Boundary Complexes and Interior Points of the Polytopes

This chapter describes how the structure of a polytope of dimension n consisting of points of the boundary complex including a set of faces from zero to n - 1 and a set of interior points that are not belonging to the boundary complex is considered. The value is equal to the number of elements of the boundary complex, which the given element belongs, having dimension one greater than the given element of the boundary complex is denoted coefficient incidence of the given element. It is proven that the coefficient incidence of an element of dimension i of the boundary complex of an n - cube and n - simplex is equal to the difference of the dimension of the cube or simplex n and the dimension of this element i. The incidence coefficient of elements of n – cross - polytopes is substantially higher than this difference.

Drosophila Dosage Compensation Loci Associate with a Boundary-Forming Insulator Complex

ABSTRACT Chromatin entry sites (CES) are 100- to 1,500-bp elements that recruit male-specific lethal (MSL) complexes to the X chromosome to upregulate expression of X-linked genes in male flies. CES contain one or more ∼20-bp GA-rich sequences called MSL recognition elements (MREs) that are critical for dosage compensation. Recent studies indicate that CES also correspond to boundaries of X-chromosomal topologically associated domains (TADs). Here, we show that an ∼1,000-kDa complex called the late boundary complex (LBC), which is required for the functioning of the Bithorax complex boundary Fab-7, interacts specifically with a special class of CES that contain multiple MREs. Mutations in the MRE sequences of three of these CES that disrupt function in vivo abrogate interactions with the LBC. Moreover, reducing the levels of two LBC components compromises MSL recruitment. Finally, we show that several of the CES that are physically linked to each other in vivo are LBC interactors.

A Combinatorial Proof of a Formula for Betti Numbers of a Stacked Polytope

For a simplicial complex $\Delta$, the graded Betti number $\beta_{i,j}({\bf k}[\Delta])$ of the Stanley-Reisner ring ${\bf k}[\Delta]$ over a field ${\bf k}$ has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if $\Delta$ is the boundary complex of a $d$-dimensional stacked polytope with $n$ vertices for $d\geq3$, then $\beta_{k-1,k}({\bf k}[\Delta])=(k-1){n-d\choose k}$. We prove this combinatorially.

Algebraic shifting and strongly edge decomposable complexes

International audience Let $\Gamma$ be a simplicial complex with $n$ vertices, and let $\Delta (\Gamma)$ be either its exterior algebraic shifted complex or its symmetric algebraic shifted complex. If $\Gamma$ is a simplicial sphere, then it is known that (a) $\Delta (\Gamma)$ is pure and (b) $h$-vector of $\Gamma$ is symmetric. Kalai and Sarkaria conjectured that if $\Gamma$ is a simplicial sphere then its algebraic shifting also satisfies (c) $\Delta (\Gamma) \subset \Delta (C(n,d))$, where $C(n,d)$ is the boundary complex of the cyclic $d$-polytope with $n$ vertices. We show this conjecture for strongly edge decomposable spheres introduced by Nevo. We also show that any shifted simplicial complex satisfying (a), (b) and (c) is the algebraic shifted complex of some simplicial sphere.

Delannoy numbers and Legendre polytopes

International audience We construct an $n$-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of $(x-1)/2$ in the $n$-th Legendre polynomial. We show that the non-central Delannoy numbers count all faces in the lexicographic pulling triangulation that contain a point in a given open quadrant. We thus provide a geometric interpretation of a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago. The polytopes we construct are closely related to the root polytopes introduced by Gelfand, Graev, and Postnikov. \par No construisons un polytope de dimension $n$ dont le complexe de bord est comprimé et dont les nombres de faces dans toute triangulation "en tirant des sommets'' sont les coefficients des puissances de $(x-1)/2$ dans le $n$-ième polynôme de Legendre. Nous montrons que les nombres centraux de Delannoy comptent toutes les faces dans la triangulation "en tirant des sommets'' en ordre lexicographique qui contiennent un point dans un certain quadrant ouvert. Ainsi nous produisons une interprétation géométrique d'une relation entre les nombres de Delannoy centraux et les polynômes de Legendre, notée il y a 50 ans. Nos polytopes sont reliés intimement aux polytopes de racines introduits par Gelfand, Graev, et Postnikov.