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In dimension 3, Kantor and Sarkaria (2003) have shown that $k=4$ works for every polytope. But this does not imply that every $k>4$ works as well. We here study the values of $k$ for which the result holds showing that:\n  1. It contains all composite numbers.\n  2. It is an additive semigroup.\n  These two properties imply that the only values of $k$ that may "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1304.7296","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-26T22:19:37Z","cross_cats_sorted":[],"title_canon_sha256":"d67706883433eacd22782434eb0bd03a22e36b1c23033c9f53e7f4999d616f4f","abstract_canon_sha256":"d8f99b3bec1b1463a92a03eb02e2097544bcafaf0ac042e960fefcd3142ee671"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:26.287469Z","signature_b64":"M8fySh56c1ZW1ubFX6Iffw2LY1zGMrh6Mvo5MkuTRpbXIMpzZt4GrQEI4chOgWYGrBs7LyZJn7NDcB9i7cObCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9e99db49dcbf1196f96819cdacf958e5afc9bf6570c2c3c21e671a08263763f9","last_reissued_at":"2026-05-18T02:41:26.287056Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:26.287056Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Unimodular triangulations of dilated 3-polytopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Francisco Santos, G\\\"unter M. Ziegler","submitted_at":"2013-04-26T22:19:37Z","abstract_excerpt":"A seminal result in the theory of toric varieties, due to Knudsen, Mumford and Waterman (1973), asserts that for every lattice polytope $P$ there is a positive integer $k$ such that the dilated polytope $kP$ has a unimodular triangulation. In dimension 3, Kantor and Sarkaria (2003) have shown that $k=4$ works for every polytope. But this does not imply that every $k>4$ works as well. We here study the values of $k$ for which the result holds showing that:\n  1. It contains all composite numbers.\n  2. 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