2022/06/14 by Foisy, Joel, Galván, Luis Ángel Topete, Knowles, Evan +3
#05C10 #57K10 #57M15 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2206.06877
A graph is intrinsically projectively linked (IPL) if its every embedding in projective space contains a nonsplit link. Some minor-minimal IPL graphs have been found previously. We determine that no minor-minimal IPL graphs on 16 edges exists and identify new minor-minimal IPL graphs by applying Δ-Y exchanges to K7-2e. We prove that for a nonouter-projective-planar graph G, G+K2 is IPL and describe the necessary and sufficient conditions on a projective planar graph G such that G+K2 is IPL. Lastly, we deduce conditions for f(G + K2) to have no nonsplit link, where G is projective planar, K2 = \w0,w1\, and f(G + K2) is the embedding onto ℝP3 with f(G) in z=0, w0 above z=0, and w1 below z=0 such that every edge connecting w0,w1 to G avoids the boundary of the 3-ball, whose antipodal points are identified to obtain projective space.