{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:WRLDC4WNIKLBOJTWEHQFDMSGQA","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"2c108d58b4a5dd2943832df83d2dc5f43ffa9f27c562db67a5b091d3e9244c0f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-06-14T00:45:10Z","title_canon_sha256":"e44ab4fd2924e3df8da61e84916a48c34eac658ac1d2e0b0e9957ab7f772031f"},"schema_version":"1.0","source":{"id":"1806.05313","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.05313","created_at":"2026-05-18T00:10:21Z"},{"alias_kind":"arxiv_version","alias_value":"1806.05313v2","created_at":"2026-05-18T00:10:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.05313","created_at":"2026-05-18T00:10:21Z"},{"alias_kind":"pith_short_12","alias_value":"WRLDC4WNIKLB","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_16","alias_value":"WRLDC4WNIKLBOJTW","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_8","alias_value":"WRLDC4WN","created_at":"2026-05-18T12:33:01Z"}],"graph_snapshots":[{"event_id":"sha256:0f8ce0d1aa25392f056bfd18f5f5f24b7140170d4fa9d1da4333016307ff804d","target":"graph","created_at":"2026-05-18T00:10:21Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We show that every $\\mathbb{Z}$-torsion free knot module is realized by a ribbon 2-knot with group of geometric dimension at most 2, and give some partial results on the characterization of the knot modules of fibred ribbon 2-knots.","authors_text":"Jonathan A. Hillman","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-06-14T00:45:10Z","title":"Knot modules and ribbon 2-knots"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.05313","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1db6140d801edf7be99f3cfc547ac1fc2bc2f138bcaf8477429ea0d7c829c9b5","target":"record","created_at":"2026-05-18T00:10:21Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"2c108d58b4a5dd2943832df83d2dc5f43ffa9f27c562db67a5b091d3e9244c0f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-06-14T00:45:10Z","title_canon_sha256":"e44ab4fd2924e3df8da61e84916a48c34eac658ac1d2e0b0e9957ab7f772031f"},"schema_version":"1.0","source":{"id":"1806.05313","kind":"arxiv","version":2}},"canonical_sha256":"b4563172cd429617267621e051b2468009a4123295187d5dde0fab7c55ae8ef2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b4563172cd429617267621e051b2468009a4123295187d5dde0fab7c55ae8ef2","first_computed_at":"2026-05-18T00:10:21.648328Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:21.648328Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TIKFn8U8exy/irZlVGmn2AUrNCz/hMFR3LqaSZtGEUZ14b9v73VLFnthuSfUv7lRqVXeaepCVUAwagpPtTpDDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:21.649005Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.05313","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1db6140d801edf7be99f3cfc547ac1fc2bc2f138bcaf8477429ea0d7c829c9b5","sha256:0f8ce0d1aa25392f056bfd18f5f5f24b7140170d4fa9d1da4333016307ff804d"],"state_sha256":"52ad10899d50665c6b777d067f4f16c710623d62012569c2545241c96b7428a7"}