Notes on Reeb graphs of real algebraic functions which may not be planar

TL;DR

This paper explores the properties of Reeb graphs derived from real algebraic functions, especially focusing on those that cannot be embedded as planar graphs, advancing understanding of their topological and algebraic structures.

Contribution

The study investigates non-planar Reeb graphs of real algebraic functions, extending prior work that mainly dealt with planar graphs and embedding properties.

Findings

Identifies classes of Reeb graphs that are non-planar.

Provides methods for constructing real algebraic functions with specified Reeb graphs.

Highlights differences between planar and non-planar Reeb graphs in algebraic settings.

Abstract

The Reeb graph of a smooth function is a graph being a natural quotient space of the manifold of the domain and the space of all connected components of preimages. Such a combinatorial and topological object roughly and compactly represents the manifold. Since the proposal by Sharko in 2006, reconstructing nice smooth functions and the manifolds from finite graphs in such a way that the Reeb graphs are the graphs has been important. The author has launched new studies on this, discussing construction of real algebraic functions. We concentrate on Reeb graphs we cannot realize as (natural) planar graphs here. Previously the graphs were planar and embedded in the plane naturally.