{"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. We also provide upper bounds for t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.1845","kind":"arxiv","version":1},"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"}