Abstract. We use the Cayley Trick to study polyhedral subdivisions of the product of two simplices. For arbitrary (fixed) l ≥ 2, we show that the numbers of regular and non-regular triangulations of ∆ l × ∆ k grow, respectively, as k Θ(k) and 2 Ω(k2). For the special case of ∆ 2 × ∆ k, we relate triangulations to certain class of lozenge tilings. This allows us to compute the exact number of triangulations up to k = 15, show that the number grows as e βk2 /2+o(k 2) where β ≃ 0.32309594 and prove that the set of all triangulations is connected under geometric bistellar flips. The latter has as a corollary that the toric Hilbert scheme of the determinantal ideal of 2×2 minors of a 3×k matrix is connected, for every k. We include “Cayley Trick pictures ” of all the triangulations of ∆ 2 × ∆ 2 and ∆ 2 × ∆ 3, as well as one non-regular triangulation of ∆ 2 × ∆ 5 and another of ∆ 3 × ∆ 3.

We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation.

Abstract. For a finite abelian group G ⊂ GL(n, k), let Yθ be the coherent component of the moduli space of θ-stable representations of the McKay quiver. We calculate the G-equivariant k[x1,...,xn]-module parameterized by each point of Yθ via Gröbner bases. In the case Mθ ∼ = G-Hilb, we show that G-Hilb may be reducible and its coherent component Yθ ∼ = Hilb G may be nonnormal, giving examples for G in GL(3, k) and GL(6, k) respectively. The latter answers a question of Nakamura. 1.

Abstract. Regular triangulations of products of lattice polytopes are constructed with the additional property that the dual graphs of the triangulations are bipartite. The (weighted) size difference of this bipartition is a lower bound for the number of real roots of certain sparse polynomial systems by recent results of Soprunova and Sottile [Adv. Math. 204(1):116–151, 2006]. Special attention is paid to the cube case. 1.

We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include intersections of lattices with convex sets, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.

This paper studies two related subjects. One is some combinatorics arising from linear projections of polytopes and fans of cones. The other is quotient varieties of toric varieties. The relation is that projections of polytopes are related to quotients of projective toric varieties and projection of fans

(without boundary) is with no doubt one of the most prominent tasks in topology ever since Poincaré’s fundamental work [88] on ≪l’analysis situs≫ appeared in 1904. There are various ways for constructing 3-manifolds, some of which that are general enough to yield all 3-manifolds (orientable or nonorientable) and some that produce only particular types or classes of examples. According to Moise [73], all 3-manifolds can be triangulated. This implies that there are only countably many distinct combinatorial (and therefore at most so many different topological) types that result from gluing together tetrahedra. Another way to obtain 3-manifolds is by starting with a solid 3-dimensional polyhedron for which surface faces are pairwise identified (see, e.g., Seifert [98] and Weber and Seifert [118]). Both approaches are rather general and, on the first sight, do not give much control on the kind of manifold we can expect as an outcome. However, if we want to determine the topological type of some given triangulated 3-manifold, then small or minimal triangulations

. Given an affine projection : P ! Q of a d-polytope P onto a polygon Q, it is proved that the poset of proper polytopal subdivisions of Q which are induced by has the homotopy type of a sphere of dimension d \Gamma 3 if maps all vertices of P into the boundary of Q. This result, originally conjectured by Reiner, is an analogue of a result of Billera, Kapranov and Sturmfels on cellular strings on polytopes and explains the significance of the interior point of Q present in the counterexample to their generalized Baues conjecture, constructed by Rambau and Ziegler. 1. Introduction Motivated by their theory of fiber polytopes [6] [18, Lecture 9], Billera and Sturmfels have associated to any affine projection of convex polytopes : P ! Q the Baues poset !(P ! Q) of proper polytopal subdivisions of Q which are induced by . This poset reduces to the poset of proper cellular strings [7] on P with respect to , if dim(Q) = 1, and can be described in general as the poset of proper ...

. Monotone paths on zonotopes and the natural generalization to maximal chains in the poset of topes of an oriented matroid or arrangement of pseudo-hyperplanes are studied with respect to a kind of local move, called polygon move or flip. It is proved that any monotone path on a d-dimensional zonotope with n generators admits at least d2n=(n \Gamma d + 2)e \Gamma 1 flips for all n d+2 4 and that for any fixed value of n \Gamma d, this lower bound is sharp for infinitely many values of n. In particular, monotone paths on zonotopes which admit only three flips are constructed in each dimension d 3. Furthermore, the previously known 2-connectivity of the graph of monotone paths on a polytope is extended to the 2-connectivity of the graph of maximal chains of topes of an oriented matroid. An application in the context of Coxeter groups of a result known to be valid for monotone paths on simple zonotopes is included. 1. Introduction Let P be a d-dimensional polytope in R ...

Abstract. We study flip graphs of (pseudo-)triangulations whose maximum vertex degree is bounded by a constant k. In particular, we consider (pseudo-)triangulations of sets of n points in convex position in the plane and prove that their flip graph is connected if and only if k> 6; the diameter of the flip graph is O(n 2). We also show that for general point sets flip graphs of minimum pseudo-triangulations can be disconnected for k ≤ 9, and flip graphs of triangulations can be disconnected for any k. 1