Cyclopermutohedron: geometry and topology

Abstract

The face poset of the permutohedron realizes the combinatorics of linearly ordered partitions of the set [InlineEquation not available: see fulltext.]. Similarly, the cyclopermutohedron is a virtual polytope that realizes the combinatorics of cyclically ordered partitions of the set [InlineEquation not available: see fulltext.]. The cyclopermutohedron was introduced by the second author by motivations coming from configuration spaces of polygonal linkages. In the paper we prove two facts: (a) the volume of the cyclopermutohedron equals zero, and (b) 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.