Regular projections of graphs with at most three double points

TL;DR

This paper proves that a knotted generic immersion of a planar graph into the plane must have at least three double points, and that certain embeddings with up to three double points are free of complex links.

Contribution

It establishes a lower bound of three double points for knotted immersions of planar graphs and characterizes embeddings with few double points as free of Hopf links and trefoil knots.

Findings

Knotted immersions require at least three double points.

Embeddings with up to three double points are free of Hopf links and trefoil knots.

A criterion for total freedom of embeddings based on double points and link types.

Abstract

A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space to the 2-space. In this paper we show that if a generic immersion of a planar graph is knotted then the number of double points of the immersion is more than or equal to three. To prove this, we also show that an embedding of a graph obtained from a generic immersion of the graph (does not need to be planar) with at most three double points is totally free if it contains neither a Hopf link nor a trefoil knot.