Unimodular triangulations of dilated 3-polytopes
Francisco Santos, Günter M. Ziegler · Transactions of the Moscow Mathematical Society · 2014
A seminal result in the theory of toric varieties, by Knudsen, Mumford and Waterman (1973), asserts that for every lattice polytope P there is a positive integer k such that the dilated polytope kP has a unimodular triangulation.In dimension 3, Kantor and Sarkaria (2003) have shown that k = 4 works for every polytope.But this does not imply that every k > 4 works as well.We here study the values of k for which the result holds, showing that:(1) It contains all composite numbers.(2) It is an additive semigroup.These two properties imply that the only values of k that may not work (besides 1 and 2, which are known not to work) are k ∈ {3, 5, 7, 11}.With an ad-hoc construction we show that k = 7 and k = 11 also work, except in this case the triangulation cannot be guaranteed to be "standard" in the boundary.All in all, the only open cases are k = 3 and k = 5.