minimal fillings of surfaces
Let be a closed orientable surface of genus . A set of pairwise non-homotopic simple closed curves on is called a minimal filling, if is a topological disc. The size of a filling is defined as the number of its elements. For , we prove that the maximum (resp. minimum) size of a minimal filling of is (resp. , if and , if ) and we show that for and for each , there exists a minimal filling of of size .
Furthermore, we study geometric intersection number of curves in a minimal filling. For , we show that for a minimal filling of size , the geometric intersection numbers satisfy , and for each such there exists a minimal filling such that .
Key words and phrases:Surface, filling, fat graph
2000 Mathematics Subject Classification:Primary 57M15; Secondary 05C10
A set of pairwise non-homotopic simple closed curves on a closed oriented surface is called a filling of , if the complement is a disjoint union of topological discs. It is assumed that the curves in a filling are in minimal position, i.e., for , the geometric intersection number is
Filling systems of closed surfaces have become increasingly important in the study of the mapping class group of surfaces and the moduli space of hyperbolic surfaces through the systolic function, in particular. The study of filling system has its origins in the work of Thurston [Thu86]. The subset of moduli space of genus consisting of the hyperbolic surfaces filling systoles is called Thurston set. Thurston proposed as a candidate for a spine of , but its proof is incomplete. More recently, fillings of surfaces has been studied by Schmutz Schaller [SS99], Aougab [AH15], Parlier [FP16], Sanki [San17] and others.
To each filling system, we can associate two numbers: these are , the number of curves in the filling system which we call the size of a filling and , the number of discs in the complement. Of particular importance is the simplest filling system–a minimal filling, where . Euler’s equation implies that , when is a minimal filling. The mapping class group acts on the set of fillings with the same number of components in the complement. In [AH15] [Theorem 1.1], Aougab and Huang have proved that the number of -orbits of minimally intersecting fillings of size two satisfies
where . Note that, means .
The complexity of a filling is defined as the number of simple closed curves intersecting only once. Then for a filling pair , with equality if and only if is a minimal filling [AH15] [Theorem 1.2]. In [San17] [Theorem 1.4], Sanki has shown that for every pair of integers with , there exists a filling pair of such that the complement is a disjoint union of topological discs. Moreover, there is no minimal filling pair of closed surface of genus .
In [APP11] [Theorem 1], the authors have proved that, a filling of size with pairwise intersecting no more than times satisfies and moreover, the inequality is sharp. Further, if is a size filling set of systolic curves of a closed hyperbolic surface of genus , then and there exist hyperbolic surfaces of genus with a filling set of systolic curves (Theorem 3 [APP11]).
In this paper, we study minimal fillings and their geometric intersection numbers on oriented closed surfaces . We define,
Therefore, we focus on the growth of . We prove the theorem below:
, for all .
The proof of Theorem 1.1 is constructive and the main ingredients are basic topological graph theoretic arguments, cellular decomposition of surfaces and Euler’s equation.
Let be an integer. For each satisfying , there exists a minimal filling of with .
Finally, we study geometric intersection numbers of the curves in minimal fillings of .
Let be a minimal filling of . Then we have
Furthermore, for each satisfying , there exists a minimal filling of size such that
2. Fat graphs
In this section, we recall some definitions from graph theory, in particular fat graphs. Next, we define two binary operations on fat graphs so-called plumbing and connected sum. Finally, we conclude the section with Proposition 2.4 and Proposition 2.5, which are essential in the subsequent sections.
A graph is a triple where,
is a set of directed edges containing an even number of elements.
is an equivalence relation on and
is a fixed point free involution on which maps a directed edge to the edge with the reverse direction.
A fat graph structure on is a permutation on whose cycles correspond to cyclic order on the equivalence classes of .
The set of equivalence classes of is the vertex set of . We assume that degree of each vertex of a fat graph is at least . A fat graph is called decorated if the degree of each vertex is an even integer. A cycle in a fat graph is called standard if every two consecutive edges in the cycle are opposite to each other in the cyclic order at their shared vertex. Given a fat graph , the set of boundary components is denoted by , which corresponds to the set of cycles of the permutation (see Lemma 2.4 in [San17]).
By thickening the edges of a fat graph one obtain a unique topological surface and its genus is called the genus of the fat graph which we denote by .
Consider the following fat graph on , given as follows:
The set of equivalence classes of is , where
The fixed point free involution is defined by for
The permutation is given by .
It is easy to see that has one boundary component and standard cycles of lengths and . Therefore , where is some minimal filling triple of .
2.1. Plumbing of fat graphs
Let be two fat graphs and , be two undirected edges of respectively. The plumbing of and along the edges is the fat graph defined by the following (for a local picture of a plumbing, we refer Figure 2):
We split into two edges and into . Define,
The involution is as usual, for all .
Now, we define the equivalence classes of , equivalently the set of vertices: let . Define, , where
Define where .
Let be the cyclic order at a vertex . Then we define and further, . The fat graph structure is given by
The graph is said to be obtained by plumbing and along the edges .
Let be fat graphs and be two edges of respectively. If , , then the number of boundary components in is given by
Let and be the sets of boundary components of and respectively. There are four cases to be considered.
Consider, are in the same boundary component of , say and are in . Then one can write and , where are paths in the boundary (Figure 2). Then we have, where . Hence, we have
In this case, we consider and . We write, and . We have
where , which implies that
Consider and . Similarly, as in Case 1 and Case 2, we can write , . Then the boundary of the new fat graph is given by
where . This implies that
Here, assume and . This case is the same as Case 3, after interchange the role of and , hence we are done. ∎
2.2. Connected sum of Fat graphs
Motivated by the construction used in the proof of Theorem 1.4 in [San17], we define a binary operation on fat graphs called connected sum. Let , be two -regular fat graphs. Let and be two vertices of and respectively. Let be the cyclic orders at and respectively. We define the new graph , called the connected sum of and at the vertices and , which is described as follows:
The set of directed edges is given by,
where and are the set of directed edges of and respectively and the are defined by (see Figure 3 for a local picture),
The fixed point free involution is defined as usual for all .
Let , where and are the set of vertices and respectively. We define , where
Then the set equivalence classes of the relation is
If is the cyclic order at the vertex , then we define is the cyclic order at the vertex . Thus
For a fat graph and , we define as follows:
Let be two -regular fat graphs with boundary components, standard cycles (simple) and genus . Let and be two vertices of and respectively. Suppose that for for some . Then the connected sum of and at has the following properties:
The number of boundary components in is given by
The number of standard cycles in is .
The genus of is given by
Let us assume that and . Without loss of generality, we take for some directed paths on the boundary of . We begin by considering the case
the arguments in the remaining cases are similar.
In this case, we have assumed that the directed paths and are in different boundary components of . Therefore, we assume that