One-Dimensional Projections and Monotone Paths on Cross-Polytopes and Other Regular Polytopes

Alexander Black, Jesús De Loera

Published 2021-02-02Version 1

Projections of regular convex polytopes have been studied for a long time (see work by Coxeter, Petrie, Schl\"afli, others). One-dimensional linear projections may not appear interesting at first sight, but a linear functional orders the vertices of a polytope and yields an orientation of the graph of the polytope in question. The monotone paths of a polytope are the directed paths on the oriented one-skeleton from the minimum to the maximum. Here we revisit projections for regular polytopes from the lens of monotone paths. Billera and Sturmfels introduced a construction in the early 1990s called the monotone path polytope, which is a polytope whose face lattice encodes the combinatorial structure of certain monotone paths on a polytope. They computed the monotone path polytopes of simplices and cubes, which yield a combinatorial cube and permutahedron respectively. However, the question remains: What are the monotone path polytopes of the remaining regular polytopes? We answer that question in this paper for cross-polytopes, the dodecahedron, the icosahedron, and partially for the 24-cell, 120-cell, and the 600-cell.