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The moduli space of $n$ chains in the plane with generic side lengths that terminate on a fixed line is a smooth, closed manifold of dimension $n-1$. This manifold is also equipped with a locally standard action of $\mathbb{Z}_2^{n-1}$. The…
This paper introduces an inductively defined tree notation for all the faces of polytopes arising from a simplex by truncations. This notation allows us to view inclusion of faces as the process of contracting tree edges. Our notation…
We apply the combinatorial theory of spherical varieties to characterize the momentum polytopes of polarized projective spherical varieties. This enables us to derive a classification of these varieties, without specifying the open orbit,…
We show how to construct homology bases for certain CW complexes in terms of discrete Morse theory and cellular homology. We apply this technique to study certain subcomplexes of the half cube polytope studied in previous works. This…
Triangulations of a product of two simplices and, more generally, of root polytopes are closely related to Gelfand-Kapranov-Zelevinsky's theory of discriminants, to tropical geometry, tropical oriented matroids, and to generalized…
The aim of this paper is to study alcoved polytopes, which are polytopes arising from affine Coxeter arrangements. This class of convex polytopes includes many classical polytopes, for example, the hypersimplices. We compare two…
A notion of "radially monotone" cut paths is introduced as an effective choice for finding a non-overlapping edge-unfolding of a convex polyhedron. These paths have the property that the two sides of the cut avoid overlap locally as the cut…
In this paper we give several criteria for the edge polytope of a fundamental FHM-graph to possess a regular unimodular triangulation in terms of some simple data of the the graph. We further apply our criteria to several examples of graphs…
Starting from the Maxwell's equations and without resort to the paraxial approximation, we derive equations describing stationary (1+1)-dimensional beams propagating at an arbitrary direction in an optical crystal with cubic symmetry and…
Cubic complexes appear in the theory of finite type invariants so often that one can ascribe them to basic notions of the theory. In this paper we begin the exposition of finite type invariants from the `cubic' point of view. Finite type…
For each finite Coxeter group $W$ and each standard Coxeter element of $W$, we construct a triangulation of the $W$-permutahedron. For particular realizations of the $W$-permutahedron, we show that this is a regular triangulation induced by…
For any finite set $\A$ of $n$ points in $\R^2$, we define a $(3n-3)$-dimensional simple polyhedron whose face poset is isomorphic to the poset of ``non-crossing marked graphs'' with vertex set $\A$, where a marked graph is defined as a…
This monograph starts with an upper triangular matrix with integer entries and 1's on the diagonal. It develops from this a spectrum of structures, which appear in different contexts, in algebraic geometry, representation theory and the…
We establish a stable homotopy-theoretic version of a recent result of Farber and Weinberger on the fibrewise topological complexity of sphere bundles and prove, by closely parallel methods, a similar result for real, complex and…
We investigate polyhedral $2k$-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex {\it $k$-Hamiltonian} if it contains the full $k$-skeleton of the polytope. Since the case of the cube is well…
By taking quotients of a certain tiling of hyperbolic plane / space by certain group actions, we obtain geometric polyhedra / cellulations with interesting symmetries and incidence structure.
Generalized associahedra are a well-studied family of polytopes associated to a finite-type cluster algebra and choice of starting cluster. We show that the generalized associahedra constructed by Padrol, Palu, Pilaud, and Plamondon,…
Tropical polytopes are images of polytopes in an affine space over the Puiseux series field under the degree map. This viewpoint gives rise to a family of cellular resolutions of monomial ideals which generalize the hull complex of Bayer…
Recent breakthroughs in the study of scattering amplitudes have uncovered profound and unexpected connections with combinatorial geometry. These connections range from classical structures -- such as polytopes, matroids, and Grassmannians…
We consider bases for the cohomology space of regular semisimple Hessenberg varieties, consisting of the classes that naturally arise from the Bialynicki-Birula decomposition of the Hessenberg varieties. We give an explicit combinatorial…