Curves on surfaces with prescribed pairwise intersection numbers

2025/08/22 by Öztürk, Ferit
#57K20 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2508.16323

Abstract

Given an ordered sequence of N-choose-2 integers, we give necessary and sufficient conditions to have an ordered collection of N simple closed curves on a torus such that the algebraic pairwise intersections of those curves are the given integers. We also present partial answers towards realizing curve systems on a surface with minimal genus, given the intersection numbers.

Citations

• Crossing lemmas for k-systems of arcs
• Curves on the torus intersecting at most k times
• The number of crossings in multigraphs with no empty lens
• Packing curves on surfaces with few intersections
• Arcs intersecting at most once

Related

• Crossing number of curves on surfaces
• Algebraic k-systems of curves
• A topological viewpoint on curves via intersection
• Large Collections of Curves Pairwise Intersecting Exactly Once
• Self-intersection numbers of curves on the punctured torus