Triangulations with very few geometric bistellar neighbors

Abstract

We are interested in a notion of elementary change between triangulations of a point configuration, the so-called bistellar flips, introduced by Gel'fand, Kapranov, and Zelevinski. We construct sequences of triangulations of point configurations in dimension 3 with n2 + 2n + 2 vertices and only 4n - 3 geometric bistellar flips (for every even integer n), and of point configurations in dimension 4 with arbitrarily many vertices and a bounded number of flips. This drastically improves previous examples and seems to be evidence against the conjecture that any two triangulations of a point configuration can be joined by a sequence of flips.