Curves on surfaces with prescribed pairwise intersection numbers

TL;DR

This paper establishes conditions for constructing collections of simple closed curves on a torus with specified intersection numbers and explores minimal genus realizations for such curve systems.

Contribution

It provides necessary and sufficient conditions for curve arrangements on a torus with prescribed intersection data and offers partial solutions for minimal genus realizations.

Findings

Characterization of intersection numbers for curves on a torus

Conditions for realizing intersection data on minimal genus surfaces

Partial results on minimal genus curve system realizations

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.