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On the crossing number of arithmetic curve systems

Sebastian Baader, Claire Burrin, Luca Studer

Abstract

We show that the family of systoles of hyperbolic surfaces associated with congruence lattices in $\mathrm{SL}_2(\mathbb{Z})$ have asymptotically minimal crossing number.

On the crossing number of arithmetic curve systems

Abstract

We show that the family of systoles of hyperbolic surfaces associated with congruence lattices in have asymptotically minimal crossing number.
Paper Structure (4 sections, 6 theorems, 30 equations)

This paper contains 4 sections, 6 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Lower bound on the crossing number
  3. Counting intersections of systoles
  4. Subsystems with optimal crossing number

Key Result

Theorem 1

The family of systoles $\mathcal{C}(N)$ of the surfaces $X(N)$ satisfies Moreover, every other family $\mathcal{D}(N)$ of pairwise non-homotopic simple closed curve systems in the same surface $X(N)$ with $|\mathcal{D}(N)|=|\mathcal{C}(N)|$ satisfies

Theorems & Definitions (12)

  • Theorem 1
  • Proposition 1
  • proof : Proof of Proposition \ref{['doublepoints']}
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark
  • ...and 2 more