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This paper provides upper bounds for $\\mbox{Len}(K)$ of a nontrivial knot $K$ in terms of its crossing number $c(K)$ as follows:\n  $\\mbox{Len}(K) \\leq \\min \\left\\{ \\frac{3}{4}c(K)^2 + 5c(K) + \\frac{17}{4}, \\, \\frac{5}{8}c(K)^2 + \\frac{15}{2}c(K) + \\frac{71}{8} \\right\\}.$\n  The ropelength of a knot is the quotient of its length by its thickness, the radius of the largest embedded normal tube around the knot. We also provide upper bounds for t"},"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":"1411.1845","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-11-07T06:44:35Z","cross_cats_sorted":[],"title_canon_sha256":"dba2828a38537a97e96182344da0a077885ffe03f447be657db00b655722bbef","abstract_canon_sha256":"4517b63f4192c3676c992641e44f746a7fa5d43a1fb628447b5b981a0ee43736"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:25.810624Z","signature_b64":"e52VccoZSW1AkGXuW5SXDemWbdCuITqbHSCBwEGJWcwc1xmt6y01LIblzqdgoeyp9jwJPeWxmEXt2db461+1Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e4ee307bf4e7370cb050a086a9f8948e291d09366771550c81fb006a41dd217a","last_reissued_at":"2026-05-18T02:38:25.809969Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:25.809969Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minimum lattice length and ropelength of knots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Hyoungjun Kim, Kyungpyo Hong, Seungsang Oh, Sungjong No","submitted_at":"2014-11-07T06:44:35Z","abstract_excerpt":"Let $\\mbox{Len}(K)$ be the minimum length of a knot on the cubic lattice (namely the minimum length necessary to construct the knot in the cubic lattice). This paper provides upper bounds for $\\mbox{Len}(K)$ of a nontrivial knot $K$ in terms of its crossing number $c(K)$ as follows:\n  $\\mbox{Len}(K) \\leq \\min \\left\\{ \\frac{3}{4}c(K)^2 + 5c(K) + \\frac{17}{4}, \\, \\frac{5}{8}c(K)^2 + \\frac{15}{2}c(K) + \\frac{71}{8} \\right\\}.$\n  The ropelength of a knot is the quotient of its length by its thickness, the radius of the largest embedded normal tube around the knot. 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