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Filling with separating curves

Bhola Nath Saha, Bidyut Sanki

Abstract

A pair $(α, β)$ of simple closed curves on a closed and orientable surface $S_g$ of genus $g$ is called a filling pair if the complement is a disjoint union of topological disks. If $α$ is separating, then we call it as separating filling pair. In this article, we find a necessary and sufficient condition for the existence of a separating filling pair on $S_g$ with exactly two complementary disks. We study the combinatorics of the action of the mapping class group $\M$ on the set of such filling pairs. Furthermore, we construct a Morse function $\mathcal{F}_g$ on the moduli space $\mathcal{M}_g$ which, for a given hyperbolic surface $X$, outputs the length of shortest such filling pair with respect to the metric in $X$. We show that the cardinality of the set of global minima of the function $\mathcal{F}_g$ is the same as the number of $\M$-orbits of such filling pairs.

Filling with separating curves

Abstract

A pair of simple closed curves on a closed and orientable surface of genus is called a filling pair if the complement is a disjoint union of topological disks. If is separating, then we call it as separating filling pair. In this article, we find a necessary and sufficient condition for the existence of a separating filling pair on with exactly two complementary disks. We study the combinatorics of the action of the mapping class group on the set of such filling pairs. Furthermore, we construct a Morse function on the moduli space which, for a given hyperbolic surface , outputs the length of shortest such filling pair with respect to the metric in . We show that the cardinality of the set of global minima of the function is the same as the number of -orbits of such filling pairs.
Paper Structure (10 sections, 25 theorems, 35 equations, 15 figures)

This paper contains 10 sections, 25 theorems, 35 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Fat Graphs
  3. Classification of curves on fat graphs
  4. Existence of minimal separating filling pairs
  5. Existence of minimal separating filling pairs for odd genera
  6. Counting of Mapping Class Group Orbits.
  7. Oriented filling pair and filling permutation
  8. A lower bound of TEXT.
  9. Upper bound of N(g)
  10. The length function

Key Result

Theorem 1.2

There exists a minimally intersecting separating filling pair on $S_g$ if and only if g is even and $g\geq4$.

Figures (15)

  • Figure 2.1: Local picture of the surface obtained from a fat graph.
  • Figure 2.2: Example of fat graph and associated surface.
  • Figure 3.1: Filling pair $(\alpha, \beta)$ on $S_2$
  • Figure 3.2: Fat graph corresponding to the filling pair $(\alpha, \beta)$.
  • Figure 3.3: The normal matrix $M(\alpha,\beta)$.
  • ...and 10 more figures

Theorems & Definitions (59)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Lemma 2.6
  • ...and 49 more