Rotation systems and simple drawings in surfaces

TL;DR

This paper explores the relationship between rotation systems and simple graph drawings on surfaces, demonstrating that not all rotation systems can be realized by simple drawings on any fixed surface.

Contribution

It extends the understanding of rotation systems beyond the plane to all surfaces, showing the existence of non-realizable rotation systems in these contexts.

Findings

Not all rotation systems are realizable as simple drawings on a given surface.

The result holds for all fixed surfaces, not just the plane.

Provides insight into the limitations of representing graphs on surfaces.

Abstract

Every simple drawing of a graph in the plane naturally induces a rotation system, but it is easy to exhibit a rotation system that does not arise from a simple drawing in the plane. We extend this to all surfaces: for every fixed surface $\Sigma$, there is a rotation system that does not arise from a simple drawing in $\Sigma$.