Hamiltonian submanifolds of regular polytopes

TL;DR

This paper studies special subcomplexes called k-Hamiltonian manifolds within regular polytopes, determining existence results for surfaces and 4-manifolds, including a new highly symmetric example with 16 vertices.

Contribution

It provides new existence results for 1-Hamiltonian surfaces and 2-Hamiltonian 4-manifolds in regular polytopes, including a novel example with high symmetry.

Findings

Existence of 1-Hamiltonian surfaces is now fully determined for all regular polytopes.

Constructed a new 16-vertex 2-Hamiltonian 4-manifold with automorphism group of order 128.

All regular cases with fewer than 20 vertices or up to 9-dimensional polytopes are now resolved.

Abstract

We investigate polyhedral $2k$-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex {\it $k$-Hamiltonian} if it contains the full $k$-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called {\it super-neighborly triangulations}) we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the $d$-dimensional cross polytope. These are the "regular cases" satisfying equality in Sparla's inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of 7 copies of $S^2 \times S^2$. By this example all regular cases of $n$ vertices with $n < 20$ or, equivalently, all cases of regular $d$-polytopes with $d\leq 9$ are now decided.