Cyclopermutohedron: geometry and topology

European Journal of Mathematics Cyclopermutohedron: geometry and topology Ilia Nekrasov 0 1 2 Gaiane Panina 0 1 2 Alena Zhukova 0 1 2 0 Faculty of Liberal Arts and Sciences, St. Petersburg State University , Universitetskaya nab. 7-9, St. Petersburg 199034 , Russia 1 Mathematics and Mechanics Faculty, St. Petersburg State University , Universitetsky pr. 28, Stary Peterhof 198504 , Russia 2 Mathematics Subject Classification 51M20 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 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. The present research is supported by RFBR, research Project No. 15-01-02021. The first author is also supported by JSC “Gazprom Neft”. The third author is also supported in part by the Young Russian Mathematics Award. Permutohedron; Virtual polytope; Discrete Morse theory; Abel polynomial - The standard permutohedron n (see [ 7 ]) is defined as the convex hull of all points in Rn that are obtained by permuting the coordinates of the point (1, 2, . . . , n). It has the following properties: (I) (a) The k-faces of n are labeled by ordered partitions of the set [n] = {1, 2, . . . , n} into n − k non-empty parts. (b) A face F of n is contained in a face F iff the label of F refines the label of F . Here and in the sequel, by a refinement we mean an order preserving refinement. For instance, the label ({1, 3} {5, 6} {4} {2}) refines the label ({1, 3} {5, 6} {2, 4}) but does not refine ({1, 3} {2, 4} {5, 6}). (II) (III) n is an (n − 1)-dimensional polytope. n is a zonotope, that is, the Minkowski sum of line segments qi j , whose defining vectors are {ei −e j }1 i< j n, where {ei }1 i n are the standard orthonormal basis vectors. By analogy, we replace the linear order by cyclic order and build up the following regular1 cell complex CPn+1 [ 3 ], see Fig. 1. (I) Assume that n > 2. For k = 0, . . . , n − 2, the k-dimensional cells (k-cells, for short) of the complex CPn+1 are labeled by (all possible) cyclically ordered partitions of the set [n + 1] = {1, . . . , n + 1} into n − k + 1 non-empty parts. (II) A (closed) cell F contains a cell F whenever the label of F refines the label of F . Here we again mean order-preserving refinement. The cyclopermutohedron CPn+1 is a virtual polytope whose face poset is combinatorially isomorphic to complex CPn+1. More details will be given in Sect. 2.1; for a complete presentation see [ 3 ]. In the paper we study geometry and topology of the cyclopermutohedron. Before we formulate the main result some explanation is needed. The cyclopermutohedron is a virtual polytope, that is, the Minkowski difference of two convex polytopes. 1 To define a regular cell complex, it suffices to list all closed cells of the complex together with the incidence relations. A detailed discussion on virtual polytopes can be found in the survey [ 4 ]. One of the messages of the survey is that virtual polytopes inherit almost all properties and structures of convex polytopes: the volume (together with its polynomiality property), normal fan, face poset, etc. However, virtual polytopes do not inherit the convexity property and therefore may appear as counter-intuitive: (a) The volume of a virtual polytope, although well-defined, can be negative, see [ 2,4 ]. The volume also can turn to zero, even if the virtual polytope does not degenerate. (b) The face poset of a virtual polytope is also well-defined. However, it is not necessarily isomorphic to a combinatorial sphere. So one can expect non-zero homologies in all dimensions. The main results of the paper are: Theorem 1.1 The volume of the cyclopermutohedron CPn+1 equals zero. Theorem 1.2 The homology groups of the face poset of the cyclopermutohedron CPn+1 are free abelian groups. Their ranks are: ⎧ rk(Hk (CPn+1, Z)) = ⎪⎪⎪⎪⎨ n k , 2n + ⎪⎪ ⎪⎪⎩ 0, , Let us understand the meaning of these theorems for the toy example n = 3, that is, for CP4. The complex CP4 (and therefore, the face poset of the cyclopermutohedron) is the graph with six vertices and twelve edges, see Fig. 2, left. Its Betti numbers are 1 and 7. The cyclopermutohedron CP4 (computed in [ 3 ]) can be represented by a closed polygon, whereas its area (that is, two-dimensional volume) equals the integral of the winding number against the Lebesgue measure (see [ 4 ]). In other words, in this case the volume equals “sum of areas of six small triangles minus the area of the hexagon” i (...truncated)