Petrie Schemes

Abstract

Abstract. Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grünbaum-Dress polyhedra, possess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes.

Representing the sporadic Archimedean polyhedra as abstract polytopes

“…One such question is the determination of whether or not the given polytope has acoptic Petrie schemes 1 , a question related to understanding under what conditions a polyhedron will have Petrie polygons that form simple closed curves. First, we require some denitions; we will follow the second author's [Wil06]. A Petrie polygon of a polyhedron is a sequence of edges of the polyhedron where any two consecutive elements of the sequence have a vertex and face in common, but no three consecutive edges share a common face.…”

“…We borrow this terminology from Branko Grünbaum who coined the term acoptic (from the Greek κoπτ ω, to cut) to describe polyhedral surfaces with no selfintersections (cf. [Grü94,Grü97,Grü99,Wil06]). Let {σ 1 , σ 2 , .…”

Problems on polytopes, their groups, and realizations

“…Petrie schemes are combinatorial substructures in polytopes that generalize Petrie polygons (see [63,64]). Let P be an abstract n-polytope, let F be its set of flags, and let γ i (i = 0, .…”

On two types of Z-monodromy in triangulations of surfaces

“…Zigzags are known also as Petrie paths [2] or closed left-right paths [5,10]. Similar objects in simplicial complexes and abstract polytopes are investigated in [4,11]. An embedded graph is called z-knotted if it contains a single zigzag.…”