The order and chain polytopes, studied by Richard P. Stanley (1986), form a pair of Ehrhart equivalent polytopes associated to a given finite poset. A conjecture by Takayuki Hibi and Nan Li from 2016 states that the f -vector of the chain polytope dominates the f -vector of the order polytope. In this paper we prove a stronger form of that conjecture for a special class of posets. More precisely, we show that the f -vectors increase monotonically over an admissible family of chain-order polytopes for such posets.

Improved algorithms for computing (partial and full) exterior algebraic shifts of hypergraphs and simplicial complexes are presented. The main benefit is in positive characteristic. Experiments with an implementation in OSCAR with various inputs such as bipartite graphs and triangulations of two and three dimensional manifolds show that the method considerably extends for which simplicial complexes exterior algebraic shifts can be computed in practice.

Improved algorithms for computing (partial and full) exterior algebraic shifts of hypergraphs and simplicial complexes are presented. The main benefit is in positive characteristic. Experiments with an implementation in OSCAR are reported.

We investigate geometric embeddings among several classes of generalised cone complexes and algorithms, e.g., to compute their homology. Interesting cases arise from moduli spaces of tropical curves. Specifically, via an explicit computation, we show that the tropical honeycomb curves form a contractible sub-locus in the moduli of all tropical K4-curves.

Abstract Smooth tropical cubic surfaces are parametrized by maximal cones in the unimodular secondary fan of the triple tetrahedron. There are $$344\, 843 \,867$$ 344 843 867 such cones, organized into a database of $$14\,373\,645$$ 14 373 645 symmetry classes. The Schläfli fan gives a further refinement of these cones. It reveals all possible patterns of lines on tropical cubic surfaces, thus serving as a combinatorial base space for the universal Fano variety. This article develops the relevant theory and offers a blueprint for the analysis of big data in tropical geometry.

We present an interactive game which challenges a single player to match 3-dimensional polytopes to their planar nets. It is open source, and it runs in standard web browsers

We develop a tropical analogue of the simplex algorithm for linear programming. In particular, we obtain a combinatorial algorithm to perform one tropical pivoting step, including the computation of reduced costs, in $O(n(m+n))$ time, where $m$ is the number of constraints and $n$ is the dimension.

Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NP-hard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results.

polymake is a software package designed for the study of the combinato rics and the geometry of convex polytopes and polyhedra. It offers access to a wide variety of algorithms and tools within a common framework. As a key design feature it allows to incorporate the functionality of a great variety of other software packages in a modular way. polymake is open source software; it is freely available on the Internet at \url{http://www.math.tu-berlin.de/diskregeom/polymake/}. AMS Subject Classification (2000): 52-04 (52Bxx)