus, S = ∪iV̄i and G = ∪i(V̄i − Vi) is a graph whose edges are

geodesic segments.

The dual cell decomposition V ′ of V is by definition constructed by joining two vertices P1 ∈ F

and P2 ∈ F by a geodesic segment, one for each common edge of the corresponding Voronoi cells

V1 and V2. The collection V
′ is itself a cell decomposition of S with the property that each of its

cells is inscribed in a unique circle: the collection of such circles is a Delaunay circle pattern for

V ′. Conversely, if we start with a cell decomposition V ′ of S which has a Delaunay circle pattern,

by joining the centers of adjacent circles we get a Voronoi cell decomposition of S whose centers

correspond to the vertices of V ′.

A Delaunay decomposition of a constant curvature surface is a cellular decomposition such that

the boundary of each face is a geodesic polygon which is inscribed in a circular disc, and these

discs have no vertices in their interiors.

The Poincare-dual decomposition of a Delaunay decomposition with the centers of the circles

as vertices and geodesics edges is a Voronoi cell decomposition. A Delaunay type circle pattern is

the circle pattern formed by the circles of a Delaunay decomposition. We will allow the surface

to have cone-like singularities in the hyperbolic metric at the vertices of Delauney decomposition,

and centers of the circles:

A k-cone cell is a hyperbolic polygon with interior cone point obtained from a collection of

hyperbolic polygons glued together cyclically around a common vertex, by isometric identification

of edges, such that the sum of angles at the vertex is k.

A cellular decomposition of a surface with n-cone singularities is a collection {Cj} of cone kj-cone

cells such that each side of each cell has been glued to a unique side of another cell (possibly the

same), by hyperbolic isometry.

From a Delaunay type circle pattern, one may obtain the following data:

• A cell decomposition Σ of a 2-dimensional manifold.

• For each edge e of Σ, the exterior (respectively interior) intersection angle θe (respectively

θe
∗ := π − θe). Thus, 0 < θe, θ

∗
e < π.

• For each face Σ, the cone angle Φf > 0 of the cone-like singularity at the center of the circle

corresponding to f. If there is no cone-like singularity at the center, then Φf = 2π.


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Surface databases based on Teichmüller theory 79

Note that the cone angle θv at a vertex v of Σ is determined by the intersection angles θe:

θv = Σθe (1)

where the sum is over all edges e around v.

Next, we present Theorem 1.8 (ii) on [20] and [5] for the case of a closed oriented surface, the

main tool that we use on this paper.

Theorem 4.1 (Springborn). Let Σ be a cell decomposition of a closed oriented surface. Suppose

the interior intersection angles are prescribed by a function θ∗ ∈ (0, π)E0 on the set E0 of edges.

Let Φ ∈ (0, ∞)F be a function on the set F of faces, which prescribe, the cone angle corresponding

to a face. A hyperbolic circle pattern corresponding to this data exists if and only if the following

condition is satisfied:

If F′ ⊆ F is any nonempty set of faces and E′ ⊆ E0 is the set of edges which are incident with

any face f ∈ F′, then

Σf∈F′Φf < Σe∈E′2θe
∗, (2)

If it exist, the circle pattern is unique up to hyperbolic isometry.

The above observations can be integrated into a system of linear inequalities L(G, σ) to solve

the Delaunay realization problem for a Delaunay graph G′ embedded in S2 with edge-labeling σ,

dual to G, as a consequence of Springborn theorem.

Below L(G, σ) is defined :

L(G, σ) =

⎧
⎪⎨

⎪⎩

2πQ(i) < Σ12j=12P(i, j)θ
∗
j , for each i ∈ {1, 2, . . . , 255}

Σk∈Jv(π − θk
∗) = π, for v = 1, 2, . . . , 6

0 < θj
∗ < π, for j = 1, 2, . . . , 12

(3)

where Jv is the set of edges incident with vertex v, Q(i) is the number of faces in the subset i

of faces, and P(i, j) is the characteristic function, which is 1 if the edge j belongs to the subset i

and 0 otherwise.

If L(G, σ) has a solution then, by Springborn theorem there is a hyperbolic circle pattern

inducing a Delaunay cell decomposition isomorphic to G′. Hence, the two-fold covering space of

the sphere that realize one of the solutions is a Delaunay triangulated Riemann surface of genus

two whose dual graph projects to G′, solving the Delaunay problem.

The above system can be solved using commercial computer programs that work even with

thousand of constraints [4]. However, we do not need to do so for the genus two case because

we can reduce the linear system that we obtain substantially, which help us to understand the

structure of the solutions.


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5 Solutions for the genus 2 case

To approach the problem of finding which of the 20 generic graphs that we found for the genus

two case are realizable, we used the package Convex [6] which solves linear systems using symbolic

algebra and allows the computation of exact solution.

Solving the reduced system, with only angle equalities and the angle constraints of the type

0 ≤ θ ≤ 1, we obtained that there are at most 10 generic graphs which are realizable. Then, we

showed that the face constraint of the systems associated to generic graphs for the genus two case

are consequence of the equations and angle constraints of the systems, which allow us to claim

that there are exactly 10 realizable generic graphs associated to Riemann surfaces of genus two.

Proposition 5.1. There are at most ten genus two generic Delaunay realizable graphs.

As an additional conclusion from our computation, we can say that the face labelling of generic

graphs is unique, e.g. Delaunay realizable generic graphs are uniquely determined by their labelling.

Then, it is convenient to relabel the realizable generic graphs by ascending lexicographic ordering

as d1, d2, . . . d10, Table 1.

Generic Delaunay realizable graphs face labels

d1 444444

d2 554433

d3 555522

d4 654432

d5 663333

d6 664422

d7 754332

d8 773322

d9 854322

d10 943332

Table 1: Generic Delaunay realizable graphs

5.1 Independence of face constraints for generic linear systems

In this section, we will prove that all the 10 Delaunay realizable generic graphs are determined by

six angle inequalities and six linear equations, corresponding to an angle equality for each vertex

of a dual graph: from these all face inequalities follow, and are thus redundant. This radically

improves our ability to understand these polytopes and corresponding realization solutions. Each

polytope is obtained as a cube cut by hyperplanes in R6.


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Surface databases based on Teichmüller theory 81

Our main result on this section is that for each Delaunay realizable generic graph the angle

constraints can be deduced from angle constraints. In other words, each system L(di, σ) is face

constraint independent.

5.2 Face inequalities for more than 3 triangles

If G is a generic graph then its dual G◦ determines a triangulation T of S2 that has 255 non-empty

subsets of triangles, since T has 8 triangles. Let Ci, i = 1, 2, . . . , 255 be the collection of non-empty

subsets of T and let Ei be the set of labels of edges which belong to Ci. The face constraint for

the linear program L(G, σ) corresponding to Ci is

Q(i) <
12∑

j=1

P(i, j)θ∗j (4)

where θj
∗ is the normalized exterior angle for the labeled edge j (we divided each angle by π to get

0 < θj
∗ < 1), and Q(i) = card(Ci). Recall that P(i, j) = 1 when the edge labeled j ∈ Ei belongs

to one of the triangles in the subset Ci, where Ei ⊂ {1, . . . , 12}.

From the above constraint, we can say that

Q(i) <
12∑

j=1

P(i, j)θ∗j =
12∑

j=1

θ∗j −
∑

j̸∈Ei

θ∗j

The following proposition reduces drastically the number of constraint that we need to consider

[2].

Proposition 5.2. Let G be a Delaunay realizable generic graph with labelling σ corresponding to a

hyperelliptic Riemann surface of genus g. Then, the system L(G, σ) associated to G is independent

of all face constraints corresponding to subsets of triangles with more than 2g − 1 triangles.

An immediate consequence of the above proposition is that in the genus two case, polytopes

associated to a Delaunay realizable graph can only depend on face constraints corresponding to

subsets with one, two or three triangles.

Corollary 5.1. Let G be a Delaunay realizable generic graph with labelling σ corresponding to a

hyperelliptic Riemann surface of genus 2. Then, the system L associated to G is independent of all

face constraints corresponding to subsets of triangles with more than 3 triangles.

The solutions of the angles systems associated to all generic graphs who are feasible are given

on Table 2.

Proposition 5.3. The linear system L(di, σ) for i = 1, 2, . . . , 10 is independent of its face con-

straints.


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1 2 3 4 5 6 7 8 9 10 11 12 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10

P1 1 1 1 1 1 1 1 0 1 0 1 0 + + + +

P2 1 1 1 1 1 1 0 1 0 1 1 0 + + + + + + +

P3 1 1 0 1 1 1 1 1 1 1 0 1 + + + + + +

P4 0 1 1 0 1 1 1 1 1 1 0 1 +

P5 1 0 1 1 1 0 1 1 1 0 1 1 + +

P6 0 1 1 1 0 1 1 1 0 1 1 1 +

P7 1 1 0 1 0 1 1 0 1 1 1 1 + + + + + + + + +

P8 1 0 1 0 1 1 0 1 1 1 1 1 + +

P9 0 1 0 1 1 1 0 1 1 1 1 1 + + + + + + + + +

P10 1 1 0 1 1 1 0 1 1 0 1 1 + + + + + +

P11
1
2

1 1 1 1 1
2

1 1 1
2

1
2

1
2

1
2

+

P12 1 1
1
2

1
2

1 1 1
2

1
2

1 1 1
2

1
2

+ + + + + + + + + +

P13
1
2

1
2

1 1
2

1
2

1 1 1
2

1 1
2

1 1 +

P14 1
1
2

1
2

1 1
2

1
2

1
2

1 1
2

1 1 1 + + +

P15 1
1
2

1
2

1 1 1 1
2

1
2

1 1 1
2

1
2

+ + + + + + +

P16 1 1
1
2

1
2

1 1
2

1
2

1
2

1 1 1 1
2

+ + + +

P17 1
1
2

1
2

1 1 1
2

1
2

1
2

1 1 1 1
2

+ + + +

P18 1 1 0 1 1 1
1
2

1
2

1
2

1 1
2

1 + + +

P19 1 1 0 1 1
1
2

1
2

1
2

1
2

1 1 1 + +

P20 1 1
1
2

1
2

1
2

1 0 1 1 1 1 1
2

+

P21 1
1
2

1
2

1 1
2

1 0 1 1 1 1 1
2

+

P22 1 1 0 1
1
2

1 0 1 1
2

1 1 1 +

Table 2: On this table, the 22 vertices P1, P2, . . . P22, of the solutions of all angle systems for
generic graphs of genus two is presented. Each vertex Pi is a point in R

12, whose coordinates are
on columns 2 to 13. The entry corresponding to the vertex Pi and the column dj is filled with +
if the vertex Pi is a vertex of the polytope of solutions corresponding to the generic graph dj.


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Surface databases based on Teichmüller theory 83

Figure 5: The solutions, φ, of top linear systems, L(G), are modified to get solutions for the
bottom graphs system by adding ±δ to φ solutions of the original graph which are on the circuit
and assigning δ to the new red edge.

Proof. By Corollary 1, we only need to prove the statement for all face constraints which corre-

sponds to sets with 1, 2 or 3 triangles. In addition, to check that this linear system is independent

of any given linear constraint, one can just verify that the 22 vertices given on Table 2 satisfy the

constraints, which in our case can be done by simple inspection.

5.3 Independence of face constraints for general genus two CE-graphs

Proposition 5.4. The solution of the system L(G) is face independent for any Delaunay realizable

graph G corresponding to a Riemann surface of genus two.

The basic idea to prove the above proposition is to use induction on the number k of contractions

that are needed to obtain G from a generic graph and notice that if G is a Delunay realisable graph

which is obtained by k + 1 contraction moves then there exist a solution φi of the system L(G)

such that one of the vertices of G can be expanded to obtain a graph G′ which is closer to the

generic level and the solution of φi can be modified to obtain a solution of G
′ which is as close as

we want to φi. See Figure 5 and Figure 6.


84 Armando Rodado Amaris, Gina Lusares CUBO
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Figure 6: The solution, φ, of top linear system L(G) is modified to get solutions for the bottom
graph system by adding ±δ to φ solutions of the original graph which are on the circuit and
assigning δ to the new two red edges.

6 Compactification of T2, Mg and polytope complexes

Figure 4 not only shows how the generic Delaunay realizable graphs are connected by Whitehead

moves, but the connections among the polytope complex P2, which have been described on Table 2.

From the geometric point of view, P2 is a covering of Teichmüller space T2 from which the Moduli

space of Riemann surfaces M2 can be obtained as the interior of the quotient space determined by

the groups of symmetries shown on Figure 4.

Polytope complexes, as the one described for genus 2, also exist for the hyperelliptic locus of any

genus g ≥ 3. Let Pg be the polytope complex Pg obtained for g ≥ 2. The mathematical theory

relating the polytope complex Pg, Tg and Mg is not yet completely developed. However, there is

no reason why we could not take advantage of this structure for applications. For a background

on polytopes see [8, 26].

7 Polytope complexes for indexing databases

It is desirable to have indexing systems for databases of surfaces which mirror the internal structure

of the space where the potential members to be included in the database belongs to. This can be

done for surfaces of genus two base on the polytope complex P2 whose vertices are given on Table


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Surface databases based on Teichmüller theory 85

2.

The internal structure of the Teichmüller space for Riemann surfaces of genus two is given by

the polytope complex P2 and the knowledge of its structure which is represented on Figure 4.

Then, if an open database is build on top of P2, a person who have classified a surface S
⋆ of genus

two, using the descriptor Dθ, could include S
⋆ in the database and annotate the database with

new additional features. In addition, since the hyperbolic surface that is associated to Dθ⋆, where

θ⋆ is determined by the circle pattern angles corresponding to S⋆, we can build and support both

new surface knowledge and the applications of surfaces theory.

We think that the design of the structure of a database for surfaces of genus g could be enhanced

by building on sound combinatorics and hyperbolic geometry, and propose to include the next steps

in its design:

(1) Find Cg, the combinatorics of the hyperelliptic locus of Tg by generating all non-isomorphic

generic graphs

(2) Construct the polytope complex Pg by solving the linear programs of the form L(G)

(3) Determine the map of the database structure, using the generic realizable graphs as vertices,

and connecting these vertices by edges corresponding to Whitehead moves

(4) Finally, a new surface S entering the database would be indexed by the descriptor Dθ(S).

In addition to the above, to construct the deeper layers of Cg, we could build all non-isomorphic

graphs which can be obtained by contraction moves from a graph on the generic layer of Cg.

However, graphs on deeper layer of C(g)-non-generic- can be viewed as well as elements of the

polytope Cg.

8 Conclusion

We have shown a framework to develop databases based on polytope complexes which arise by

considering the combinatorics and geometry of the Teichmüller space for closed surfaces of a given

genus. Databases supported on the mathematical structures that we have described are desirable

because they can be used to study deformation of surfaces in real applications, and supported by

rich mathematical results generations of mathematicians have developed. However, our description

is not complete because we do not know a canonical cell decomposition of Tg for general g. Then,

further theoretical and computational tools are needed in this area. In particular, research on

the applications of canonical decomposition of the moduli space of Riemann surfaces which have

punctures or boundaries [17] for the development of databases is desirable.

We hope to encourage collaboration between researchers with different backgrounds by building

database of surfaces having the indexing system that we have proposed. Teichmüller and moduli


86 Armando Rodado Amaris, Gina Lusares CUBO
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space theories are in the heart of surfaces theory, modelling and its applications. Databases of

surfaces which mirror the amazing structure of Mg and Tg should be one of the tools that support

the increasingly complex study of surfaces.

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