Cyclopermutohedron: geometry and topology

TL;DR

This paper explores the geometry and topology of the cyclopermutohedron, a virtual polytope related to cyclically ordered partitions, revealing its zero volume and non-trivial homology groups.

Contribution

It proves the volume of the cyclopermutohedron is zero and characterizes the homology groups of its face poset, providing a new topological understanding.

Findings

Volume of the cyclopermutohedron equals zero

Homology groups of the face poset are non-zero free abelian groups

A formula for the ranks of these homology groups is provided

Abstract

The face poset of the permutohedron realizes the combinatorics of linearly ordered partitions of the set $[n]=\{1,...,n\}$. Similarly, the cyclopermutohedron is a virtual polytope that realizes the combinatorics of cyclically ordered partitions of the set $[n+1]$. The cyclopermutohedron was introduced by the third author by motivations coming from configuration spaces of polygonal linkages. In the paper we prove two facts: (1) the volume of the cyclopermutohedron equals zero, and (2) the homology groups $H_k$ for $k=0,...,n-2$ of the face poset of the cyclopermutohedron are non-zero free abelian groups. We also present a short formula for their ranks.