Crossing number of curves on surfaces

Abstract

We consider systems of simple closed curves on surfaces and their total number of intersection points, their so-called crossing number. For a fixed number of curves, we aim to minimise the crossing number. We determine the minimal crossing number of up to 12 curves on a surface of genus 2 and prove that minimising systems are unique up to homeomorphisms of the surface and isotopies of curves.

Figures (4)

• Figure 1: Curves $\alpha_1, \dots, \alpha_{2g-3}$,$\beta_1, \dots, \beta_g$,$\gamma_1, \dots, \gamma_g$,$\delta_1, \dots, \delta_g$
• Figure 2: Curves $\delta_1, \dots, \delta_{11}$
• Figure 3: Minimal system of 12 curves
• Figure 4: Two different minimal systems of 14 curves in a surface of genus 4

Theorems & Definitions (51)

• Theorem 1.1
• Proposition 3.1
• proof
• Lemma 3.2
• proof
• Corollary 3.3
• Lemma 4.1
• proof
• Definition 4.2
• Lemma 4.3