{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:5KT7FMVK6LHLOZZ7YWQCQHVB6X","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":"3b74b31dcd93b1cbdde19e496307b27c05cc5cd0c7ed60dce5d990e03d376e2a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2026-06-01T15:37:31Z","title_canon_sha256":"9e001a8d5b045bcd1ef17c23a6375bfcfcc7bcc59d5b51046d9a312f7fc8456f"},"schema_version":"1.0","source":{"id":"2606.02390","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.02390","created_at":"2026-06-02T03:04:57Z"},{"alias_kind":"arxiv_version","alias_value":"2606.02390v1","created_at":"2026-06-02T03:04:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.02390","created_at":"2026-06-02T03:04:57Z"},{"alias_kind":"pith_short_12","alias_value":"5KT7FMVK6LHL","created_at":"2026-06-02T03:04:57Z"},{"alias_kind":"pith_short_16","alias_value":"5KT7FMVK6LHLOZZ7","created_at":"2026-06-02T03:04:57Z"},{"alias_kind":"pith_short_8","alias_value":"5KT7FMVK","created_at":"2026-06-02T03:04:57Z"}],"graph_snapshots":[{"event_id":"sha256:9ba52210033ca26936c839a4469d7e96f52aa452069db6691a3a705d007e0258","target":"graph","created_at":"2026-06-02T03:04:57Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.02390/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Every knot $K \\subset S^3$ that admits a symmetric union presentation bounds an immersed ribbon disk in $S^3$, while the converse is an open problem due to Christoph Lamm. The symmetric ribbon number $r_s(K)$ of $K$ is the minimum number of ribbon singularities in any symmetric ribbon disk bounded by $K$. In this paper, we undertake a systematic investigation of symmetric ribbon numbers of knots with at most 12 crossings. Along the way, we exhibit novel lower bounds for $r_s(K)$ arising from knot determinants, Alexander polynomials, Jones polynomials, and Kauffman polynomials.","authors_text":"Alexander Zupan, Anok Timothy, Bishop Placke, Eric Corrado, Nick Starns, Sajid Raihan Akash, Sam Sanketh","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2026-06-01T15:37:31Z","title":"Symmetric ribbon numbers of low-complexity knots"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02390","kind":"arxiv","version":1},"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:a2bc073508053ff62c9759e88143bb9329fdd84953ceb71e686a4810db5341f4","target":"record","created_at":"2026-06-02T03:04:57Z","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":"3b74b31dcd93b1cbdde19e496307b27c05cc5cd0c7ed60dce5d990e03d376e2a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2026-06-01T15:37:31Z","title_canon_sha256":"9e001a8d5b045bcd1ef17c23a6375bfcfcc7bcc59d5b51046d9a312f7fc8456f"},"schema_version":"1.0","source":{"id":"2606.02390","kind":"arxiv","version":1}},"canonical_sha256":"eaa7f2b2aaf2ceb7673fc5a0281ea1f5ddc05d8a3b01605e3451bb70315bb26e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eaa7f2b2aaf2ceb7673fc5a0281ea1f5ddc05d8a3b01605e3451bb70315bb26e","first_computed_at":"2026-06-02T03:04:57.907387Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T03:04:57.907387Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PVpsj4jfjiUu5ciJSAZaM5ApNHC7FVIOGECO3pFpJxYg0MKnyTVPynxP25JQLuECiWy0xQR5OswsASjTiIUhAw==","signature_status":"signed_v1","signed_at":"2026-06-02T03:04:57.907800Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.02390","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a2bc073508053ff62c9759e88143bb9329fdd84953ceb71e686a4810db5341f4","sha256:9ba52210033ca26936c839a4469d7e96f52aa452069db6691a3a705d007e0258"],"state_sha256":"61ab3dfe4dc48e77bbb530a013487d009d1cd5b77e17b7a1c56868f7ea8743d8"}