Curves on surfaces with prescribed pairwise intersection numbers

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.

arXiv comment20 pages, 3 figures