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Classification of Reflexive Polyhedra in Three Dimensions

M. Kreuzer, H. Skarke

TL;DR

This work completes the algorithmic framework for classifying reflexive polyhedra and applies it to three dimensions, where 4319 reflexive polytopes yield K3 surfaces in toric varieties. By organizing the search around 16 maximal ambient spaces defined by IP-weight systems and a refined notion of minimality, the authors enumerate all reflexive subpolytopes via sublattices, verify mirror symmetry through dual vertex pairing data, and demonstrate that the complete 3D set forms a single connected web under containment. Notable results include two mirror pairs with Picard numbers 1 and 19 and explicit realizations as quartics in P^3 and elliptically fibered K3 models, illustrating the rich algebraic structure encoded in toric polytopes. The catalog and methodologies developed here provide a scalable framework for higher-dimensional classifications and have implications for toric Calabi–Yau constructions, dualities, and F-theory applications.

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.

Classification of Reflexive Polyhedra in Three Dimensions

TL;DR

This work completes the algorithmic framework for classifying reflexive polyhedra and applies it to three dimensions, where 4319 reflexive polytopes yield K3 surfaces in toric varieties. By organizing the search around 16 maximal ambient spaces defined by IP-weight systems and a refined notion of minimality, the authors enumerate all reflexive subpolytopes via sublattices, verify mirror symmetry through dual vertex pairing data, and demonstrate that the complete 3D set forms a single connected web under containment. Notable results include two mirror pairs with Picard numbers 1 and 19 and explicit realizations as quartics in P^3 and elliptically fibered K3 models, illustrating the rich algebraic structure encoded in toric polytopes. The catalog and methodologies developed here provide a scalable framework for higher-dimensional classifications and have implications for toric Calabi–Yau constructions, dualities, and F-theory applications.

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.

Paper Structure

This paper contains 10 sections, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Basic definitions
  4. Strategy and results
  5. The algorithm
  6. General outline and existing results
  7. New definitions of minimality of polyhedra
  8. Finding subpolyhedra
  9. Normal forms of polyhedra
  10. Sublattices

Figures (1)

  • Figure 1: Applying the classification algorithm to $\Delta({\mathbb P}_{(1,2,3)})$