Using sutured Floer homology (in short SFH) I will define a polytope inside the second relative cohomology group of a sutured manifold. This is a generalization of the dual Thurston norm polytope of a link-complement studied by Ozsvath and Szabo using link Floer homology. The polytope is maximal dimensional under certain conditions. Moreover, surface decompositions correspond to the faces of th...

We introduce the interval order polytope of a digraph D as the convex hull of interval order inducing arc subsets of D. Two general schemes for producing valid inequalities are presented. These schemes have been used implicitly for several polytopes and they are applied here to the interval order polytope. It is shown that almost all known classes of valid inequalities of the linear ordering po...

A theorem of Howe states that every 3-dimensional lattice polytope P whose only lattice points are its vertices, is a Cayley polytope, i.e. P is the convex hull of two lattice polygons with distance one. We want to generalize this result by classifying 3-dimensional lattice polytopes without interior lattice points. The main result will be, that they are up to finite many exceptions either Cayl...

By δ and wk denote the minimum degree and minimum degree-sum (weight) of a k-vertex path in a given graph, respectively. For every 3-polytope, w2 6 13 (Kotzig, 1955) and w3 6 21 (Ando, Iwasaki, Kaneko, 1993), where both bounds are sharp. For every 3-polytope with δ > 4, we have sharp bounds w2 6 11 (Lebesgue, 1940) and w3 6 17 (Borodin, 1997). Madaras (2000) proved that every triangulated 3-pol...

Let P be a convex polytope containing the origin, whose dual is a lattice polytope. Hibi’s Palindromic Theorem tells us that if P is also a lattice polytope then the Ehrhart δ-vector of P is palindromic. Perhaps less well-known is that a similar result holds when P is rational. We present an elementary lattice-point proof of this fact.

The Hard Lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope. One can construct the intersection cohomology combinatorially from the polytope, hence it is well defined even for nonrational polytopes when there is no variety associated to it. We prove the Hard Lefschetz theorem for the intersection cohomology of a gener...

In this paper we study some geometrical questions about the polytope of bi-capacities. For this, introduce concept pointed order polytope, a natural generalization polytopes. Basically, is that takes advantage relation partially ordered set and such there relevant element in structure. We which are vertices polytopes sort out simple way to determine whether two adjacent. also general form its f...

A compressed polytope is an integral convex polytope any of whose reverse lexicographic initial ideals is squarefree. A sufficient condition for a (0, 1)-polytope to be compressed will be presented. One of its immediate consequences is that the class of compressed (0, 1)-polytopes includes (i) hypersimplices, (ii) order polytopes of finite partially ordered sets, and (iii) stable polytopes of p...

A positroid is a special case of realizable matroid that arose from the study totally nonnegative part Grassmannian by Postnikov. In this paper, we facets its polytope and independent set polytope. This allows one to describe bases sets directly decorated permutation, bypassing use Grassmann necklace. We also criterion for determining whether given cyclic interval flat or not using then show ho...

We investigate the question of whether any d-colorable simplicial d-polytope can be octahedralized, i.e., subdivided to a d-dimensional geometric cross-polytopal complex. give positive answer in dimension 3, with additional property that octahedralization introduces no new vertices on boundary polytope.