Characterization of affine links in the projective space

TL;DR

This paper characterizes affine links in real projective 3-space, showing they are precisely those whose complement's fundamental group contains an element of order two, linking topology and algebraic properties.

Contribution

It provides a complete topological characterization of affine links in projective space based on the fundamental group's properties.

Findings

Affine links correspond to fundamental groups with elements of order two.

A projective link is affine iff its complement's fundamental group contains a non-trivial element of order two.

The main theorem establishes a clear algebraic-topological criterion for affineness.

Abstract

A projective link is a smooth closed 1-submanifold of the real projective space of dimension three. A projective link is said to be affine if it is isotopic to a link, which does not intersect some projective plane. The main result: a projective link is affine if and only if the fundamental group of its complement contains a non-trivial element of order two.