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Crossing number inequalities for curves on surfaces

Alfredo Hubard, Hugo Parlier

TL;DR

The paper addresses crossing numbers for drawings of non-homotopic curves on orientable surfaces, proving a lower bound of cr$(\Gamma) \gtrsim (m \log m)^2$ for a family of $m$ distinct curves and extending the result to general surfaces with a bound depending on $|\chi|$. The authors leverage the Koebe–Andreev–Thurston uniformization to work in a hyperbolic metric, apply circle packing to realize a length/area framework, and derive length-based lower bounds via averaging to obtain the main crossing inequality. They then translate these curve results into corollaries for arcs and graphs (via arc-to-curve translations) and provide explicit upper bounds on the size of arc/curve families with bounded pairwise intersections. Finally, they demonstrate near-optimality of the bounds through combinatorial pants models and explicit pants-based constructions, yielding tight growth rates up to constants on both arcs and curves across surfaces. These results settle questions of Pach–Tardos–Toth about crossing numbers for non-homotopic edges and establish optimal growth rates on orientable surfaces, enriching the understanding of topological graph drawings.

Abstract

We prove that, as $m$ grows, any family of $m$ homotopically distinct closed curves on a surface induces a number of crossings that grows at least like $(m \log m)^2$. We use this to answer two questions of Pach, Tardos and Toth related to crossing numbers of drawings of multigraphs where edges are required to be non-homotopic. Furthermore, we generalize these results, obtaining effective bounds with optimal growth rates on every orientable surface.

Crossing number inequalities for curves on surfaces

TL;DR

The paper addresses crossing numbers for drawings of non-homotopic curves on orientable surfaces, proving a lower bound of cr for a family of distinct curves and extending the result to general surfaces with a bound depending on . The authors leverage the Koebe–Andreev–Thurston uniformization to work in a hyperbolic metric, apply circle packing to realize a length/area framework, and derive length-based lower bounds via averaging to obtain the main crossing inequality. They then translate these curve results into corollaries for arcs and graphs (via arc-to-curve translations) and provide explicit upper bounds on the size of arc/curve families with bounded pairwise intersections. Finally, they demonstrate near-optimality of the bounds through combinatorial pants models and explicit pants-based constructions, yielding tight growth rates up to constants on both arcs and curves across surfaces. These results settle questions of Pach–Tardos–Toth about crossing numbers for non-homotopic edges and establish optimal growth rates on orientable surfaces, enriching the understanding of topological graph drawings.

Abstract

We prove that, as grows, any family of homotopically distinct closed curves on a surface induces a number of crossings that grows at least like . We use this to answer two questions of Pach, Tardos and Toth related to crossing numbers of drawings of multigraphs where edges are required to be non-homotopic. Furthermore, we generalize these results, obtaining effective bounds with optimal growth rates on every orientable surface.

Paper Structure

This paper contains 14 sections, 15 theorems, 49 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The proof of the crossing theorem for curves
  4. Two corollaries: bounding sets of curves and a crossing theorem for graphs
  5. From curves to arcs
  6. Curves and arcs which pairwise intersect at most $k$ times
  7. Optimality of the bounds
  8. Curves and arcs on a combinatorial pair of pants
  9. Counting curves and arcs on a pair of pants
  10. Arcs
  11. Curves
  12. Counting on a general surface and final estimates
  13. Arcs on a punctured sphere
  14. The case of curves

Key Result

Theorem 1.1

Let $G$ be a graph with $n$ vertices and $m$ edges drawn on a closed surface of genus $g\geq 0$ such that no two edges are homotopic. Then, for $m \geq e^6(|\chi|+1)$, its crossing number satisfies

Figures (6)

  • Figure 1: From an arc to a curve
  • Figure 2: Intersections
  • Figure 3: Constructing a pair of pants $P$ with $12$ squares
  • Figure 4: The first segments of arcs on $P$ (front on left, back on right)
  • Figure 5: The first segments of arcs on $P$ (front on left, back on right)
  • ...and 1 more figures

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 4.1: Theorem \ref{['thm:graphs']}
  • Lemma 4.2
  • ...and 12 more