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When $\\mathcal{C}(a,b,c,\\phi)$ is nonsingular,it defines a polynomial knot. We determine all possible knot diagrams when $\\phi$ varies. Let $a,b,c$ be integers, $a$ is odd, $(a,b)=1$, we show that one can list all possible knots $\\mathcal{C}(a,b,c,\\phi)$ in$\\tilde{\\mathcal{O}}(n^2)$ bit operations, with $n=abc$."},"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":"1512.07766","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.SC","submitted_at":"2015-12-24T09:23:25Z","cross_cats_sorted":[],"title_canon_sha256":"a453cd7c134df91fa67ad84e5d9d0784d751fb980c5b83f393f50b7e6a8f5ba1","abstract_canon_sha256":"f2bb2bc2d76620170bc82ae15a6536f20774e7af97a73e57e7944cb6a9c1c033"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:25.043797Z","signature_b64":"NFS86W0mESObJBMkoBYS380HdiwTS7mAfvAnK7JomnxHch28G02hk6sNQKhDF5bwO6xQ+C55PhZM4qM36QVlAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f4c577d51e772f0c05a59c31d622cb04e3d99279f1957c7b277cfde0a0bc335","last_reissued_at":"2026-05-18T00:44:25.043175Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:25.043175Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Computing Chebyshev knot diagrams","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.SC","authors_text":"C Tran (UPMC, D Pecker (IMJ-PRG, Fabrice Rouillier (OURAGAN, IMJ-PRG, IMJ-PRG), P.-V Koseleff (OURAGAN, UPMC)","submitted_at":"2015-12-24T09:23:25Z","abstract_excerpt":"A Chebyshev curve $\\mathcal{C}(a,b,c,\\phi)$ has a parametrization of the form$ x(t)=T\\_a(t)$; \\ $y(t)=T\\_b(t)$; $z(t)= T\\_c(t + \\phi)$, where $a,b,c$are integers, $T\\_n(t)$ is the Chebyshev polynomialof degree $n$ and $\\phi \\in \\mathbb{R}$. When $\\mathcal{C}(a,b,c,\\phi)$ is nonsingular,it defines a polynomial knot. We determine all possible knot diagrams when $\\phi$ varies. Let $a,b,c$ be integers, $a$ is odd, $(a,b)=1$, we show that one can list all possible knots $\\mathcal{C}(a,b,c,\\phi)$ in$\\tilde{\\mathcal{O}}(n^2)$ bit operations, with $n=abc$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.07766","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":"1512.07766","created_at":"2026-05-18T00:44:25.043265+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.07766v2","created_at":"2026-05-18T00:44:25.043265+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.07766","created_at":"2026-05-18T00:44:25.043265+00:00"},{"alias_kind":"pith_short_12","alias_value":"N5GFO7KR45ZP","created_at":"2026-05-18T12:29:32.376354+00:00"},{"alias_kind":"pith_short_16","alias_value":"N5GFO7KR45ZPBQC2","created_at":"2026-05-18T12:29:32.376354+00:00"},{"alias_kind":"pith_short_8","alias_value":"N5GFO7KR","created_at":"2026-05-18T12:29:32.376354+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/N5GFO7KR45ZPBQC2LHBR2YRMWB","json":"https://pith.science/pith/N5GFO7KR45ZPBQC2LHBR2YRMWB.json","graph_json":"https://pith.science/api/pith-number/N5GFO7KR45ZPBQC2LHBR2YRMWB/graph.json","events_json":"https://pith.science/api/pith-number/N5GFO7KR45ZPBQC2LHBR2YRMWB/events.json","paper":"https://pith.science/paper/N5GFO7KR"},"agent_actions":{"view_html":"https://pith.science/pith/N5GFO7KR45ZPBQC2LHBR2YRMWB","download_json":"https://pith.science/pith/N5GFO7KR45ZPBQC2LHBR2YRMWB.json","view_paper":"https://pith.science/paper/N5GFO7KR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.07766&json=true","fetch_graph":"https://pith.science/api/pith-number/N5GFO7KR45ZPBQC2LHBR2YRMWB/graph.json","fetch_events":"https://pith.science/api/pith-number/N5GFO7KR45ZPBQC2LHBR2YRMWB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/N5GFO7KR45ZPBQC2LHBR2YRMWB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/N5GFO7KR45ZPBQC2LHBR2YRMWB/action/storage_attestation","attest_author":"https://pith.science/pith/N5GFO7KR45ZPBQC2LHBR2YRMWB/action/author_attestation","sign_citation":"https://pith.science/pith/N5GFO7KR45ZPBQC2LHBR2YRMWB/action/citation_signature","submit_replication":"https://pith.science/pith/N5GFO7KR45ZPBQC2LHBR2YRMWB/action/replication_record"}},"created_at":"2026-05-18T00:44:25.043265+00:00","updated_at":"2026-05-18T00:44:25.043265+00:00"}