On approximability by embeddings of cycles in the plane

TL;DR

This paper establishes a criterion for when a piecewise linear map of a circle into the plane can be approximated by embeddings, based on derivatives and winding properties, extending previous results and criteria.

Contribution

It provides a new criterion for approximability by embeddings of circle maps into the plane, generalizing Minc's results and analyzing derivatives and winding numbers.

Findings

A criterion based on derivatives and winding numbers determines approximability.

The van Kampen obstruction is complete for segment maps to the plane.

Criteria are extended to certain graph maps without high-degree vertices.

Abstract

We obtain a criterion for approximability by embeddings of piecewise linear maps of a circle to the plane, analogous to the one proved by Minc for maps of a segment to the plane. Theorem. Let S be a triangulation of a circle with s vertices. Let f be a simplicial map of the graph S to the plane. The map f is approximable by embeddings if and only if for each i=0,...,s the i-th derivative of the map f (defined by Minc) neither contains transversal self-intersections nor is the standard winding of degree greater than 1. We deduce from the Minc result the completeness of the van Kampen obstruction to approximability by embeddings of piecewise linear maps of a segment to the plane. We also generalize these criteria to simplicial maps of a graph without vertices of degree >3 to a circle.