{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:DWBRR5OE2CT4JE7226LXS7OXKZ","short_pith_number":"pith:DWBRR5OE","canonical_record":{"source":{"id":"1405.1947","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-05-08T14:43:07Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"1abff8736409b36f3aa989896c5bae489afca5197953c3435975a855bcec3628","abstract_canon_sha256":"eeedbe5fb97e6fb4033e2d514008b4c4deb6938f07ad9a554803a2efd826d755"},"schema_version":"1.0"},"canonical_sha256":"1d8318f5c4d0a7c493fad797797dd75666c9197107261daf5954e12c574f6890","source":{"kind":"arxiv","id":"1405.1947","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.1947","created_at":"2026-05-18T01:25:16Z"},{"alias_kind":"arxiv_version","alias_value":"1405.1947v2","created_at":"2026-05-18T01:25:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.1947","created_at":"2026-05-18T01:25:16Z"},{"alias_kind":"pith_short_12","alias_value":"DWBRR5OE2CT4","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"DWBRR5OE2CT4JE72","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"DWBRR5OE","created_at":"2026-05-18T12:28:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:DWBRR5OE2CT4JE7226LXS7OXKZ","target":"record","payload":{"canonical_record":{"source":{"id":"1405.1947","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-05-08T14:43:07Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"1abff8736409b36f3aa989896c5bae489afca5197953c3435975a855bcec3628","abstract_canon_sha256":"eeedbe5fb97e6fb4033e2d514008b4c4deb6938f07ad9a554803a2efd826d755"},"schema_version":"1.0"},"canonical_sha256":"1d8318f5c4d0a7c493fad797797dd75666c9197107261daf5954e12c574f6890","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:16.251396Z","signature_b64":"Ol6AKGHLblsamvWmgELqg7DTjfk+HTAoqVQDcptCL7kqWtZzZDbbcP+aHXjQ5pIr3puzVVqDtKMlO5f1gXvqCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1d8318f5c4d0a7c493fad797797dd75666c9197107261daf5954e12c574f6890","last_reissued_at":"2026-05-18T01:25:16.250967Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:16.250967Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1405.1947","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:25:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Fgdcl9z8JTuRCNULQioVRqFfkuP7XYOY5tPTxWfbMOhoR9mP6Ke5dp3v3vaBkBDaAQ7Fpjtnp4fugEhQMvL3DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T00:02:27.273483Z"},"content_sha256":"3190371b0a5030ba778e1df8bb451e0d0fe782dabaf98af0b32cd73af6cd965a","schema_version":"1.0","event_id":"sha256:3190371b0a5030ba778e1df8bb451e0d0fe782dabaf98af0b32cd73af6cd965a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:DWBRR5OE2CT4JE7226LXS7OXKZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lin-Wang type formula for the Haefliger invariant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.GT","authors_text":"Keiichi Sakai","submitted_at":"2014-05-08T14:43:07Z","abstract_excerpt":"In this paper we study the Haefliger invariant for long embeddings $\\mathbb{R}^{4k-1}\\hookrightarrow\\mathbb{R}^{6k}$ in terms of the self-intersections of their projections to $\\mathbb{R}^{6k-1}$, under the condition that the projection is a generic long immersion $\\mathbb{R}^{4k-1}\\looparrowright\\mathbb{R}^{6k-1}$. We define the notion of \"crossing changes\" of the embeddings at the self-intersections and describe the change of the isotopy classes under crossing changes using the linking numbers of the double point sets in $\\mathbb{R}^{4k-1}$. This formula is a higher-dimensional analogue to t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.1947","kind":"arxiv","version":2},"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":"2026-05-18T01:25:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+XKZz4gQ+bA9e+u4S5Mxcb+wMGYArXScNnsK24lQBChT7SQ2s99lwSekNbVQ8vyYIe4adqoJck4IWUyAYABGAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T00:02:27.273836Z"},"content_sha256":"ca5d7a99735998b8c9f41db164ecd4f62028bd4aa7b8945ba8662764bb249840","schema_version":"1.0","event_id":"sha256:ca5d7a99735998b8c9f41db164ecd4f62028bd4aa7b8945ba8662764bb249840"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DWBRR5OE2CT4JE7226LXS7OXKZ/bundle.json","state_url":"https://pith.science/pith/DWBRR5OE2CT4JE7226LXS7OXKZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DWBRR5OE2CT4JE7226LXS7OXKZ/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":"2026-06-23T00:02:27Z","links":{"resolver":"https://pith.science/pith/DWBRR5OE2CT4JE7226LXS7OXKZ","bundle":"https://pith.science/pith/DWBRR5OE2CT4JE7226LXS7OXKZ/bundle.json","state":"https://pith.science/pith/DWBRR5OE2CT4JE7226LXS7OXKZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DWBRR5OE2CT4JE7226LXS7OXKZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:DWBRR5OE2CT4JE7226LXS7OXKZ","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":"eeedbe5fb97e6fb4033e2d514008b4c4deb6938f07ad9a554803a2efd826d755","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-05-08T14:43:07Z","title_canon_sha256":"1abff8736409b36f3aa989896c5bae489afca5197953c3435975a855bcec3628"},"schema_version":"1.0","source":{"id":"1405.1947","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.1947","created_at":"2026-05-18T01:25:16Z"},{"alias_kind":"arxiv_version","alias_value":"1405.1947v2","created_at":"2026-05-18T01:25:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.1947","created_at":"2026-05-18T01:25:16Z"},{"alias_kind":"pith_short_12","alias_value":"DWBRR5OE2CT4","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"DWBRR5OE2CT4JE72","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"DWBRR5OE","created_at":"2026-05-18T12:28:25Z"}],"graph_snapshots":[{"event_id":"sha256:ca5d7a99735998b8c9f41db164ecd4f62028bd4aa7b8945ba8662764bb249840","target":"graph","created_at":"2026-05-18T01:25:16Z","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":"In this paper we study the Haefliger invariant for long embeddings $\\mathbb{R}^{4k-1}\\hookrightarrow\\mathbb{R}^{6k}$ in terms of the self-intersections of their projections to $\\mathbb{R}^{6k-1}$, under the condition that the projection is a generic long immersion $\\mathbb{R}^{4k-1}\\looparrowright\\mathbb{R}^{6k-1}$. We define the notion of \"crossing changes\" of the embeddings at the self-intersections and describe the change of the isotopy classes under crossing changes using the linking numbers of the double point sets in $\\mathbb{R}^{4k-1}$. This formula is a higher-dimensional analogue to t","authors_text":"Keiichi Sakai","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-05-08T14:43:07Z","title":"Lin-Wang type formula for the Haefliger invariant"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.1947","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":"sha256:3190371b0a5030ba778e1df8bb451e0d0fe782dabaf98af0b32cd73af6cd965a","target":"record","created_at":"2026-05-18T01:25:16Z","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":"eeedbe5fb97e6fb4033e2d514008b4c4deb6938f07ad9a554803a2efd826d755","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-05-08T14:43:07Z","title_canon_sha256":"1abff8736409b36f3aa989896c5bae489afca5197953c3435975a855bcec3628"},"schema_version":"1.0","source":{"id":"1405.1947","kind":"arxiv","version":2}},"canonical_sha256":"1d8318f5c4d0a7c493fad797797dd75666c9197107261daf5954e12c574f6890","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1d8318f5c4d0a7c493fad797797dd75666c9197107261daf5954e12c574f6890","first_computed_at":"2026-05-18T01:25:16.250967Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:16.250967Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ol6AKGHLblsamvWmgELqg7DTjfk+HTAoqVQDcptCL7kqWtZzZDbbcP+aHXjQ5pIr3puzVVqDtKMlO5f1gXvqCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:16.251396Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.1947","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3190371b0a5030ba778e1df8bb451e0d0fe782dabaf98af0b32cd73af6cd965a","sha256:ca5d7a99735998b8c9f41db164ecd4f62028bd4aa7b8945ba8662764bb249840"],"state_sha256":"233d52753a88b43a6d4c8c1416d44fa23666767ecc7b603101a2352b323c64b5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rgNT/1z1GZQcsjltOZwuWX5AIUJ828H/cl88gMTlDynp+Wwxva1AYHd6D7ivfdgbKEPVOFyuC66mMDomliswCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T00:02:27.275802Z","bundle_sha256":"ae0582b28c482f881e712eddc4e06e56c7c3b947db2650115a987030161b3b90"}}