Orthogonal surfaces

TL;DR

This paper explores the structure of orthogonal surfaces in higher dimensions, focusing on their characteristic points, dominance orders, and conditions for polytopal realizability, revealing complex behaviors beyond the well-understood 2D and 3D cases.

Contribution

It investigates the properties of cp-orders in higher-dimensional orthogonal surfaces and identifies conditions for their polytopal realizability, extending understanding beyond generic cases.

Findings

cp-orders can lack key face lattice properties in general

extra conditions may ensure cp-orders are face lattices

certain polytopes are realizable on orthogonal surfaces

Abstract

Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial already the three dimensional case has a rich structure with connections to Schnyder woods, planar graphs and 3-polytopes. Our objective is to detect more of the structure of orthogonal surfaces in four and higher dimensions. In particular we are driven by the question which non-generic orthogonal surfaces have a polytopal structure. We study characteristic points and the cp-orders of orthogonal surfaces, i.e., the dominance orders on the characteristic points. In the generic case these orders are (almost) face lattices of polytopes. Examples show that in general cp-orders can lack key properties of face lattices. We investigate extra requirements which may help to have cp-orders which are face lattices. Finally, we turn the focus and ask for the realizability of polytopes on orthogonal surfaces. There are criteria which prevent large classes of simplicial polytopes from being realizable. On the other hand we identify some families of polytopes which can be realized on orthogonal surfaces.