Coverage

The database contains lattice polygons and rational polygons in the following ranges, up to affine unimodular equivalence. Here i is the number of interior lattice points. Polygons that can be realized in ℝ × [0, 1] (thin polygons) are only included where stated. Searches tell whether their results are complete.

Please let me know if you think references are missing!

Denominator	Polygons	References
1	all polygons with i = 0	[Kho97], [Koe91], [Sch03], [BS26, Theorem 1.1], [SB24a]
	all polygons with i = 1–30	[Cas12], [Spr24], [OEIS, A322343]; for i = 1 also [Bat85], [Rab89], [Koe91]; for i = 2 also [WD12]
	all polygons with at most 50 lattice points, including thin ones	[Koe91], [BS26, Section 7], [SB24e], [OEIS, A371917]
2	all polygons with i = 0–9	[BS26, Theorem 1.1], [SB24a], [SB24c], [HHK24]; for i = 1 also [OEIS, A379887]
	maximal polygons with i = 0–40	[BS26, Classifications 4.5 and 6.7], [SB24a], [SB24c]; for i = 0 also [AKW17]
3	all polygons with i = 0–2	[BS26, Theorem 1.1], [SB24a], [SB24d]; for i = 1 also [OEIS, A379887]
	maximal polygons with i = 0–5	[BS26, Classification 4.5 and Section 7], [SB24a], [SB24d]
4	all polygons with i = 0–1	[BS26, Theorem 1.1], [SB24a], [SB24b]; for i = 1 also [OEIS, A379887]
5	all polygons with i = 0	[BS26, Theorem 1.1], [SB24a]
	maximal polygons with i = 1	[BS26, Classification 5.5], [SB24b]
	LDP polygons with i = 1	[BS26, Classification 5.6], [SB24b], [OEIS, A379894]; see also [KKN10], [HHS26], [LDP]
6–10	maximal polygons with i = 0–1	[BS26, Classifications 4.5 and 5.5], [SB24a], [SB24b]
11–21	maximal polygons with i = 0	[BS26, Classification 4.5], [SB24a]

Additionally, the database contains all lattice polygons in a square of side length at most 8, i.e. all lattice polygons equivalent to a subpolygon of [0, 8]², including thin ones: [BK13], [BS26, Table 2.13], [SB24f], [OEIS, A374975].

References

[AKW17] G. Averkov, J. Krümpelmann, S. Weltge: Notions of maximality for integral lattice-free polyhedra: the case of dimension three. Math. Oper. Res. 42(4), 1035–1062 (2017). https://doi.org/10.1287/moor.2016.0836
[Bat85] V. V. Batyrev: Higher-dimensional toric varieties with ample anticanonical class. PhD thesis, Moscow State University (1985).
[BK13] G. Brown, A. M. Kasprzyk: Small polygons and toric codes. J. Symbolic Comput. 51, 55–62 (2013). https://doi.org/10.1016/j.jsc.2012.07.001
[BS26] M. Bohnert, J. Springer: Classifying rational polygons with small denominator and few interior lattice points. Discrete Comput. Geom. 76, 1238–1265 (2026). https://doi.org/10.1007/s00454-026-00822-0
[Cas12] W. Castryck: Moving out the edges of a lattice polygon. Discrete Comput. Geom. 47(3), 496–518 (2012). https://doi.org/10.1007/s00454-011-9376-2
[HHK24] G. Hamm, J. Hofscheier, A. Kasprzyk: Classification and Ehrhart theory of denominator 2 polygons (2024). https://arxiv.org/abs/2411.19183
[HHS26] D. Hättig, J. Hausen, J. Springer: Classifying log del Pezzo surfaces with torus action. Rev. Mat. Complut. 39(1), 1–74 (2026). https://doi.org/10.1007/s13163-025-00526-8
[Kho97] A. G. Khovanskiĭ: Newton polytopes, curves on toric surfaces, and inversion of Weil's theorem. Uspekhi Mat. Nauk 52(6), 113–142 (1997). https://doi.org/10.1070/RM1997v052n06ABEH002156
[KKN10] A. M. Kasprzyk, M. Kreuzer, B. Nill: On the combinatorial classification of toric log del Pezzo surfaces. LMS J. Comput. Math. 13, 33–46 (2010). https://doi.org/10.1112/S1461157008000387
[Koe91] R. J. Koelman: The number of moduli of families of curves on toric surfaces. PhD thesis, University of Nijmegen (1991).
[LDP] J. Springer: ldp-database, a database of log del Pezzo surfaces with torus action. https://www.math.uni-tuebingen.de/forschung/algebra/ldp-database/
[OEIS] OEIS Foundation Inc.: The On-Line Encyclopedia of Integer Sequences. https://oeis.org
[Rab89] S. Rabinowitz: A census of convex lattice polygons with at most one interior lattice point. Ars Combin. 28, 83–96 (1989).
[SB24a] J. Springer, M. Bohnert: Rational polygons with no interior lattice points. Zenodo (2024). https://doi.org/10.5281/zenodo.13838991
[SB24b] J. Springer, M. Bohnert: Rational polygons with exactly one interior lattice point. Zenodo (2024). https://doi.org/10.5281/zenodo.13839216
[SB24c] J. Springer, M. Bohnert: Half-integral polygons with few interior lattice points. Zenodo (2024). https://doi.org/10.5281/zenodo.13928298
[SB24d] J. Springer, M. Bohnert: 3-rational polygons with few interior lattice points. Zenodo (2024). https://doi.org/10.5281/zenodo.13960791
[SB24e] J. Springer, M. Bohnert: Lattice polygons with at most 70 lattice points. Zenodo (2024). https://doi.org/10.5281/zenodo.13959996
[SB24f] J. Springer, M. Bohnert: Lattice subpolygons of a square with given sidelength. Zenodo (2024). https://doi.org/10.5281/zenodo.13838476
[Sch03] J. Schicho: Simplification of surface parametrizations — a lattice polygon approach. J. Symbolic Comput. 36(3–4), 535–554 (2003). https://doi.org/10.1016/S0747-7171(03)00094-4
[Spr24] J. Springer: RationalPolygons.jl. https://github.com/justus-springer/RationalPolygons.jl
[WD12] X. Wei, R. Ding: Lattice polygons with two interior lattice points. Math. Notes 91(5–6), 868–877 (2012). https://doi.org/10.1134/S0001434612050343