{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:5IGC5RQHYAS6E7LGYYLPEXWSHT","short_pith_number":"pith:5IGC5RQH","canonical_record":{"source":{"id":"1607.01722","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-07-06T17:42:08Z","cross_cats_sorted":[],"title_canon_sha256":"a8dfb456a8b081f629d2abc0a02870b36a5ad6a7b659e609009ea3411749b695","abstract_canon_sha256":"0f47cf802000969ef240bc2641b740f4cfe6ceb004fa28a83c97329acdd36067"},"schema_version":"1.0"},"canonical_sha256":"ea0c2ec607c025e27d66c616f25ed23cf5de932bc30df1ab833d48a8e37fa67a","source":{"kind":"arxiv","id":"1607.01722","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.01722","created_at":"2026-05-18T01:11:25Z"},{"alias_kind":"arxiv_version","alias_value":"1607.01722v1","created_at":"2026-05-18T01:11:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.01722","created_at":"2026-05-18T01:11:25Z"},{"alias_kind":"pith_short_12","alias_value":"5IGC5RQHYAS6","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_16","alias_value":"5IGC5RQHYAS6E7LG","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_8","alias_value":"5IGC5RQH","created_at":"2026-05-18T12:30:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:5IGC5RQHYAS6E7LGYYLPEXWSHT","target":"record","payload":{"canonical_record":{"source":{"id":"1607.01722","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-07-06T17:42:08Z","cross_cats_sorted":[],"title_canon_sha256":"a8dfb456a8b081f629d2abc0a02870b36a5ad6a7b659e609009ea3411749b695","abstract_canon_sha256":"0f47cf802000969ef240bc2641b740f4cfe6ceb004fa28a83c97329acdd36067"},"schema_version":"1.0"},"canonical_sha256":"ea0c2ec607c025e27d66c616f25ed23cf5de932bc30df1ab833d48a8e37fa67a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:25.632036Z","signature_b64":"eYJ0J320lyVuiogO+SWAONtZDZAr2hy7a+MRsYy2lV6Iu6tvJzdjEiXXnDMwOneh0VbcefZJnNMDHpBOdV0OBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ea0c2ec607c025e27d66c616f25ed23cf5de932bc30df1ab833d48a8e37fa67a","last_reissued_at":"2026-05-18T01:11:25.631600Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:25.631600Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1607.01722","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:11:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4cY9FczuDT1LX+MPnktIIDhGiDkHyNIpC1gG2kALzGiQX07cfFx45yveM8PxkN3VhAxmf+YUDOnbV23zqgI2CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T11:26:52.759496Z"},"content_sha256":"0a03f87e64f2b85d63fca94f933f031a6ada1e77009a674c17097b6c74839f38","schema_version":"1.0","event_id":"sha256:0a03f87e64f2b85d63fca94f933f031a6ada1e77009a674c17097b6c74839f38"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:5IGC5RQHYAS6E7LGYYLPEXWSHT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Cochran's $\\beta^i$ invariants via twisted Whitney towers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Jim Conant, Peter Teichner, Rob Schneiderman","submitted_at":"2016-07-06T17:42:08Z","abstract_excerpt":"We show that Tim Cochran's invariants $\\beta^i(L)$ of a $2$-component link $L$ in the $3$--sphere can be computed as intersection invariants of certain 2-complexes in the $4$--ball with boundary $L$. These 2-complexes are special types of twisted Whitney towers, which we call {\\em Cochran towers}, and which exhibit a new phenomenon: A Cochran tower of order $2k$ allows the computation of the $\\beta^i$ invariants for all $i\\leq k$, i.e. simultaneous extraction of invariants from a Whitney tower at multiple orders. This is in contrast with the order $n$ Milnor invariants (requiring order $n$ Whi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.01722","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:11:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GaDSPtOrZO/yYZosSTPFRbD6qVvXaRvStBC+0DAoeLJ2O+g/ZuviMM0jAqcHZz7tDPb7rgTi+2sgT0jfCkoOBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T11:26:52.759845Z"},"content_sha256":"b293647acc634464d0ac1922c504be990f675148f4be39abd0da1b690d2f8e1d","schema_version":"1.0","event_id":"sha256:b293647acc634464d0ac1922c504be990f675148f4be39abd0da1b690d2f8e1d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5IGC5RQHYAS6E7LGYYLPEXWSHT/bundle.json","state_url":"https://pith.science/pith/5IGC5RQHYAS6E7LGYYLPEXWSHT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5IGC5RQHYAS6E7LGYYLPEXWSHT/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-23T11:26:52Z","links":{"resolver":"https://pith.science/pith/5IGC5RQHYAS6E7LGYYLPEXWSHT","bundle":"https://pith.science/pith/5IGC5RQHYAS6E7LGYYLPEXWSHT/bundle.json","state":"https://pith.science/pith/5IGC5RQHYAS6E7LGYYLPEXWSHT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5IGC5RQHYAS6E7LGYYLPEXWSHT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:5IGC5RQHYAS6E7LGYYLPEXWSHT","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":"0f47cf802000969ef240bc2641b740f4cfe6ceb004fa28a83c97329acdd36067","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-07-06T17:42:08Z","title_canon_sha256":"a8dfb456a8b081f629d2abc0a02870b36a5ad6a7b659e609009ea3411749b695"},"schema_version":"1.0","source":{"id":"1607.01722","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.01722","created_at":"2026-05-18T01:11:25Z"},{"alias_kind":"arxiv_version","alias_value":"1607.01722v1","created_at":"2026-05-18T01:11:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.01722","created_at":"2026-05-18T01:11:25Z"},{"alias_kind":"pith_short_12","alias_value":"5IGC5RQHYAS6","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_16","alias_value":"5IGC5RQHYAS6E7LG","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_8","alias_value":"5IGC5RQH","created_at":"2026-05-18T12:30:01Z"}],"graph_snapshots":[{"event_id":"sha256:b293647acc634464d0ac1922c504be990f675148f4be39abd0da1b690d2f8e1d","target":"graph","created_at":"2026-05-18T01:11:25Z","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 Tim Cochran's invariants $\\beta^i(L)$ of a $2$-component link $L$ in the $3$--sphere can be computed as intersection invariants of certain 2-complexes in the $4$--ball with boundary $L$. These 2-complexes are special types of twisted Whitney towers, which we call {\\em Cochran towers}, and which exhibit a new phenomenon: A Cochran tower of order $2k$ allows the computation of the $\\beta^i$ invariants for all $i\\leq k$, i.e. simultaneous extraction of invariants from a Whitney tower at multiple orders. This is in contrast with the order $n$ Milnor invariants (requiring order $n$ Whi","authors_text":"Jim Conant, Peter Teichner, Rob Schneiderman","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-07-06T17:42:08Z","title":"Cochran's $\\beta^i$ invariants via twisted Whitney towers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.01722","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":"sha256:0a03f87e64f2b85d63fca94f933f031a6ada1e77009a674c17097b6c74839f38","target":"record","created_at":"2026-05-18T01:11:25Z","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":"0f47cf802000969ef240bc2641b740f4cfe6ceb004fa28a83c97329acdd36067","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-07-06T17:42:08Z","title_canon_sha256":"a8dfb456a8b081f629d2abc0a02870b36a5ad6a7b659e609009ea3411749b695"},"schema_version":"1.0","source":{"id":"1607.01722","kind":"arxiv","version":1}},"canonical_sha256":"ea0c2ec607c025e27d66c616f25ed23cf5de932bc30df1ab833d48a8e37fa67a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ea0c2ec607c025e27d66c616f25ed23cf5de932bc30df1ab833d48a8e37fa67a","first_computed_at":"2026-05-18T01:11:25.631600Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:11:25.631600Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eYJ0J320lyVuiogO+SWAONtZDZAr2hy7a+MRsYy2lV6Iu6tvJzdjEiXXnDMwOneh0VbcefZJnNMDHpBOdV0OBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:11:25.632036Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.01722","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0a03f87e64f2b85d63fca94f933f031a6ada1e77009a674c17097b6c74839f38","sha256:b293647acc634464d0ac1922c504be990f675148f4be39abd0da1b690d2f8e1d"],"state_sha256":"fc853a7de9c0313dba930ff3e82e34d2aecfad520ea9eb60c11ef64052373c85"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XBvFFwPI6WufN379dngNlyXVNrtxfBcQ3DqCXVRDCXpg+tH8I3gKJ8+JuzfKiOnTyI7e61XabwFhohhaKUtTBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T11:26:52.761918Z","bundle_sha256":"6471d1bad4acf27a31f46be83ebf6fbe3f9cd25590933b542f6570f29c6dc4a5"}}