{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2004:D4ORI4HHXA64PNP6EAKGXIYV2D","short_pith_number":"pith:D4ORI4HH","schema_version":"1.0","canonical_sha256":"1f1d1470e7b83dc7b5fe20146ba315d0da73cf032eeff8075ebba99da80cf88c","source":{"kind":"arxiv","id":"math/0407050","version":4},"attestation_state":"computed","paper":{"title":"On Generalized Knot Groups","license":"","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Sam Nelson, Xiao-Song Lin","submitted_at":"2004-07-04T18:45:03Z","abstract_excerpt":"Generalized knot groups G_n(K) were introduced first by Wada and Kelly independently. The classical knot group is the first one G_1(K) in this series of finitely presented groups. For each natural number n, G_1(K) is a subgroup of G_n(K) so the generalized knot groups can be thought of as extensions of the classical knot group. For the square knot SK and the granny knot GK, we have an isomorphism $G_1(SK)\\cong G_1(GK)$. From the presentations of G_n(SK) and G_n(GK), for n>1, it seems unlikely that G_n(SK) and G_n(GK) would be isomorphic to each other. We are able to show that for many finite g"},"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":"math/0407050","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2004-07-04T18:45:03Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"50c7ef485b3edc41afb48ce8cee82f7db99b6bd8934e8e9bd257ce9abf04d35b","abstract_canon_sha256":"d8edf0e894c8e38300e9e344eae372637d65dd27a02ad635289d379255a77419"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:13:33.520693Z","signature_b64":"07ojsOZs17q6QRm9LI4cgSvS9m/nW2CIm5kAsIS+ikQJ5bWCD22k3dsmzqN0Vg4pk4phZdUJq3In686yC6ZHAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1f1d1470e7b83dc7b5fe20146ba315d0da73cf032eeff8075ebba99da80cf88c","last_reissued_at":"2026-07-04T15:13:33.520325Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:13:33.520325Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Generalized Knot Groups","license":"","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Sam Nelson, Xiao-Song Lin","submitted_at":"2004-07-04T18:45:03Z","abstract_excerpt":"Generalized knot groups G_n(K) were introduced first by Wada and Kelly independently. The classical knot group is the first one G_1(K) in this series of finitely presented groups. For each natural number n, G_1(K) is a subgroup of G_n(K) so the generalized knot groups can be thought of as extensions of the classical knot group. For the square knot SK and the granny knot GK, we have an isomorphism $G_1(SK)\\cong G_1(GK)$. From the presentations of G_n(SK) and G_n(GK), for n>1, it seems unlikely that G_n(SK) and G_n(GK) would be isomorphic to each other. We are able to show that for many finite g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0407050","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0407050/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"math/0407050","created_at":"2026-07-04T15:13:33.520389+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0407050v4","created_at":"2026-07-04T15:13:33.520389+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0407050","created_at":"2026-07-04T15:13:33.520389+00:00"},{"alias_kind":"pith_short_12","alias_value":"D4ORI4HHXA64","created_at":"2026-07-04T15:13:33.520389+00:00"},{"alias_kind":"pith_short_16","alias_value":"D4ORI4HHXA64PNP6","created_at":"2026-07-04T15:13:33.520389+00:00"},{"alias_kind":"pith_short_8","alias_value":"D4ORI4HH","created_at":"2026-07-04T15:13:33.520389+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/D4ORI4HHXA64PNP6EAKGXIYV2D","json":"https://pith.science/pith/D4ORI4HHXA64PNP6EAKGXIYV2D.json","graph_json":"https://pith.science/api/pith-number/D4ORI4HHXA64PNP6EAKGXIYV2D/graph.json","events_json":"https://pith.science/api/pith-number/D4ORI4HHXA64PNP6EAKGXIYV2D/events.json","paper":"https://pith.science/paper/D4ORI4HH"},"agent_actions":{"view_html":"https://pith.science/pith/D4ORI4HHXA64PNP6EAKGXIYV2D","download_json":"https://pith.science/pith/D4ORI4HHXA64PNP6EAKGXIYV2D.json","view_paper":"https://pith.science/paper/D4ORI4HH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0407050&json=true","fetch_graph":"https://pith.science/api/pith-number/D4ORI4HHXA64PNP6EAKGXIYV2D/graph.json","fetch_events":"https://pith.science/api/pith-number/D4ORI4HHXA64PNP6EAKGXIYV2D/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/D4ORI4HHXA64PNP6EAKGXIYV2D/action/timestamp_anchor","attest_storage":"https://pith.science/pith/D4ORI4HHXA64PNP6EAKGXIYV2D/action/storage_attestation","attest_author":"https://pith.science/pith/D4ORI4HHXA64PNP6EAKGXIYV2D/action/author_attestation","sign_citation":"https://pith.science/pith/D4ORI4HHXA64PNP6EAKGXIYV2D/action/citation_signature","submit_replication":"https://pith.science/pith/D4ORI4HHXA64PNP6EAKGXIYV2D/action/replication_record"}},"created_at":"2026-07-04T15:13:33.520389+00:00","updated_at":"2026-07-04T15:13:33.520389+00:00"}