1. No category

On intersection graphs of stable curves, appearing as
boundary points of origami curves
Michael Maier
1
∗
Introduction
Origamis will play a central role in this article. They are a special kind of surfaces, which occur
by sticking a finite number of unit squares together according to special rules. If you provide them
with special complex structures, the result is a family of Riemann surfaces of the same topological
type and therewith points in the corresponding moduli space. All together they form an algebraic
curve called origami curve.
More specifically origami curves are a special kind of Teichm¨
uller curves, that is they are the
images of geodesic discs in the moduli space. If we trace back Lochak’s approach, who invented
the name origami, one can regard them as generalisations of ”dessins d’enfants” (see [Lo]).
If one extends these curves to the boundary of the moduli space, the result is a projective curve.
Then the added points correspond to stable curves. The boundary of the moduli space can be
separated in strata, which are determined by the topological typeis of the stable curves. The
latter can be characterised by an intersection graph: Each irreducible component corresponds to a
vertex, each singularity to an edge. This edge, as ordinary double point, connects the two vertices
corresponding to the two components on whom the point lies or is a loop if it is shared by only
one component. Furthermore the graph has a natural N0 vertex weight, which assigns to a vertex
the geometric genus of the corresponding component.
The aim of this article is to discover, which boundary strata of the moduli space admit intersections
with origami curves. The answer is given by the following theorem.
Theorem 1. An undirected graph with at least one edge is the intersection graph of a stable curve
appearing as a boundary point of an origami curve, iff it is finite, connected and bridge free. Every
N0 -vertex weight assigning the geometric genus to each component can be realized this way.
A general introduction to origamis and origami curves as well as references to other authors working on origamis and Teichm¨
uller curves can be found for example in [Sch1] and [Moe].
This article is based on my diplom thesis, which was written at the Universit¨at Karlsruhe and
advised by Professor Frank Herrlich and Dr. Gabriela Schmith¨
usen.
∗ [email protected]
1
1 INTRODUCTION
2
This work is divided in five sections and organized as follows: In the second section the definitions and basic facts are introduced. A defining attribute of origamis is that they correspond to
finite unbranched coverings p : O∗ → E ∗ , where E ∗ is the punctured torus and the closure O of
O∗ is a closed surface.
The boundary points of the origami curve C(O) are stable curves that arise by contracting the
preimage of a simple closed geodesic (according to the hyperbolic metric) on E ∗ . This will be
explained in the third section and is used for proving the results of this article.
The fourth section shows the easy implication of the proof of Theorem 1, namely that each intersection graph is of the given kind.
The fifth section deals with the other implication. We construct a suitable origami for each appropriate graph. To accomplish this one shows that each of these graphs has an N vertex weight
with special properties. With the help of that weight one constructs the origami as desired.
Special thanks go to Professor Herrlich, because this work would not have been possible without
his mentoring beyond my diplom thesis. He always had an open door and took time to reread
and constructively correct my concept. Further I am indebted to PD Dr. Stefan K¨
uhnlein for
his ideas on the graph theoretical part of the fifth section. By using Menger’s Theorem the proof
became much more elegant. Last but not least I want to thank Dr. Gabriela Schmith¨
usen for
her corrections and annotations both on the subject and form. These helpful ideas made the text
much more readable and far better understandable. Without this help the publication would not
have been possible.
3
2 DEFINITIONS
2
Definitions
This work discusses boundary strata, which appear as boundary points of origami curves. They can
be described as intersection graphs. Therefore we recall some basic facts about graphs, origamis
and stable curves.
2.1
Graphs
Further information can be found in standard textbooks like [ClHo] or [Ai]. We distinguish between
undirected graphs, as they occur in the statement of Theorem 1, and digraphs, which will be used
in the proof.
Definition 1
1. By a directed graph or digraph we will understand a tuple G = (V, E, Φ) consisting of a set
of vertices V, a set of edges E with V ∩ E = ∅ and a map Φ : E −→ V × V.
2. A trail of length p ∈ N0 is a tuple t = (v0 , e1 , v1 , ..., ep , vp ) ∈ V × (E × V)p with
Φ(ei ) = (vi−1 , vi ) and vi 6= vj for i 6= j, except possibly v0 = vp (closed trail ).
3. Two trails t1 = (v0 , e1 , v1 , ..., ep , vp ), t2 = (v0′ , e′1 , v1′ , ..., e′p′ , vp′ ′ ) are said to be edge-disjoint,
iff ei 6= e′j for i = 1, . . . , p, j = 1, . . . , p′ .
4. Let M be a set. An M weight of the vertices (respectively the edges) is a map Ψ : V −→ M
(respectively Ψ : E −→ M ).
5. By an undirected graph we will understand as above a tuple G = (V, E, Φ) consisting of a set
of vertices V, a set of edges E with V ∩ E = ∅ and a map
Φ : E −→ (V×V)/∼. Here ∼ is defined by (a, b)∼(c, d) :⇔ (a = c and b = d) or (a = d and b = c).
The definitions 1.2-1.4 apply analogously for undirected graphs.
6. A bridge is an edge e with Φ(e)=(v, v ′ ) for which each trail from v to v ′ contains e.
A graph is called bridge free, if none of its edges is a bridge.
7. The following map ν is called the valence map of an undirected graph
ν:V
v
−→ N0
7→ |{e ∈ E | Φ(e) = [(v, v ′ )]∼ , v ′ ∈ V}| + |{e ∈ E | Φ(e) = [(v, v)]∼ }|
8. The class of pairs (G, φ) consisting of an undirected, finite, bridge free, connected graph G
with at least one edge together with an N0 edge weight φ on G will be denoted by G.
4
2 DEFINITIONS
2.2
Origamis
Definition and remark 2.
1. An origami is a finite set of disjoint unit squares Qi := [0, 1]2 in R2 (i = 1, . . . , m) together
with a gluing rule, identifying each right side with exactly one left side (by translation) and
each upper one with a lower one, such that a compact connected surface O of genus g is
obtained. We mark the n points of O which are covered by the vertices of the squares. O∗ is
defined by O∗ := O \ {marked points}.
2. In order to draw origamis in a simple way we will specify the gluing rule by either joining
the edges directly or such that the edge is glued to the next free opposite edge. If that is not
possible we will mark the two edges by a common label.
Example:
•
•
⋆
⋆
⋆
•
•
•
⋆
⋆
2
1
2
1
Here the origami is a surface of genus 2 with 2
marked points (•, ⋆).
In this case by gluing the
surface according to the
labels we obtain a surface
of again genus 2 with 1
marked point.
3. One may equivalently define origamis as unramified finite coverings O∗ −→E ∗ of the oncepunctured torus E ∗ = (R2 \ Z2 )/Z2 .
We use this in the following item to get complex structures on O, when we are given complex
structures on E.
4. For each τ ∈ H we obtain a complex structure on E by mapping the unit square onto the
parallelogram Pτ with the vertices 0, 1, τ + 1, τ ∈ C. This gives us a Riemann surface in
the moduli space M1,1 (marked point (0, 0) covering 0) biholomorphic to the elliptic curve
C/(Z + τ Z). Therefore we obtain the map ϕE : H −→ M1,1 .
More generally we may take an origami O of genus g and n marked points (covering the
vertices of the squares). We can lift the complex structure of ϕE (τ )∗ on E ∗ via the
unramified covering O∗ −→ E ∗ and obtain a complex structure on O∗ . This makes O∗ a
Riemann surface which we call ϕO (τ ).
We obtain the map ϕO : H −→ Mg,n which assigns to τ ∈ H an n-punctured Riemann
surface of genus g.
5
2 DEFINITIONS
5. Let π : Tg,n −→ Mg,n be the natural projection from the Teichm¨
uller space Tg,n to the moduli
⊂
- Tg,n
space Mg,n . Then ϕO descends from a holomorphic isometric inclusion ϕf
O : H
(according to the hyperbolic and the Teichm¨
uller metric), such that π ◦ ϕf
=
ϕ
O
O . The
image of ϕf
then
is
a
Teichm¨
u
ller
disc,
denoted
by
∆(O).
The
canonical
projection
maps
O
this Teichm¨
uller disc to π(∆(O)) = ϕO (H) which turns out to be an algebraic curve called
origami curve C(O) (see e.g. [Sch2]).
⊂
⊂
--
∆(O)
H
⊂
ϕfO
ϕO
π |∆(O)
--
?
?
C(O)
2.3
⊆Tg,n
-
⊂
⊆-
π
?
?
Mg,n
Stable curves
In this section we will introduce stable complex algebraic curves. Further information can be
found in [Be1], [Be2], [Be3], [DeMu], [Mu] and [HaMo].
Definition 3.
1. An n-punctured stable curve of genus g is a projective connected curve X of arithmetic genus
g over C together with a finite set {p1 , . . . , pn } ⊂ X, fulfilling the following properties:
– The finite set D ⊂ X of singularities of X is disjoint from {p1 , . . . , pn }.
– Every singularity c ∈ D is an ordinary double point.
– For each irreducible component Xi of X with Xi ∼
= P1C the total number of singularities
counted in multiplicities (that is singularities in the inner of a component are counted
twice) and the number of punctures is greater than 2.
2. The intersection graph Γ(X) of a stable curve X with l ≥ 1 irreducible components X1 , . . . , Xl
and the set D of singularities on X is the undirected graph with
V = {X1 , . . . , Xl }, E = D and
Φ : E −→ V × V/ ∼
(Xi , Xj )∼ : ∃ i, j ∈ {1, . . . , l} i 6= j : c ∈ Xi ∩ Xj
c 7→
(Xi , Xi )∼ : ∃! i ∈ {1, . . . , l} : c ∈ Xi
Φ is well defined, as each c ∈ D is an ordinary double point and hence lies on at most 2
irreducible components.
Remark 4. In addition we obtain a natural vertex weight by assigning to each vertex the geometric
genus of the corresponding irreducible component.
These stable curves of genus g with n marked points form the boundary of the moduli space Mg,n
and define the Deligne-Mumford compactification M g,n of the moduli space.
6
3 BOUNDARY POINTS OF ORIGAMI CURVES
3
Boundary points of origami curves
In this section we will describe the degeneration of the surface which results by moving along the
origami curve towards the boundary of the moduli space.
More precisely we show:
Lemma 1. Let O be an origami and p : O∗ → E ∗ the associated, finite, unramified covering.
Then each boundary point of the origami curve C(O) is a stable curve, which arises by contracting
the full preimage of a not null-homotopic, simple closed path on E.
For further information especially on T g,n see [HeSch]. A more general statement for surfaces with
translation structures can be found in [Sch2].
For our purpose we use the following description of the Teichm¨
uller space Tg,n : We are given a
reference surface R of genus g with n marked points p1 , . . . , pn . The punctured surface R∗ arises
from the reference surface by removing the marked points.


X Riemann surface of genus g, with n marked points, ,

f : R −→ X diffeomorphism, mapping the
∼
Tg,n = (X, f ) 

marked points of R onto the marked points of X
[X, f ] ∼ [X ′ , f ′ ] :⇔ f ′ ◦ f −1 homotopic to a biholomorphic map.
As above X ∗ is the punctured surface associated to X. More about Teichm¨
uller spaces can be
found in [ImTa] and [Ab2].
From now on we will think of a not null-homotopic, simple closed path γ on X as a simple closed
path on X ∗ , which is not null-homotopic on X. By deforming γ in its homotopy class if necessary
such that it does not contain any puncture.
If we take any not null-homotopic, simple closed path γ on R, then for each (X, f ) ∈ Tg,n we
get a not null-homotopic, simple closed path f (γ) on X. If g ≥ 1, n ≥ 1 the punctured Riemann
surface X ∗ has the universal covering H and so there is a hyperbolic metric on it. Therefore in
the homotopy class of f (γ) there is a unique geodesic. That way for each such γ we obtain a
continuous map lγ : Tg,n −→ R>0 with
lγ (X, f ) = ”hyperbolic length of the geodesic in the homotopy class of f (γ)”.
From now on we will take an origami O of genus g and n marked points as reference surface. Using
the notation of 2.2.5 on page 5 we get the following diagram:
lγ ∆(O) ⊆ Tg,n
R>0
H
⊂
⊂
ϕfO
ϕO
-
π
?
? - ?
?
C(O) ⊆ Mg,n
Now we are interested in the boundary points of C(O) in M g,n , that is the stable curves that occur
in the closure of the origami curve in the compactified moduli space. Hence we have to take the
appropriate boundaries of all spaces that occur in the diagram and extend the maps continously.
For the upperhalf plane we take Q ∪ {∞} as boundary together with the horocycle topology (see
[Bea]). For Tg,n we take as closure the augmented Teichm¨
uller space (as introduced by Abikoff,
see [Ab1]) and for the moduli space Mg,n we take the Deligne-Mumford compactification.
The map π : Tg,n −→ Mg,n has a natural extension π : T g,n −→ M g,n .
7
3 BOUNDARY POINTS OF ORIGAMI CURVES
∆(O) ⊆
- T g,n
H ∪ Q ∪ {∞}
ϕfO
ϕO
-
lγ R≥0 ∪ {∞}
π
?
?
? ?
C(O) ⊆ M g,n
For each r ∈ Q ∪ {∞} we may choose a neighbourhood Ur ⊂ H ∪ Q ∪ {∞} of r such that
Ur \ {r} ⊂ H and π|ϕf
O (Ur ) is a homeomorphism. We define
lγr := lγ ◦ π −1 |ϕfO (Ur ) : ϕO (Ur ) −→ R≥0 ∪ {∞}
Lemma 2. Let O be an origami of genus g with n vertices, r ∈ Q ∪ {∞},
p : O∗ → E ∗ the corresponding, finite and unramified covering and ϕO : H → Mg,n the map
belonging to the origami curve C(O) := ϕO (H) (Mg,n , Tg,n , lγr as above). Then we have:
a) If O = E up to homotopy there is a unique, simple closed, not null-homotopic path γ(r) on
r
E such that lγ(r)
(ϕE (r)) = 0. For any closed path γ not homotopic to a power of γ(r) we
r
obtain lγ (ϕE (r)) 6= 0.
b) In general for each closed path γ on O it holds that
lγr (ϕO (r)) = 0 ⇔ p(γ) is freely homotopic to a power of γ(r).
c) ϕO (r) is a stable curve topologically obtained by contracting a systemS
{γ1 , . . . , γk } of simple
closed, not null-homotopic, pairwise not homotopic pathes such that γi = p−1 (γ(r)), that
is by contracting the full preimage of γ(r).
Proof:
a) For O = E we have g = n = 1 and therefore T1,1 = H, M1,1 = H/SL2 (Z). Without loss
of generality we can assume r = ∞, as any element of Q can be mapped to ∞ by an element of SL2 (Z). On the other hand SL2 (Z) acts on π1 (E ∗ ) up to conjugation such that
A(r)
lγr = lA(γ) ∀A ∈ SL2 (Z).
ϕE (τ ) is covered by D = {z ∈ C | |z| < 1} for each τ ∈ H. We fix the sequence τm := m · i
converging to ∞. For symmetry reasons the covering can be chosen such that the fundamental domain is bounded by the geodesics between zm , −z m , −zm , z m ∈ ∂D (see [Sch2]):
−z m
zm
γ ′′
γ′
zm
−zm
As the covering of elliptic curves is well known, we have the two not null-homotopic closed
geodesics γ ′ , γ ′′ on E whose lift is connecting the opposite sides of the boundary. γ ′ belongs
to the real axis and γ ′′ to the imaginary axis by our chart map.
8
3 BOUNDARY POINTS OF ORIGAMI CURVES
For m → ∞ it follows that zm → i and therefore the fundamental domain degenerates as
indicated in the following picture for m1 < m2 < m3 :
−z m3
zm3
−z m2
zm2
−z m1
zm1
−zm1
z m1
z m2
−zm2
−zm3
z m3
From this we see directly:
m→∞
m→∞
lγ∞′ (ϕE (τm )) −→ lγ∞′ (ϕE (∞)) = 0 and lγ∞′′ (ϕE (τm )) −→ lγ∞′′ (ϕE (∞)) = ∞
Now we take any closed path γ not homotopic to a power of
γ(∞) = γ ′ . As l∞ depends only on the geodesic in the homotopy
class, we can assume that γ is a geodesic. Now we take a closer
look at the once-punctured cylinder which is obtained by cutting
E along γ ′ . Equivalently it is possible to regard the punctured
cylinder as a once-punctured square whose left side is glued to
the right one.
γ′
γ ′′
γ′
γ′
γ ′′
γ
If γ intersects γ ′ it follows that lγ (τm ) ≥ lγ ′′ (τm ) as γ ′′ is
the shortest connection between the two lifts of γ ′ . Therefore
m→∞
lγ∞ (τm ) −→ ∞ and in particular lγ∞ (∞) 6= 0.
γ′
If γ does not intersect γ ′ , it is completely contained in the oncepunctured cylinder. As γ is not null-homotopic and not homotopic
to a power of γ ′ , it is not in the subgroup generated by γ ′ and
γ ′′ . So γ orbits the puncture. But as it is not null-homotopic
on E it has to orbit the cylinder as well. So it intersects itself
and therefore is not simple. According
√ to [Ya] for a geodesic that
intersects √
itself we have 4 · ln(1 + 2) ≤ l(γ). Therefore we find
τ →∞
4 · ln(1 + 2) ≤ lγ∞ (ϕE (τ )) −→ lγ∞ (ϕE (∞)) 6= 0.
γ′
γ
γ′
9
3 BOUNDARY POINTS OF ORIGAMI CURVES
b) Recall that ϕE (τ )∗ is a Riemann surface covered by the Riemann surface ϕO (τ )∗ . Let
H −→ ϕE (τ )∗ be the universal covering. By the theorem of the universal covering we obtain
the following commutative diagram:
H
∗
ϕO (τ )
p?
?
ϕE (τ )∗
Let γ be a closed path on O. So p(γ) is λ times a closed path on E and we call the
corresponding simple path γE . Therefore we have lγ (ϕO (τ )) = lp(γ) (ϕE (τ )) = λ · lγE (ϕE (τ ))
r
and hence lγr (ϕO (r)) = lp(γ)
(ϕE (r)) = λ · lγrE (ϕE (r)), which together with a) leads directly
to the assertion.
c) From b) we know, that degeneration arises by contracting the full preimage p−1 (γ(r)) of γ(r).
The connected components of the preimage do not intersect and they are not homotopic to
each other on O∗ . So we can conclude, that the obtained singularities are ordinary double
points. Each irreducible component is bordered by at least two singularities, as none of the
preimages is null-homotopic on O, and by at least one puncture, as it covers E. So ϕO (r)
is, as a point in M g,n , a stable curve of arithmetic genus g with n punctures.
3.1
Examples
1. For each r ∈ Q ∪ {∞} the origami E induces a stable curve that has the intersection graph
G0 := Γ(ϕE (r)) with one vertex, the irreducible component of genus 0, and with a loop.
τ →r
0
G0
2. For the following origami O one has more than one intersection graph.
For example G1 := Γ(ϕE (∞)) 6= G2 := Γ(ϕE (0)).
2
1
1
2
2
1
1
2
τ →∞
1
G1
τ →0
0
0
G2
10
4 OCCURING INTERSECTION GRAPHS
4
Occuring intersection graphs
In the following we will take a closer look at the type of the intersection graph of the boundary
points of an origami curve.
Lemma 3. Let O be an origami and r ∈ Q ∪ {∞}. Then Γ(ϕO (r)) is a finite, connected, bridge
free, undirected graph with at least one edge.
Proof: By definition Γ(ϕO (r)) is an undirected graph and has at least one edge because O covers
E. Therefore ϕO (r) has at least one geodesic of length zero. As an origami is connected the
degeneration leads to a connected stable curve and hence the intersection graph is also connected.
To complete the proof, it remains to show that Γ(ϕO (r)) is bridge free. This was first observed
by M. M¨oller and G. Schmith¨
usen (unpublished).
Assume there is an origami O and r ∈ Q ∪ {∞} for which Γ(ϕO (r)) has a bridge. Then there
is a connected component γˆ of p−1 (γ(r)) separating O. That is we have two open connected
components X1 , X2 such that X1 ∪· X2 = O \ γˆ .
For any m ∈ N we get an origami O′ by dividing each square of O in m2 small squares. Then the
lift of γˆ separates O′ . We choose m such that the intersection of the lift of γˆ and any single square
is connected on O′ . So without loss of generality we can assume, that this is the case on O.
X1
X2
As the degeneration of E only depends on the homotopy class of γ(r) and as γ(r) is simple closed,
we can think of it as the straight line with slope r on E and the same for γˆ . In the case r 6= ∞
let X1 be the part above γˆ , X2 the part below γˆ . We see that each square passed by γˆ has no
entire bottom side contained in X1 . But at least for one square its upper side is entirely in X1 .
Obviously for each square that totally lies in X1 the lower and upper side are entirely contained in
this part. Hence it is not possible to glue the upper and lower sides of the squares that lie entirely
in X1 , since there are more upper sides than lower sides. Therefore such an O can not exist.
In the case r = ∞ it is the same, but for left and right instead of lower and upper sides, if we take
X1 right of γˆ .
5 CONSTRUCTION OF AN ORIGAMI TO A GIVEN INTERSECTION GRAPH
5
11
Construction of an origami to a given intersection graph
The converse implication of the theorem is that for every pair (G, φ) ∈ G, where G is as in Def.
1.8, G is a graph with properties as in the definition and φ is any N0 vertex weight, there exists an
origami O and an element r ∈ Q ∪ {∞} such that Γ(ϕO (r)) is isomorphic to G and the geometric
genus of each component is the value of φ at the related vertex.
The proof will be constructive and consists of the following steps:
5.1 Introduction of three ”elementary” generating graph transformations.
5.2 According to 5.1 one can find a special N edge weight.
5.3 Development of a prototype for the components of an origami.
5.4 Modification of the construction.
5.1
Graph transformations
As a first step we try to get a better understanding for the finite, bridge free and connected graphs
with at least one edge. We will see that each of these graphs can be created out of the graph G0 ,
which consists of one vertex and one loop, by the following three graph transformations. One can
easily check that every graph finitely generated by the three transformations out of G0 is finite,
bridge free and connected.
α) Adding a loop:
β) Adding a vertex on an edge (possibly a loop):
γ) Adding an edge between two vertices:
The idea is to take a finite graph G as above and to simplify it step-by-step by removing vertices
and edges until we get G0 . The operations α) and β) can be easily inverted, as long as there
remains another vertex respectively another edge. So we can reduce G to a loop free graph, where
every vertex has at least valence 3 (valence 1 would mean that G is not bridge free).
Now we want to invert γ) in order to reduce the graph further. The problem is to verify, that
we can find an edge such that the resulting graph is still bridge free. Our consideration will be
completed by the following lemma.
5 CONSTRUCTION OF AN ORIGAMI TO A GIVEN INTERSECTION GRAPH
12
Lemma 4. Let G be a loop free, finite, connected, bridge free and undirected graph where
ν(v) ≥ 3 ∀v ∈ V. Then there exist adjacent vertices a, b ∈ V, which are connected by two
additional edge-disjoint trails.
Proof: If G has no cut of capacity 2, then the number of edges separating any two distinct points
is greater than 2. (Of course there is no cut of capacity 0 as the graph is connected and none
of capacity 1 as it is bridge free.) Then Menger’s theorem ([Bo], Theorem 5. on page 52) tells
us that for each two vertices there are at least 3 edge-disjoint trails connecting both. Especially
every two adjacent vertices are connected by two additional edge-disjoint trails.
Otherwise there are finitely many cuts of capacity 2 and we can take one (V′ , V \ V′ ) with the
property, that there is no cut (V′′ , V \ V′′ ) of capacity 2 such that V′′ ⊂ V′ . Let G′ be the full
subgraph defined by V′ . Then G′ is connected and it is bridge free, since V′ was chosen minimal.
Let v1 , v2 ∈ V′ be the vertices adjacent to V \ V′ in G.
Assume first that v1 = v2 . Then ν(v1 ) ≥ 3 in G′ , as G is loop free and else there would be the
properly ”smaller” cut ( V′ \ {v1 } , V \ V′ ∪ {v1 } ) of capacity 2 (in addition this proves, that
|V′ | > 1). For the same reason G′ has no cut of capacity 2 as this cut would be also a cut of capacity 2 in G. Then as above every two vertices in V′ are connected by at least three edge-disjoint
trails in G′ and therefore in G.
In the other case v1 6= v2 , there can only be cuts of capacity 2 in G′ separating v1 from v2 , as
there is no properly ”smaller” cut of capacity 2. These cuts disappear if we add an edge between
v1 and v2 . Therefore the obtained graph has only cuts of capacity 3 or more. Again by Menger’s
theorem each pair of vertices is connected by three edge-disjoint trails. As there is a trail in G
connecting v1 and v2 with vertices in V \ V′ , every two points of V′ are connected in G by three
edge-disjoint trails.
13
5 CONSTRUCTION OF AN ORIGAMI TO A GIVEN INTERSECTION GRAPH
e and the associated N edge weight
G
5.2
As seen in the last section, one can construct finite, connected, bridge free and undirected graphs
with at least one edge out of the graph G0 by using the graph transformations α, β and γ. Now we
e Further more we create a flow (without
will use this to orient our graph and obtain a digraph G.
source and sink) which is a special kind of N edge weight ψ satisfying (1).
Lemma 5. For every finite, connected, bridge free and undirected graph G there exists a digraph
e and an N edge weight ψ : E
e = E −→ N, such that G can be obtained from G
e by disregarding the
G
directions of the edges and such that the following holds:
X
X
e=V
ψ(e) = 0 ∀v ∈ V
(1)
ψ(e) −
|
e∈E
e
pr2 (Φ(e))
=v
{z
incoming edges
}
|
e∈E
e
pr1 (Φ(e))
=v
{z
outgoing edges
}
Proof:
e is uniquely determined and ψ : E
e = E = {e} −→ N, e 7→ 1 fulfils
For G0 this is clearly true, as G
(1), as ψ(e) − ψ(e) = 0
According to the last section it remains to show that one can get a suitable digraph and an N
edge weight if one of the transformations α), β) and γ) is applied. This can be done as follows:
α) Adding a loop:
β) Adding a vertex on an edge (possibly a loop):
γ) Adding an edge between two vertices:
a3
a2
2a3
a4
2a2
2a3−1
2a4
2a4+1
2a2+1
a1
2a1
2a1+1
b
2b
2b
1
In this case in general we have to take an intermediate step in which we double the whole N
edge weight. That way we obtain another suitable flow (this is true more generally - compare
the following remark). Then we choose a trail between the two vertices we want to connect
and raise respectively lower (depending on the direction) the weight by 1 along it.
The intermediate step was necessary to get in general another N edge weight, more precisely
that no edge weight becomes 0. In explicit constructions one should check whether it is
really necessary.
5 CONSTRUCTION OF AN ORIGAMI TO A GIVEN INTERSECTION GRAPH
e with property (1) and q =
Remark 5. If ψ is an N edge weight of G
∀e ∈ E, then
qψ : E −→ N,
a
b
14
∈ Q>0 such that b | ψ(e)
(qψ)(e) := qψ(e)
e with property (1).
is a well defined N edge weight of G
This will be used in section 5.4 for q ∈ N ⊂ Q>0 . The general statement is helpful for explicit
construction in order to lower the edge weight and therefore make the resulting origami smaller.
15
5 CONSTRUCTION OF AN ORIGAMI TO A GIVEN INTERSECTION GRAPH
5.3
r-components
In this section we generate an origami for a given weighted graph (G, φ) in G with the aid of
5.1-5.2, such that its origami curve has a boundary point with that graph as intersection graph.
The idea is to construct each vertex separately and to use the N edge weight such that the parts
fit together.
Definition 6. An r-component, for r ∈ Q ∪ {∞}, of an origami O is the closure of a connected
component of O \ p−1 (γ(r)), where p and γ(r) are defined as in Lemma 2.
Remark 7. On ϕO (r) the images of the r-components correspond exactly to the closures Xi of
the irreducible components (see Lemma 2 c) ). The r-components form a closed covering of O for
each r ∈ Q ∪ {∞}.
As we will see, one can obtain all graphs in G by considering only limits towards ∞. That is
they can be obtained as intersection graphs of the stable curve generated by contracting the full
preimage of the horizontal simple closed path γ(∞) on E.
Consequently it is sufficient to illustrate the origami in the following way: We order the squares,
starting with any square and then putting the right adjacent square directly right of it. We
continue like this until we arrive at the first square again. Then we repeat the same procedure,
starting from a new square. That way the only missing information is how to glue the upper and
lower sides of the squares. This can be described by a permutation π ∈ Sn (where n is the number
of squares).
For example:
4
5
3
1
2
1
2
π(1)
π(2)
2
1
2
3
3
1
3
3
4
4
π(3)
4
4
π(4)
5
5
π(5)
π=
1
4
2
5
3
1
4
6
5
2
6
3
∈ S6
2
6
6
1
6
π(6)
The full preimage of γ(∞) decomposes into three connected components γ1 , γ2 , γ3 and there are
two ∞-components (see picture). As in general, each component contains as many upper as lower
halves of squares.
γ1
@
@
2
4
3
γ2
1
3
4
2
γ3
1
∞-component 1
∞-component 2
16
5 CONSTRUCTION OF AN ORIGAMI TO A GIVEN INTERSECTION GRAPH
Now we look at one vertex v ∈ V of a given graph G and for the moment we disregard its N0
vertex weight, that is its genus. Then we have l ∈ N0 loops and m ∈ N0 incoming and outgoing
edges ei which are not loops. We assign an ai ∈ Z \ {0} to each of the ei using an N edge weight ψ
with property (1), such that ai = ψ(ei ) if ei is directed towards v and ai = −ψ(ei ) else. So we get
m
P
ai = 0. Using this we can define the associated ∞-component X ′ (l, a1 , . . . , am ) in most cases
i=1
as it is shown below (the remaining cases will be treated in chapter 5.4). In this illustration we
assume, that a1 , ..., ai > 0 and ai+1 , ..., am < 0.
X ′ (l, a1 , . . . , am ):
v and adjacent edges:
a2
a1
B
B
...
B
B
B?
N ..
rv
.
6
B
B
B
... B
BN
l
ai
ai+1
|
l
l
?
am
{z
...
a1
...
..
.
}|
l
{z
...
a2
...
}
...
6 l−ai+1
z
}|
{
...
z
|
...
{z }
?
..
.
6
ai
−am
}|
...
{






l





The upper and lower half squares fit to the half squares of the adjacent components because of
the N edge weight. The squares are glued together as indicated above.
l
?






In the case m = 0, that is if there is only one vertex, we construct
.
.
l
6
l
′
.
?
X (l) as indicated in the picture:





6 l
Remark 8. As mentioned above the definition of X ′ (l, a1 , . . . , am ) is useful for the construction
of ∞-components, but not sufficient. Apart from the disregarded genus, it is in general false that
one obtains a connected part, see the counter example below. In the case l = 0 it is possible that
the component splits apart. In some cases this problem can be solved by rearranging the edge, but
even this is not possible in every case. And it is even hard to say when this is possible. This will
be our aim in the next section.
3
A
−3
U A
r
A
4
A
U
−4
X ′ (0, 3, 4, −3, −4) :
3
A
AU
4
r
r
A
AU
−4
−3
5 CONSTRUCTION OF AN ORIGAMI TO A GIVEN INTERSECTION GRAPH
5.4
17
Genus and unit
To transform the definition of X ′ (l, a1 , . . . , am ) in order to assure that it is connected, the first
step is to glue crosswise like this:
←
→→
←
This might not be a solution, if there exists an ai with |ai | = 1 that is an incoming or outgoing
edge with weight 1, like in the following illustrations:
←
→→
←
←
→→
←
←
→→
←
or
In order to include the genus in the construction and such that the problem does not arise we recall
remark 5. It will be useful to raise the N edge weight by multiplying it with a natural number,
called the unit λ, in order to broaden the parts of the origami.
...
...
|{z}
λ
...
←
→→
←
←
→→
←
←
→→
←
This gives us a construction of (connected) ∞-components and we will use a similar approach to
control the genus of the component. As shown in lemma 6, it is possible to raise the genus to
g ∈ N by crossed gluing g times inside the component as below. In order to provide enough space
inside, the unit must be at most 2g +1. As a raise of the unit strongly increases the number of
squares of the origami, one is interested in the smallest unit in which this construction is possible
- the minimal unit. The following remark helps to get an estimate by an upper bound.
...
Remark 9. Let (G, φ) ∈ G and (V, E) = G then it follows for the minimal unit λ that:
λ ≤ 2 · max φ(v) + 2
v∈V
18
5 CONSTRUCTION OF AN ORIGAMI TO A GIVEN INTERSECTION GRAPH
So in particular, it is ensured that there is always a suitable unit. Therefore we can go on with
the construction for the components:
v and adjacent edges:
X(φ(v), l, a1 , . . . , am ):
if l = 0 if necessary (does not count towards φ(v))
B
...
B
B?
N ..
r
l
v φ(v)
.
6
B
B
B
... B
BN
am
ai+1
ai
...
{z
←
→→
←
B
a1
}|
...
..
l
.
?
6
...
|
{z
}
−ai+1
φ(v) in quantity
a2
}|
...
{
...
Defined for l ∈ N0 , m ∈ N, ai ∈ Z\{0} and
m
P
z
ai
}| {
...
...
|
...
{z
−am
?
6
..
.
←
→→
←
B
z
←
→→
←
a2
a1






l





}
ai = 0.
i=1
Lemma 6. Let (G, φ) ∈ G and Xv := X(φ(v), l, a1 , . . . , am ) as it is defined above. Then it follows:
a) The Xv together form an origami O with ϕO (∞) = G.
b) Each Xv has geometric genus φ(v).
Proof:
e and an N edge weight, such that condition
a) Like in Section 5.1-5.2 we construct a digraph G
(1) is fulfilled. Then with the construction as above we get the ∞-components Xv . For
each edge (v, v ′ ) with weight a the component Xv contains a lower half squares, and Xv′
contains a upper half squares. Together they form whole squares, and that way an origami
is obtained.
As we glued along the preimage of γ(∞), the degeneration produces edges in the intersection
graph.
b) We will use Euler’s formula to calculate the genus g using the natural triangulation on
ϕO (∞). Let V be the number of vertices, E the number of edges and F the number of faces,
then it follows:
2g − 2 = −V + E − F
(2)
19
5 CONSTRUCTION OF AN ORIGAMI TO A GIVEN INTERSECTION GRAPH
if l = 0 if necessary (does not count towards φ(v))
...
...
...
..
.
..
.
←
→→
←
l
?
6
φ(v) in quantity
...
←
→→
←
←
→→
←
...
...
?
6
...






l





First we regard the case φ(v) = 0. That means there are no internal cross gluing, and l = 0:
F =
–
m
X
i=1
– For a triangulation:
E=
|ai |
3
·F
2
– The difficult part is to calculate the number of vertices. We will separate this into two
cases:
i) Xv′ connected: There are m vertices that descend from contracted geodesics. To
calculate the rest of the vertices, we start with a set of 2F points, as each half
square had two vertices, and take a closer look at the equivalence relation induced
by the gluing. The identification of left and right sides forms pairs of vertices, so
we have F points. For the gluing of the upper and lower sides, most points form
pairs again. But there are two special points, the point in which the upper and the
lower parts of half squares are glued. These two points are glued to each other and
together with m − 2 pairs of other points (oneP
pair for each edge beside the first
m
two) they form the point p. So we have V = 12 i=1 |ai | + 2.
p b
×
×
...
...
p
b
b
p
×
...
b
p
...
...
×
bp
5 CONSTRUCTION OF AN ORIGAMI TO A GIVEN INTERSECTION GRAPH
20
ii) Xv′ not connected: Let r ∈ N be the number of connected components C1 , ..., Cr we
would obtain without cross gluing. Using part i) we can conclude that the number
of vertices would be


r
m
X
X
1X
1
|ai | + 2 =
|ai | + 2r
2
2 i=1
j=1
ai in Cj
Each cross gluing reduces the number of vertices by 2, and we need r − 1 of them.
@
@• @⋆
l
•
⋆
△
l
@ @
@• @⋆•
⋄
△ ⋄
@ @
@ @
It follows that
– The result for the genus
1
1
g = (−V + E − F ) + 1 =
2
2
•
⋆
←
→→
←
@
⋆• ⋆
@ @
@ @
m
1X
|ai | + 2
V =
2 i=1
!
m
m
m
X
1X
3X
−
|ai | + 1 = 0
|ai | − 2 +
|ai | −
2 i=1
2 i=1
i=1
For l ≥ 1 Xv′ is connected. But we have to count the number of vertices carefully because
we obtain two for each geodesic contracted inside a component.
This is because they depart
P
atP
the closure of the component. For
each
loop
we
get
|a
|
+
2 additional faces, that is
i
P
1
3
|a
|
+
3
additional
edges,
and
|a
|
+
1
additional
vertices.
As above we get g = 0.
i
i
2
2
Let φ(v) ∈ N0 : Now we will take a look at what happens at the internal crosswise gluing.
l
A
• • A•
←
→→
←
A
• ⋆ A△
l
l
l
l
•
⋆
△
A
A
•
•
•
A
A
l
As above we get two vertices less for each cross gluing which raises the genus by 1 as desired.
The geometric genus is φ(v). This proves Lemma 6.
This completes the proof of Theorem 1 and gives us a possibility for constructing origamis to a
given element of G.
In order to clarify the construction we give some examples.
21
5 CONSTRUCTION OF AN ORIGAMI TO A GIVEN INTERSECTION GRAPH
Construction examples
1. (G3 , φ3 ) ∈ G
• 0
0 •
• 0
• 0
X(0, 0, 4, −2, −2) :
?
r 0
A
U
A
−2
−2
@
@@
@@
@@
@@
@@
@@
@@
2
3
A
0
U r
A
@
@@
@@
@
2
A
U A
r 0
?
−3
1•
•
- 1
• 1
γ
4
•
•
2•
•
- 1? 3
2
•
X(0, 0, 2, 3, −5) :
@
@
@@
@@
@
············
·············
············
·············
············
2
···················
····················
···················
····················
···················
X(0, 0, 5, −4, −1) :
?
r 0
A
U
A
−1
−4
1
•
•
γ
@@
@@
@@
?
−5
5
•
1
•
4
•
-
β
-
•
•
5
•
6
1
1
?
•
6
1
β
β −1
•
•
2
1
•
1
•
•
β −1
•
-
1
-•
•
•
•
β
•
β −1
2
•
•
•
•
γ −1
1
•
•
•
γ −1
1
•
1
5.5
X(0, 0, 1, 2, −3) :
·····
············
······
············
·····
············
······
····· ············
············
···················
····················
···················
····················
···················
1
@
@@
@@
@
@@
@@
@@
@
@@
@@
@@
@@
@@
@
1
······
·······
······
·······
······
2
22
5 CONSTRUCTION OF AN ORIGAMI TO A GIVEN INTERSECTION GRAPH
•
0
2. (G4 , φ4 ) ∈ G
•
1
•
-
?
r 0
?
−1
1
1
•
•
β
1
• - •
1
•
•
α
X ′ (1, 1, −1) :
1
A
U A
r 0
A
U
A
−1
−1
1
•
2
β −1
α−1
•
•
0
•
1
1
- • - •
1
?
6
β
•
•
1
1
• - • - •
1
1
•
α
1
1
1
- • - • - •
1
1
X(0, 1, 1, −1) :
λ=5
X ′ (0, 1, 1, −1, −1) : departs!
λ=5
X ′ (0, 1, −1) :
?
r 2
λ=5
?
−1
1
2
3
4
5
································
·································
································
·································
································
HH
Y
*
H
j
H
································
·································
································
·································
································
2
1
4
3
5
•
β −1
α−1
?
6
X(0, 0, 1, 1, −1, −1) :
×
I
R
X(2, 0, 1, −1) :
································
·································
································
·································
································
×
×
R
I
I
R
································
·································
································
·································
································
REFERENCES
23
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24