PreprintCC BY-NC-ND 4.0arXiv
Polycubes and polycube ideals
Abstract
To a polycube 𝒫 in ℝn, we associate an ideal I𝒫 generated by the join--meet relations of vertices that are corners of a cuboid of 𝒫. For n=2 these are the polyomino ideals. In contrast with the planar case, we show that for n≥ 3 the ideal I𝒫 is prime if and only if 𝒫 is a cuboid. For convex polycubes in ℝ3 of uniform thickness whose vertex set is a sublattice of ℕ3, the defining relations form a quadratic Gröbner basis, and we characterize the Cohen--Macaulay ones in terms of a poset on the vertices of 𝒫; in this case the order complex of the poset is vertex decomposable. Finally, as an application, we show that the polycube algebra of a thickened L-convex polyomino is Koszul, and we characterize when it is Cohen--Macaulay.
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BibTeX
@misc{qureshi2026polycubes,
title = {{Polycubes and polycube ideals}},
author = {Ayesha Asloob Qureshi and Dharm Veer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12191},
eprint = {2610.12191},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12191},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Polycubes and polycube ideals
AU - Ayesha Asloob Qureshi
AU - Dharm Veer
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12191
UR - https://arxiv.org/abs/2610.12191
AB - To a polycube 𝒫 in ℝ^n, we associate an ideal I_𝒫 generated by the join--meet relations of vertices that are corners of a cuboid of 𝒫. For n=2 these are the polyomino ideals. In contrast with the planar case, we show that for n≥ 3 the ideal I_𝒫 is prime if and only if 𝒫 is a cuboid. For convex polycubes in ℝ^3 of uniform thickness whose vertex set is a sublattice of ℕ^3, the defining relations form a quadratic Gröbner basis, and we characterize the Cohen--Macaulay ones in terms of a poset on the vertices of 𝒫; in this case the order complex of the poset is vertex decomposable. Finally, as an application, we show that the polycube algebra of a thickened L-convex polyomino is Koszul, and we characterize when it is Cohen--Macaulay.
ER -