Classification of Reflexive Polyhedra in Three Dimensions

TL;DR

This paper completes an algorithm for classifying reflexive polyhedra and applies it to three dimensions, identifying 4319 such polyhedra related to K3 surfaces, with some containing all others as subpolyhedra.

Contribution

It provides the final details of an algorithm for classifying reflexive polyhedra and reports the complete classification in three dimensions, including the structure of their interrelations.

Findings

Identified 4319 three-dimensional reflexive polyhedra.

Discovered 16 polyhedra containing all others as subpolyhedra.

Mapped the connectivity of polyhedra based on containment.

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.