A point set whose space of triangulations is disconnected

Abstract

In this paper we explicitly construct a triangulation of a 6-dimensional point configuration of 324 points which admits no geometric bistellar operations (or flips, for short). This triangulation is an isolated element in the graph of triangulations of the point configuration. It has been a central open question in polytope combinatorics in the last decade whether point configurations exist for which this graph is not connected (see, e.g., [37, Question 1.2] and [48, Challenge 3]). We also construct a 234-dimensional polytope with 552 vertices whose graph of triangulations has an isolated element. Our construction is likely to have an impact in algebraic geometry too, via the connections between lattice polytopes and toric varieties [21, 23, 31, 43]. For example, in [2, Section 2] and [24, Section 4] the different authors study algebraic schemes based on the poset of subdivisions of an integer point configuration. The connectivity of these schemes and of the graph of triangulations are equivalent. See Section 4.3, in particular Corollary 4.9. The graph of triangulations is also related to the Baues poset, which appears in oriented matroid theory, zonotopal tilings and hyperplane arrangements, so our result has implications in these areas. ∗ Let A be a finite point set in the real affine space R. A polyhedral subdivision of A is a geometric polyhedral complex with vertices in A which covers the convex hull of A. If all the cells are simplices, then it is a triangulation. More combinatorial definitions are convenient if A is not in convex position, i.e. if some element of A is not a vertex of the convex hull. See Definitions 4.1 and 1.1 for details, and also [6], [21, Chapter 7], [36], [47, Chapter 9], or the monograph in preparation [14]. There are at least the following three ways to give a structure to the collection of all triangulations of a point configuration A: (A) Flips. Geometric bistellar operations, or flips, are the minimal changes which can be made in a triangulation of A to produce a new one (see Definition 1.3). A particular case is the familiar diagonal edge flip in two-dimensional triangulations, of frequent use in computational geometry and geometric combinatorics.