The Graph of a Convex 4-Polytope Is Flat in the Boundary Whenever It Is Linklessly Embeddable: An Answer to a Question of Izmestiev

Problem 7 of the problem list of the Oberwolfach workshop “Discrete Differential Geometry” of 2009 (collected by G. Rote; Oberwolfach Reports 6, p. 108), posed by I. Izmestiev, asks whether there is a convex 4-polytope whose 1-skeleton is linked as a graph in the boundary of the polytope, a 3-sphere, but has a different embedding that is not linked. We show that there is none. Let \(P\) be a convex 4-polytope and \(\varphi\) a piecewise linear embedding of its graph in the 3-sphere in which the boundary of every 2-face of \(P\) bounds a disc meeting the graph only in this boundary; every flat embedding is of this kind. Then \(\varphi\) is the embedding of the graph in \(\partial P\) followed by a piecewise linear homeomorphism of 3-spheres. Hence the following are equivalent: the graph of \(P\) has no minor in the Petersen family; it has an embedding without two disjoint cycles of odd linking number; its embedding in \(\partial P\) is flat; any two of its disjoint cycles bound disjoint discs in \(\partial P\). This gives the negative answer for each of the usual meanings of “linked”. The proof is a short application of a lemma of Böhme, in the form given by Robertson, Seymour and Thomas, and of Alexander's theorem on polyhedral spheres; the first statement also follows from that lemma and a uniqueness theorem of Georgakopoulos and Kim. The equivalence relies on the theorem of Robertson, Seymour and Thomas on linkless embeddings, which is deep and is not reproved here. As a consequence, the null-space embedding of the Colin de Verdière matrix that Izmestiev attaches to a 4-polytope is the boundary embedding, hence flat when the graph is linklessly embeddable; the general question of Lovász and Schrijver about null spaces of such matrices remains open. As a consistency test, which is not part of the proof, exact computations for all \(275{,}478\) combinatorial types of 4-polytopes with at most nine vertices find two disjoint cycles with non-zero linking number in the boundary exactly when the graph has a minor in the Petersen family. No novelty is claimed beyond the answer to the question. This is an unrefereed note.