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Enumeration of Lattice 3-Polytopes by Their Number of Lattice Points

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Abstract

We develop a procedure for the complete computational enumeration of lattice 3-polytopes of width larger than one, of which there are finitely many for each given number of lattice points. We also implement an algorithm for doing this and enumerate those with at most 11 lattice points (there are 216,453 of them). In order to achieve this we prove that if P is a lattice 3-polytope of width larger than one and with at least seven lattice points then it fits in one of three categories that we call boxed, spiked and merged. Boxed polytopes have at most 11 lattice points; in particular they are finitely many, and we enumerate them completely with computer help. Spiked polytopes are infinitely many but admit a quite precise description (and enumeration). Merged polytopes are computed as a union (merging) of two polytopes of width larger than one and strictly smaller number of lattice points.

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Acknowledgements

The research was supported by grants MTM2011-22792, MTM2014-54207-P (both authors) and BES-2012-058920 (M. Blanco) of the Spanish Ministry of Economy and Competitiveness, and by the Einstein Foundation Berlin (F. Santos). We thank Gabriele Balletti for sharing with us his results ([1], joint work with A. Kasprzyk) on the classification of lattice 3-polytopes with two interior points. In particular, a discrepancy between their results and a preliminary version of ours led us to correct a mistake in a first version of Theorem 3.2 and Corollary 3.3. We thank I. Soprunov and J. Soprunova for pointing us to reference [17]. We thank the anonymous referees of this paper for their thorough revision of it and the constructive comments they made.

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  1. Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, 39005, Santander, Spain

    Mónica Blanco & Francisco Santos

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  1. Mónica Blanco
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  2. Francisco Santos
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Correspondence to Mónica Blanco.

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Blanco, M., Santos, F. Enumeration of Lattice 3-Polytopes by Their Number of Lattice Points. Discrete Comput Geom 60, 756–800 (2018). https://doi.org/10.1007/s00454-017-9932-5

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  • DOI: https://doi.org/10.1007/s00454-017-9932-5

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