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The crossing number of polynomial curve systems

Sebastian Baader, Jasmin Jörg, Hugo Parlier

TL;DR

The paper determines the crossing number of polynomial-size curve systems on standard genus $g$ surfaces, showing sharp two-sided bounds for $Cr(g,[g^{1+\alpha}])$ in terms of $g$ and the growth exponent $\alpha$. It combines a known $\Omega$-type lower bound from the Hubard–HP inequality with a novel upper-bound construction built from fibre surfaces $\Sigma(p,q)$ of torus links to produce a polynomial-growth regime. The main result proves that, for large $g$, $Cr(g,[g^{1+\alpha}])$ scales as $\Theta(\alpha^2 g^{1+2\alpha} (\log g)^2)$, establishing both lower and upper bounds with explicit constants. The polynomial-growth regime is essential to capture how crossing numbers behave when the topology is fixed and the number of curves grows polynomially, revealing connections to exponential growth phenomena in the space of curves on surfaces. This work advances understanding of crossing numbers in topological graph theory and their dependence on genus, curve-count, and combinatorial constructions.

Abstract

We determine the crossing number of polynomial size curve systems on standard surfaces, in terms of the genus, up to high precision.

The crossing number of polynomial curve systems

TL;DR

The paper determines the crossing number of polynomial-size curve systems on standard genus surfaces, showing sharp two-sided bounds for in terms of and the growth exponent . It combines a known -type lower bound from the Hubard–HP inequality with a novel upper-bound construction built from fibre surfaces of torus links to produce a polynomial-growth regime. The main result proves that, for large , scales as , establishing both lower and upper bounds with explicit constants. The polynomial-growth regime is essential to capture how crossing numbers behave when the topology is fixed and the number of curves grows polynomially, revealing connections to exponential growth phenomena in the space of curves on surfaces. This work advances understanding of crossing numbers in topological graph theory and their dependence on genus, curve-count, and combinatorial constructions.

Abstract

We determine the crossing number of polynomial size curve systems on standard surfaces, in terms of the genus, up to high precision.
Paper Structure (3 sections, 1 theorem, 19 equations, 2 figures)

This paper contains 3 sections, 1 theorem, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Bounding the crossing number from below
  3. Polynomial families of curves

Key Result

Theorem 1

For all $\alpha \geq 0$, there exists a constant $N \in {\mathbb N}$, so that for all $g \geq N$

Figures (2)

  • Figure 1: Surface $\Sigma(3,4)$
  • Figure 2: Intersection of two curves in a common vertex

Theorems & Definitions (2)

  • Theorem 1
  • Remark 1