Cyclopermutohedron: geometry and topology

الأصل · EN

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 Hₖ 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.