Invariants of graph drawings in the plane

TL;DR

This paper provides an accessible overview of classical and modern results on planar graph drawings, emphasizing invariants like the van Kampen number and connecting to higher-dimensional topology and combinatorics.

Contribution

It offers simplified, elementary formulations of key invariants and results, making advanced topological concepts accessible to a broader mathematical audience.

Findings

Introduction of a mod2-valued self-intersection invariant

Connections between planar graph invariants and higher-dimensional topology

Accessible presentation of complex topological ideas

Abstract

We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results on the border of topology and combinatorics. We define a mod2-valued self-intersection invariant (i.e. the van Kampen number) and its generalizations. We present elementary formulations and arguments accessible to mathematicians not specialized in any of the areas discussed. So most part of this survey could be studied before textbooks on algebraic topology, as an introduction to starting ideas of algebraic topology motivated by algorithmic, combinatorial and geometric problems.