Cyclopermutohedron: geometry and topology
Ilia Nekrasov, Gaiane Panina, Alena Zhukova

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 . Similarly, the cyclopermutohedron is a virtual polytope that realizes the combinatorics of cyclically ordered partitions of the set . 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 for of the face poset of the cyclopermutohedron are non-zero free abelian groups. We also present a short formula for their ranks.
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