{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:J2EIRD3NCEPVHEJCI6NYYZL373","short_pith_number":"pith:J2EIRD3N","canonical_record":{"source":{"id":"math/0609055","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2006-09-03T17:24:49Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"6f534644313bfc5212bf7c5b391da572516ebe0871992a051fc8633a3aa942b3","abstract_canon_sha256":"28b14857adb41fae5ff2177f6132f066e01beb111f7a1156723bc92c23cfb8e2"},"schema_version":"1.0"},"canonical_sha256":"4e88888f6d111f539122479b8c657bfef1a41fe159788f848842da17446ecd6b","source":{"kind":"arxiv","id":"math/0609055","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0609055","created_at":"2026-05-18T02:41:31Z"},{"alias_kind":"arxiv_version","alias_value":"math/0609055v2","created_at":"2026-05-18T02:41:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0609055","created_at":"2026-05-18T02:41:31Z"},{"alias_kind":"pith_short_12","alias_value":"J2EIRD3NCEPV","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"J2EIRD3NCEPVHEJC","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"J2EIRD3N","created_at":"2026-05-18T12:25:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:J2EIRD3NCEPVHEJCI6NYYZL373","target":"record","payload":{"canonical_record":{"source":{"id":"math/0609055","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2006-09-03T17:24:49Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"6f534644313bfc5212bf7c5b391da572516ebe0871992a051fc8633a3aa942b3","abstract_canon_sha256":"28b14857adb41fae5ff2177f6132f066e01beb111f7a1156723bc92c23cfb8e2"},"schema_version":"1.0"},"canonical_sha256":"4e88888f6d111f539122479b8c657bfef1a41fe159788f848842da17446ecd6b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:31.798790Z","signature_b64":"sqhOqXMBarLjtD7WzdafHou8lwCxTXBgDZ6fgpLOJfQ7Fkt2MAar6zznkFTDUEJxDg63CF3TtveiXcBng6eJAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4e88888f6d111f539122479b8c657bfef1a41fe159788f848842da17446ecd6b","last_reissued_at":"2026-05-18T02:41:31.798447Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:31.798447Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0609055","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":"2026-05-18T02:41:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kIPPl31FF/a40034+Xn/uETutd7dXB+uqPI45JH9FjL/cTqxOvuBXXFG07tNNS5AkPOUIzr6w98q3bYuPThpAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T07:50:52.156956Z"},"content_sha256":"4a2db9867963050ae0a1f2ca934a501986297eaad980e248ac89bc22e54a95bc","schema_version":"1.0","event_id":"sha256:4a2db9867963050ae0a1f2ca934a501986297eaad980e248ac89bc22e54a95bc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:J2EIRD3NCEPVHEJCI6NYYZL373","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"High-codimensional knots spun about manifolds","license":"","headline":"","cross_cats":["math.AT"],"primary_cat":"math.GT","authors_text":"Dennis Roseman, Masamichi Takase","submitted_at":"2006-09-03T17:24:49Z","abstract_excerpt":"Using spinning we analyze in a geometric way Haefliger's smoothly knotted (4k-1)-spheres in the 6k-sphere. Consider the 2-torus standardly embedded in the 3-sphere, which is further standardly embedded in the 6-sphere. At each point of the 2-torus we have the normal disk pair: a 4-dimensional disk and a 1-dimensional proper sub-disk. We consider an isotopy (deformation) of the normal 1-disk inside the normal 4-disk, by using a map from the 2-torus to the space of long knots in 4-space, first considered by Budney. We use this isotopy in a construction called spinning about a submanifold introdu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0609055","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":"2026-05-18T02:41:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mC3mkAGVvWUH7MGJymdgFzmopusf+TowM0N8sO25GnxJG9hoUGjSaPlLeJ7YaZhbLoVjKYYjLX92doQueD4zDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T07:50:52.157664Z"},"content_sha256":"b5e85d0b60772b45f1cc30f318eabb758077140bfffbbf89a7d2fbc5dae68887","schema_version":"1.0","event_id":"sha256:b5e85d0b60772b45f1cc30f318eabb758077140bfffbbf89a7d2fbc5dae68887"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/J2EIRD3NCEPVHEJCI6NYYZL373/bundle.json","state_url":"https://pith.science/pith/J2EIRD3NCEPVHEJCI6NYYZL373/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/J2EIRD3NCEPVHEJCI6NYYZL373/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":"2026-06-09T07:50:52Z","links":{"resolver":"https://pith.science/pith/J2EIRD3NCEPVHEJCI6NYYZL373","bundle":"https://pith.science/pith/J2EIRD3NCEPVHEJCI6NYYZL373/bundle.json","state":"https://pith.science/pith/J2EIRD3NCEPVHEJCI6NYYZL373/state.json","well_known_bundle":"https://pith.science/.well-known/pith/J2EIRD3NCEPVHEJCI6NYYZL373/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:J2EIRD3NCEPVHEJCI6NYYZL373","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":"28b14857adb41fae5ff2177f6132f066e01beb111f7a1156723bc92c23cfb8e2","cross_cats_sorted":["math.AT"],"license":"","primary_cat":"math.GT","submitted_at":"2006-09-03T17:24:49Z","title_canon_sha256":"6f534644313bfc5212bf7c5b391da572516ebe0871992a051fc8633a3aa942b3"},"schema_version":"1.0","source":{"id":"math/0609055","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0609055","created_at":"2026-05-18T02:41:31Z"},{"alias_kind":"arxiv_version","alias_value":"math/0609055v2","created_at":"2026-05-18T02:41:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0609055","created_at":"2026-05-18T02:41:31Z"},{"alias_kind":"pith_short_12","alias_value":"J2EIRD3NCEPV","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"J2EIRD3NCEPVHEJC","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"J2EIRD3N","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:b5e85d0b60772b45f1cc30f318eabb758077140bfffbbf89a7d2fbc5dae68887","target":"graph","created_at":"2026-05-18T02:41:31Z","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":"Using spinning we analyze in a geometric way Haefliger's smoothly knotted (4k-1)-spheres in the 6k-sphere. Consider the 2-torus standardly embedded in the 3-sphere, which is further standardly embedded in the 6-sphere. At each point of the 2-torus we have the normal disk pair: a 4-dimensional disk and a 1-dimensional proper sub-disk. We consider an isotopy (deformation) of the normal 1-disk inside the normal 4-disk, by using a map from the 2-torus to the space of long knots in 4-space, first considered by Budney. We use this isotopy in a construction called spinning about a submanifold introdu","authors_text":"Dennis Roseman, Masamichi Takase","cross_cats":["math.AT"],"headline":"","license":"","primary_cat":"math.GT","submitted_at":"2006-09-03T17:24:49Z","title":"High-codimensional knots spun about manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0609055","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":"sha256:4a2db9867963050ae0a1f2ca934a501986297eaad980e248ac89bc22e54a95bc","target":"record","created_at":"2026-05-18T02:41:31Z","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":"28b14857adb41fae5ff2177f6132f066e01beb111f7a1156723bc92c23cfb8e2","cross_cats_sorted":["math.AT"],"license":"","primary_cat":"math.GT","submitted_at":"2006-09-03T17:24:49Z","title_canon_sha256":"6f534644313bfc5212bf7c5b391da572516ebe0871992a051fc8633a3aa942b3"},"schema_version":"1.0","source":{"id":"math/0609055","kind":"arxiv","version":2}},"canonical_sha256":"4e88888f6d111f539122479b8c657bfef1a41fe159788f848842da17446ecd6b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4e88888f6d111f539122479b8c657bfef1a41fe159788f848842da17446ecd6b","first_computed_at":"2026-05-18T02:41:31.798447Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:41:31.798447Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sqhOqXMBarLjtD7WzdafHou8lwCxTXBgDZ6fgpLOJfQ7Fkt2MAar6zznkFTDUEJxDg63CF3TtveiXcBng6eJAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:41:31.798790Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0609055","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4a2db9867963050ae0a1f2ca934a501986297eaad980e248ac89bc22e54a95bc","sha256:b5e85d0b60772b45f1cc30f318eabb758077140bfffbbf89a7d2fbc5dae68887"],"state_sha256":"2935a27fab0bc747492fdfa9f0b160cc8f3d16d3f20a0ae6e208798f8485ed50"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GtPDba/vY4slZ5GRl7Qds767nEzF5OESbNf4FPcwQPX/ucRYujodGl9oJgJxZYGvd5AI4lmUEcnOLJO0wLC6AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T07:50:52.161696Z","bundle_sha256":"6c10886b8674959bcad6ba6812e6304753f6f3d636e324a81ceae9d56dd1f7c9"}}