Related papers: Classification of Reflexive Polyhedra in Three Dim…

We classify the three-dimensional lattice polytopes with two interior lattice points. Up to unimodular equivalence there are 22,673,449 such polytopes. This classification allows us to verify, for this case only, a conjectural upper bound…

We apply combinatorial methods to a geometric problem: the classification of polytopes, in terms of Minkowski decomposability. Various properties of skeletons of polytopes are exhibited, each sufficient to guarantee indecomposability of a…

Kreuzer and Skarke famously produced the largest known database of Calabi-Yau threefolds by providing a complete construction of all 473,800,776 reflexive polyhedra that exist in four dimensions. These polyhedra describe the singular limits…

We describe an algorithm to enumerate polytopes. This algorithm is then implemented to give a complete classification of combinatorial spheres of dimension 3 with 9 vertices and decide polytopality of those spheres. In particular, we…

In previous work, we have commenced the task of unpacking the $473,800,776$ reflexive polyhedra by Kreuzer and Skarke into a database of Calabi-Yau threefolds (see http://www.rossealtman.com). In this paper, following a pedagogical…

We give coordinate-minimal geometric realizations in general position for 17 of the 20 vertex-minimal triangulations of the orientable surface of genus 3 in the 5x5x5-cube.

There exist exactly 166 4-dimensional reflexive polytopes such that the corresponding 4-dimensional Gorenstein toric Fano varieties have at worst terminal singularities in codimension 3 and their anticanonical divisor is divisible by 2. For…

We present an algorithm for the classification of triples of lattice polytopes with a given mixed volume $m$ in dimension 3. It is known that the classification can be reduced to the enumeration of so-called irreducible triples, the number…

When studying mirror symmetry in the context of K3 surfaces, the hyperkaehler structure of K3 makes the notion of exchanging Kaehler and complex moduli ambiguous. On the other hand, the metric is not renormalized due to the higher amount of…

A reflexive polytope, respectively its associated Gorenstein toric Fano variety, is called pseudo-symmetric, if the polytope has a centrally symmetric pair of facets. Here we present a complete classification of pseudo-symmetric simplicial…

We classify the 3-dimensional hyperbolic polyhedral orbifolds that contain no embedded essential 2-suborbifolds, up to decomposition along embedded hyperbolic triangle orbifolds (turnovers). We give a necessary condition for a 3-dimensional…

We construct a surprisingly large class of new Calabi-Yau 3-folds $X$ with small Picard numbers and propose a construction of their mirrors $X^*$ using smoothings of toric hypersurfaces with conifold singularities. These new examples are…

We present a general scheme for identifying fibrations in the framework of toric geometry and provide a large list of weights for Calabi--Yau 4-folds. We find 914,164 weights with degree $d\le150$ whose maximal Newton polyhedra are…

We analyze polyhedra composed of hexagons and triangles with three faces around each vertex, and their 3-regular planar graphs of edges and vertices, which we call "trihexes". Trihexes are analogous to fullerenes, which are 3-regular planar…

We prove a correspondence theorem for singular tropical surfaces in real three space, which recovers singular algebraic surfaces in an appropriate toric three-fold that tropicalize to a given singular tropical surface. Furthermore, we…

Which surfaces can be realized with two-dimensional faces of the five-dimensional cube (the penteract)? How can we visualize them? In recent work, Aveni, Govc, and Roldan, show that there exist 2690 connected closed cubical surfaces up to…

Calabi-Yau manifolds can be obtained as hypersurfaces in toric varieties built from reflexive polytopes. We generate reflexive polytopes in various dimensions using a genetic algorithm. As a proof of principle, we demonstrate that our…

A triangulation of a surface is irreducible if there is no edge whose contraction produces another triangulation of the surface. In this work we propose an algorithm that constructs the set of irreducible triangulations of any surface with…

A cubic polyhedron is a polyhedral surface whose edges are exactly all the edges of the cubic lattice. Every such polyhedron is a discrete minimal surface, and it appears that many (but not all) of them can be relaxed to smooth minimal…

Given arbitrary integers $d$ and $r$ with $d \geq 4$ and $1 \leq r \leq d + 1$, a reflexive polytope $\mathcal{P} \subset \mathbb{R}^d$ of dimension $d$ with ${\rm depth} K[\mathcal{P}] = r$ for which its dual polytope $\mathcal{P}^\vee$ is…