{"paper":{"title":"Classification of empty lattice $4$-simplices of width larger than two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Francisco Santos, \\'Oscar Iglesias Vali\\~no","submitted_at":"2017-04-24T15:48:17Z","abstract_excerpt":"A lattice $d$-simplex is the convex hull of $d+1$ affinely independent integer points in ${\\mathbb R}^d$. It is called empty if it contains no lattice point apart of its $d+1$ vertices. The classification of empty $3$-simplices is known since 1964 (White), based on the fact that they all have width one. But for dimension $4$ no complete classification is known.\n  Haase and Ziegler (2000) enumerated all empty $4$-simplices up to determinant 1000 and based on their results conjectured that after determinant $179$ all empty $4$-simplices have width one or two. We prove this conjecture as follows:"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.07299","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}