K3 polytopes appear in complements of tropical quartic surfaces. They are dual to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Exploring these combinatorial objects, we classify K3 polytopes with up to 30 vertices. Their number is 36 297 333. We study the singular loci of quartic surfaces that tropicalize to K3 polytopes. These surfaces are stable in the sense of Geometric Invariant Theory.
Funding statement: GB was partially supported by the Vetenskapsrådet grant NT:2014-3991. MP and BS acknowledge support by the Einstein Foundation Berlin, which also funded a visit of GB to TU Berlin.
We are very grateful to Michael Joswig for several inspiring discussions.We also thank Matteo Gallet, Lars Kastner and Benjamin Schröter for help with this project.
Communicated by: R. Cavalieri
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