Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Filling systems of maximum size

Rakesh Kumar, Shiv Parsad

TL;DR

The paper addresses the maximal size of filling systems on closed orientable surfaces with prescribed boundary components and shows the maximum equals $U_{g,b}=2g+b-1$ for $1\le b\le 2g-2$. It develops a fat-graph (combinatorial) framework to realize fillings, providing explicit constructions of 4-regular fat graphs that achieve the bound across the allowed range, and demonstrates sharpness for $b=1$ and $b=2$ while detailing two construction cases for $3\le b\le 2g-2$. A key contribution is a lower bound on the number of mapping class group orbits of maximal-size fillings, expressed via equivalence classes of integer partitions under the symmetric group action. The results illuminate the geometry–combinatorics interplay in the mapping class group and Teichmüller theory, and clarify the limits of the bound through a counterexample when $b>2g-2$.

Abstract

Let $S_g$ be a closed orientable surface of genus $g\geq 2$. A collection $Ω= \{ γ_1, \dots, γ_s\}$ of pairwise non-homotopic simple closed curves on $S_g$ such that $γ_i$ and $γ_j$ are in minimal position, is called a \emph{filling system} or a \emph{filling} of $S_g$ if the complement $S_g\setminus Ω$ is a disjoint union of $b$ topological discs for some $b\geq 1$. The \emph{size} of a filling system is defined as the number of its elements. We prove that the maximum size of a filling system on $S_g$ with $ 1 \leq b \leq 2g-2$ boundary components is $2g+b-1$. Furthermore, we give a lower bound on mapping class group orbits of filling systems of maximum size with $ 1 \leq b \leq g-2$ boundary components.

Filling systems of maximum size

TL;DR

The paper addresses the maximal size of filling systems on closed orientable surfaces with prescribed boundary components and shows the maximum equals for . It develops a fat-graph (combinatorial) framework to realize fillings, providing explicit constructions of 4-regular fat graphs that achieve the bound across the allowed range, and demonstrates sharpness for and while detailing two construction cases for . A key contribution is a lower bound on the number of mapping class group orbits of maximal-size fillings, expressed via equivalence classes of integer partitions under the symmetric group action. The results illuminate the geometry–combinatorics interplay in the mapping class group and Teichmüller theory, and clarify the limits of the bound through a counterexample when .

Abstract

Let be a closed orientable surface of genus . A collection of pairwise non-homotopic simple closed curves on such that and are in minimal position, is called a \emph{filling system} or a \emph{filling} of if the complement is a disjoint union of topological discs for some . The \emph{size} of a filling system is defined as the number of its elements. We prove that the maximum size of a filling system on with boundary components is . Furthermore, we give a lower bound on mapping class group orbits of filling systems of maximum size with boundary components.

Paper Structure

This paper contains 4 sections, 3 theorems, 5 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Fat graphs
  3. Filling Systems Of Maximum Size
  4. Upper Bound On Mapping Class Orbits of Maximum Size Filling

Key Result

Theorem 1.1

Let $S_g$ be a closed orientable surface of genus $g\geq2$ and let $U_{g,b}$ denotes the maximum size of filling systems on $S_{g}$ with $b$ boundary components. Then $U_{g,b}=2g+b-1$, for $1 \leq b \leq 2g-2$.

Figures (10)

  • Figure 1: $\Gamma_0$
  • Figure 2: The fat graph $\Gamma_{(g,g-1)}$
  • Figure 3: The fat graph $\Gamma_{(g,b)}$ for $g\leq b \leq 2g-2$
  • Figure 4: The fat graph $\Gamma_{(g,b)}$ for $3 \leq b \leq g-2$
  • Figure 5: $\Gamma_1$
  • ...and 5 more figures

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Example 2.2
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['orbit']}