Intrinsically Linked Graphs in Projective Space

TL;DR

This paper characterizes graphs that are intrinsically linked in real projective space, identifying all minor-minimal such graphs with low connectivity and establishing new results on Petersen-family graphs and K7 minus edges.

Contribution

It provides a complete characterization of intrinsically linked graphs in projective space with connectivity up to 2 and identifies all minor-minimal examples, including a unique Petersen-family graph.

Findings

594 graphs are minor-minimal intrinsically linked in projective space

Only one Petersen-family graph is intrinsically linked in projective space

K7 minus any two edges is minor-minimal intrinsically linked

Abstract

We examine graphs that contain a non-trivial link in every embedding into real projective space, using a weaker notion of unlink than was used by Flapan, et al. We call such graphs intrinsically linked in projective space. We fully characterize such graphs with connectivity 0,1 and 2. We also show that only one Petersen-family graph is intrinsically linked in projective space and prove that K7 minus any two edges is also minor-minimal intrinsically linked. In all, 594 graphs are shown to be minor-minimal intrinsically linked in projective space.