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We classify the harmonic knots $\\H(a,b,c)$ for $ a \\le 4. $ We study the knots $\\H (2n-1, 2n, 2n+1),$ the knots $\\H(5,n,n+1),$ and give a table of the simplest harmonic knots."},"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":"1203.4376","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2012-03-20T10:35:40Z","cross_cats_sorted":[],"title_canon_sha256":"85f2bf200b4114f37517de6e3495fcb8b5f8a29588f94d845af0288e2a971e9c","abstract_canon_sha256":"d3459fd4b3947cce1013bc68eb6737657bfdfcff445e0008984fe784317b6f24"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:29.241351Z","signature_b64":"J+TuLELeMpyX1vtSDN54Q/LogSzP36hxsABU5CpHnPC5vdijiFS0vg6dAptdjxqC82JT2rD9aBcuij7DXKlKCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"474119736e6c44234d985ba1e2020046a82d9ac25b2dffec75cafd6739c92453","last_reissued_at":"2026-05-18T02:42:29.240663Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:29.240663Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Harmonic Knots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Daniel Pecker (UPMC, IMJ, IMJ), Inria Paris-Rocquencourt), Pierre-Vincent Koseleff (UPMC","submitted_at":"2012-03-20T10:35:40Z","abstract_excerpt":"The harmonic knot $\\H(a,b,c)$ is parametrized as $K(t)= (T_a(t) ,T_b (t), T_c (t))$ where $a$, $b$ and $c$ are pairwise coprime integers and $T_n$ is the degree $n$ Chebyshev polynomial of the first kind. We classify the harmonic knots $\\H(a,b,c)$ for $ a \\le 4. $ We study the knots $\\H (2n-1, 2n, 2n+1),$ the knots $\\H(5,n,n+1),$ and give a table of the simplest harmonic knots."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.4376","kind":"arxiv","version":2},"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":"1203.4376","created_at":"2026-05-18T02:42:29.240803+00:00"},{"alias_kind":"arxiv_version","alias_value":"1203.4376v2","created_at":"2026-05-18T02:42:29.240803+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.4376","created_at":"2026-05-18T02:42:29.240803+00:00"},{"alias_kind":"pith_short_12","alias_value":"I5ARS43ONRCC","created_at":"2026-05-18T12:27:09.501522+00:00"},{"alias_kind":"pith_short_16","alias_value":"I5ARS43ONRCCGTMY","created_at":"2026-05-18T12:27:09.501522+00:00"},{"alias_kind":"pith_short_8","alias_value":"I5ARS43O","created_at":"2026-05-18T12:27:09.501522+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/I5ARS43ONRCCGTMYLOQ6EAQAI2","json":"https://pith.science/pith/I5ARS43ONRCCGTMYLOQ6EAQAI2.json","graph_json":"https://pith.science/api/pith-number/I5ARS43ONRCCGTMYLOQ6EAQAI2/graph.json","events_json":"https://pith.science/api/pith-number/I5ARS43ONRCCGTMYLOQ6EAQAI2/events.json","paper":"https://pith.science/paper/I5ARS43O"},"agent_actions":{"view_html":"https://pith.science/pith/I5ARS43ONRCCGTMYLOQ6EAQAI2","download_json":"https://pith.science/pith/I5ARS43ONRCCGTMYLOQ6EAQAI2.json","view_paper":"https://pith.science/paper/I5ARS43O","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1203.4376&json=true","fetch_graph":"https://pith.science/api/pith-number/I5ARS43ONRCCGTMYLOQ6EAQAI2/graph.json","fetch_events":"https://pith.science/api/pith-number/I5ARS43ONRCCGTMYLOQ6EAQAI2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I5ARS43ONRCCGTMYLOQ6EAQAI2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I5ARS43ONRCCGTMYLOQ6EAQAI2/action/storage_attestation","attest_author":"https://pith.science/pith/I5ARS43ONRCCGTMYLOQ6EAQAI2/action/author_attestation","sign_citation":"https://pith.science/pith/I5ARS43ONRCCGTMYLOQ6EAQAI2/action/citation_signature","submit_replication":"https://pith.science/pith/I5ARS43ONRCCGTMYLOQ6EAQAI2/action/replication_record"}},"created_at":"2026-05-18T02:42:29.240803+00:00","updated_at":"2026-05-18T02:42:29.240803+00:00"}