Geometric bistellar flips: the setting, the context and a construction

Abstract

Abstract.We give a self-contained introduction to the theory of secondary polytopes and geometric bistellar flips in triangulations of polytopes and point sets, as well as a review of some of the known results and connections to algebraic geometry, topological combinatorics, and other areas.As a new result, we announce the construction of a point set in general position with a disconnected space of triangulations. This shows, for the first time, that the poset of strict polyhedral subdivisions of a point set is not always connected. Mathematics Subject Classification (2000). Primary 52B11; Secondary 52B20.

A result on flip-graph connectivity

“…Its next-to-minimal elements will be called flips according to the definition found in [19] (definition 4.7.) and [21] (remark 1.17.). This definition identifies flips within the refinement poset of a point configuration, but other equivalent definitions are possible [18,19,21].…”

“…and [21] (remark 1.17.). This definition identifies flips within the refinement poset of a point configuration, but other equivalent definitions are possible [18,19,21]. In particular, it is more usual to define flips not as subdivisions, but as local operations instead that transform a triangulation of a point configuration into another one.…”

“…The flip-graph of a point configuration A then is the graph whose vertices are the triangulations of A and whose edges are its flips. While this definition is equivalent to the one above [18,21], it better shows that flips are also used to build triangulations incrementally [8,12].…”

Infinitesimal rigidity of polyhedra with vertices in convex position

“…exhibited two triangulations with the same set of vertices in ‫ޒ‬ 5 that cannot be connected via 2 ↔ 5 and 3 ↔ 4 bistellar moves. For an overview on geometric bistellar moves, see [Santos 2006]. …”

Pseudo-triangulations—a survey

“…This implies that one can flip much faster between triangulations of a point set if pseudo-triangulations are allowed as intermediate steps. As a side remark, we mention that an even better, linear bound is known for triangulations using the so-called geometric bistellar flips (see [57,Section 1.3]). In this case, only triangulations appear as intermediate steps, but in addition, deletions/insertions of interior vertices of degree three are allowed.…”