Classification of reflexive polyhedra in three dimensions

Abstract

We present the last missing details of our algorithm for the classification of reflexive polyhedra in arbitrary dimensions. We also present the results of an application of this algorithm to the case of three dimensional reflexive polyhedra. We get 4319 such polyhedra that give rise to K3 surfaces embedded in toric varieties. 16 of these contain all others as subpolyhedra. The 4319 polyhedra form a single connected web if we define two polyhedra to be connected if one of them contains the other.

Complete classification of reflexive polyhedra in four dimensions

“…It turns out, however, that we need far less than the above mentioned 201346 CWS for our classification scheme if we choose suitable refinements of our original definition of minimality. While there are several possibilities of doing this (see [22,24,27]), we will mention only the one that is most powerful for the case of four dimensions considered here. We call a polytope A C M^ r-maximal if A is reflexive w.r.t.…”

“…we can interpret the first n entries of the j'th column of D • U as coordinates of Vj on Mfi n est-Similarly, the lines of W • D provide coordinates for the vertices of A* on Nfi nest , whereas U and W give the corresponding coordinates on the coarsest possible lattices. The intermediate lattices are in one-to-one correspondence to subgroups of the finite lattice quotient Mfi n est/M C oarsest and can be found by decomposing D into a product of triangular matrices as described in [24,27].…”

“…For the classification of reflexive polyhedra in three dimensions [24] we used, more or less, an implementation of the algorithm outlined in the previous section already. Our programs, however, would not have worked for the four dimensional case because everything that can go wrong in large computer calculations would have gone wrong: Within a rather short time we would have run out of RAM and disk space, would have encountered numerical overflows, and even without these problems we would not have been able to obtain our results within a reasonable time.…”

“…elliptic curves. To change this situation, we developed an algorithm for the classification in arbitrary dimensions [22,23] and, as a first step, used it to find all 4319 reflexive polyhedra in three dimensions [24], which give rise to K3 manifolds. In the present work we present the results of a complete classification in four dimensions, and as a side result we can state that the corresponding moduli space of Calabi-Yau threefolds is connected (this was almost, but not quite completely shown in [25]).…”

Notes on the Roots of Ehrhart Polynomials

“…In particular, Hibi [19] showed that the coefficients a i (P ) of a lattice polytope are symmetric, i.e., a i (P ) = a n−i (P ), if and only if P is reflexive. Kreuzer and Skarke [22,23] classified all reflexive polytopes in dimensions ≤ 4. For n = 2, 3, 4 there are respectively 16; 4, 319 and 473, 800, 776 reflexive polytopes (up to unimodular equivalence).…”