{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:NPBBLSXDSYV5JYABZSYRSKTF3A","short_pith_number":"pith:NPBBLSXD","schema_version":"1.0","canonical_sha256":"6bc215cae3962bd4e001ccb1192a65d8226eb652b6a73f83503c44f303a4e2c5","source":{"kind":"arxiv","id":"1811.09726","version":1},"attestation_state":"computed","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."},"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":"1811.09726","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-11-23T23:27:50Z","cross_cats_sorted":[],"title_canon_sha256":"60e072965f457c8818eb999d7724f195a41c55f7320725c42c6a0507cadfeb11","abstract_canon_sha256":"9344683869cc2fbf0de6450de6ecee260ea6749ece6bfca02b3ea4a7a76a106f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:01.383589Z","signature_b64":"Ycp3RtMBhiOjIRTu+zzuCRC4C1cLTxxK2t8PtbB6Z1UY3Xf34GCygaZ1x7w9XHytcCyv8RrOtRfiZBNhqjfnCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6bc215cae3962bd4e001ccb1192a65d8226eb652b6a73f83503c44f303a4e2c5","last_reissued_at":"2026-05-18T00:00:01.382779Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:01.382779Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1811.09726","created_at":"2026-05-18T00:00:01.382906+00:00"},{"alias_kind":"arxiv_version","alias_value":"1811.09726v1","created_at":"2026-05-18T00:00:01.382906+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.09726","created_at":"2026-05-18T00:00:01.382906+00:00"},{"alias_kind":"pith_short_12","alias_value":"NPBBLSXDSYV5","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_16","alias_value":"NPBBLSXDSYV5JYAB","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_8","alias_value":"NPBBLSXD","created_at":"2026-05-18T12:32:40.477152+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NPBBLSXDSYV5JYABZSYRSKTF3A","json":"https://pith.science/pith/NPBBLSXDSYV5JYABZSYRSKTF3A.json","graph_json":"https://pith.science/api/pith-number/NPBBLSXDSYV5JYABZSYRSKTF3A/graph.json","events_json":"https://pith.science/api/pith-number/NPBBLSXDSYV5JYABZSYRSKTF3A/events.json","paper":"https://pith.science/paper/NPBBLSXD"},"agent_actions":{"view_html":"https://pith.science/pith/NPBBLSXDSYV5JYABZSYRSKTF3A","download_json":"https://pith.science/pith/NPBBLSXDSYV5JYABZSYRSKTF3A.json","view_paper":"https://pith.science/paper/NPBBLSXD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1811.09726&json=true","fetch_graph":"https://pith.science/api/pith-number/NPBBLSXDSYV5JYABZSYRSKTF3A/graph.json","fetch_events":"https://pith.science/api/pith-number/NPBBLSXDSYV5JYABZSYRSKTF3A/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NPBBLSXDSYV5JYABZSYRSKTF3A/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NPBBLSXDSYV5JYABZSYRSKTF3A/action/storage_attestation","attest_author":"https://pith.science/pith/NPBBLSXDSYV5JYABZSYRSKTF3A/action/author_attestation","sign_citation":"https://pith.science/pith/NPBBLSXDSYV5JYABZSYRSKTF3A/action/citation_signature","submit_replication":"https://pith.science/pith/NPBBLSXDSYV5JYABZSYRSKTF3A/action/replication_record"}},"created_at":"2026-05-18T00:00:01.382906+00:00","updated_at":"2026-05-18T00:00:01.382906+00:00"}