{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:JGC5ZJX6R5DFAYVXVRSTOSC45L","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":"fe018b2ae10eb57b1ecb686f6313270c1e9a140580c4c5aae06dd44eedd303b3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-08-29T16:33:25Z","title_canon_sha256":"95892308f6aefdc3a39c17dc782aa01e06690f4ec5c8a431390e8bdbad0f35d0"},"schema_version":"1.0","source":{"id":"1108.5640","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.5640","created_at":"2026-05-18T04:14:37Z"},{"alias_kind":"arxiv_version","alias_value":"1108.5640v1","created_at":"2026-05-18T04:14:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.5640","created_at":"2026-05-18T04:14:37Z"},{"alias_kind":"pith_short_12","alias_value":"JGC5ZJX6R5DF","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"JGC5ZJX6R5DFAYVX","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"JGC5ZJX6","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:05f3e8a5e94dd6baf700affa025b640e81463b7bc0a7eb2fcbfaa0aaf1492791","target":"graph","created_at":"2026-05-18T04:14:37Z","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 essential punctured spheres in the complement of links with distance three bridge spheres have bounded complexity. We define the operation of tangle product, a generalization of both connected sum and Conway product. Finally, we use the bounded complexity of essential punctured spheres to show that the bridge number of a tangle product is at least the sum of the bridge numbers of the two factor links up to a constant error.","authors_text":"Ryan Blair","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-08-29T16:33:25Z","title":"Bridge number and tangle products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.5640","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:f10aa40eb0b679a5e106a3b5dff8ae510a98c23e2e3a333cd589fdb3c4bd81b0","target":"record","created_at":"2026-05-18T04:14:37Z","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":"fe018b2ae10eb57b1ecb686f6313270c1e9a140580c4c5aae06dd44eedd303b3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-08-29T16:33:25Z","title_canon_sha256":"95892308f6aefdc3a39c17dc782aa01e06690f4ec5c8a431390e8bdbad0f35d0"},"schema_version":"1.0","source":{"id":"1108.5640","kind":"arxiv","version":1}},"canonical_sha256":"4985dca6fe8f465062b7ac6537485ceafbf947188eb3f594dc0be22117580f95","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4985dca6fe8f465062b7ac6537485ceafbf947188eb3f594dc0be22117580f95","first_computed_at":"2026-05-18T04:14:37.082047Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:14:37.082047Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ffxohx/Tb18tkpAcpNw8M9aKZ/8a2m01Z6riCFfC8KdaZQ+BzelOzEBAsSohKh7jmowJyFDrX3S1hw3LwhB4CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:14:37.082429Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.5640","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f10aa40eb0b679a5e106a3b5dff8ae510a98c23e2e3a333cd589fdb3c4bd81b0","sha256:05f3e8a5e94dd6baf700affa025b640e81463b7bc0a7eb2fcbfaa0aaf1492791"],"state_sha256":"462be468e4068aa04662c2faedc2c9f823e4ffd447323c088dd7b9a8ed468602"}