{"paper":{"title":"Most Graphs are Knotted","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Kazuhiro Ichihara, Thomas W. Mattman","submitted_at":"2018-11-23T23:27:50Z","abstract_excerpt":"We present four models for a random graph and show that, in each case, the probability that a graph is intrinsically knotted goes to one as the number of vertices increases. We also argue that, for $k \\geq 18$, most graphs of order $k$ are intrinsically knotted and, for $k \\geq 2n+9$, most of order $k$ are not $n$-apex. We observe that $p(n) = 1/n$ is the threshold for intrinsic knotting and linking in Gilbert's model."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.09726","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"}