{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:EKHNNSP2R47JZSJJG2E3QADSR4","short_pith_number":"pith:EKHNNSP2","canonical_record":{"source":{"id":"0907.4639","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2009-07-27T14:27:03Z","cross_cats_sorted":[],"title_canon_sha256":"fffa35cdd7a3a669c9c0cab3df2f6a5a8972d3d6cd850755f33391851a1c4bd4","abstract_canon_sha256":"121fbf4f5abae3bce45f273dd9d3ba4f7810714e2243a5f7f1c45e7a22d37609"},"schema_version":"1.0"},"canonical_sha256":"228ed6c9fa8f3e9cc9293689b800728f247320676ee930f531cd05a486ec771e","source":{"kind":"arxiv","id":"0907.4639","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0907.4639","created_at":"2026-05-18T01:22:31Z"},{"alias_kind":"arxiv_version","alias_value":"0907.4639v2","created_at":"2026-05-18T01:22:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.4639","created_at":"2026-05-18T01:22:31Z"},{"alias_kind":"pith_short_12","alias_value":"EKHNNSP2R47J","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"EKHNNSP2R47JZSJJ","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"EKHNNSP2","created_at":"2026-05-18T12:25:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:EKHNNSP2R47JZSJJG2E3QADSR4","target":"record","payload":{"canonical_record":{"source":{"id":"0907.4639","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2009-07-27T14:27:03Z","cross_cats_sorted":[],"title_canon_sha256":"fffa35cdd7a3a669c9c0cab3df2f6a5a8972d3d6cd850755f33391851a1c4bd4","abstract_canon_sha256":"121fbf4f5abae3bce45f273dd9d3ba4f7810714e2243a5f7f1c45e7a22d37609"},"schema_version":"1.0"},"canonical_sha256":"228ed6c9fa8f3e9cc9293689b800728f247320676ee930f531cd05a486ec771e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:31.002366Z","signature_b64":"fKv7VdVOVnPZMxXpFn9dCUBHtA/8qxUrIUoBznbebVTyuVZRFedYG91IEWWaN8Ehg72N0UVfOinpN9CHHhcVCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"228ed6c9fa8f3e9cc9293689b800728f247320676ee930f531cd05a486ec771e","last_reissued_at":"2026-05-18T01:22:31.001719Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:31.001719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0907.4639","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":"2026-05-18T01:22:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VRXEFls5qi9/q2Dly6DktE9lBrcbmLEdr7I6yY1ho8JlrtDMbZkUjCdDkdzFIGUvrfyNK1X6EQLo+qk0kaJ8AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T18:48:07.418260Z"},"content_sha256":"7f47a996aa5ca9a10748b730aa0582c3c41ff884bd212f2c24ebd949da150c9d","schema_version":"1.0","event_id":"sha256:7f47a996aa5ca9a10748b730aa0582c3c41ff884bd212f2c24ebd949da150c9d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:EKHNNSP2R47JZSJJG2E3QADSR4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Instanton Floer homology and the Alexander polynomial","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"P. B. Kronheimer, T. S. Mrowka","submitted_at":"2009-07-27T14:27:03Z","abstract_excerpt":"The instanton Floer homology of a knot in the three-sphere is a vector space with a canonical mod 2 grading. It carries a distinguished endomorphism of even degree,arising from the 2-dimensional homology class represented by a Seifert surface. The Floer homology decomposes as a direct sum of the generalized eigenspaces of this endomorphism. We show that the Euler characteristics of these generalized eigenspaces are the coefficients of the Alexander polynomial of the knot. Among other applications, we deduce that instanton homology detects fibered knots."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.4639","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":"2026-05-18T01:22:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y38VMAGp+48TAGC/hzsDSGkSwNAUDtcqnEeKVB+3HZbM/CXRjaF4QrzTlesYbMLez7ur8xbu7iNvRhl30O6mCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T18:48:07.418608Z"},"content_sha256":"37b75c4aef870a88f839ebd087b4208e2dccc34a1be60b44f44046561dae35ba","schema_version":"1.0","event_id":"sha256:37b75c4aef870a88f839ebd087b4208e2dccc34a1be60b44f44046561dae35ba"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EKHNNSP2R47JZSJJG2E3QADSR4/bundle.json","state_url":"https://pith.science/pith/EKHNNSP2R47JZSJJG2E3QADSR4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EKHNNSP2R47JZSJJG2E3QADSR4/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":"2026-06-29T18:48:07Z","links":{"resolver":"https://pith.science/pith/EKHNNSP2R47JZSJJG2E3QADSR4","bundle":"https://pith.science/pith/EKHNNSP2R47JZSJJG2E3QADSR4/bundle.json","state":"https://pith.science/pith/EKHNNSP2R47JZSJJG2E3QADSR4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EKHNNSP2R47JZSJJG2E3QADSR4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:EKHNNSP2R47JZSJJG2E3QADSR4","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":"121fbf4f5abae3bce45f273dd9d3ba4f7810714e2243a5f7f1c45e7a22d37609","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2009-07-27T14:27:03Z","title_canon_sha256":"fffa35cdd7a3a669c9c0cab3df2f6a5a8972d3d6cd850755f33391851a1c4bd4"},"schema_version":"1.0","source":{"id":"0907.4639","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0907.4639","created_at":"2026-05-18T01:22:31Z"},{"alias_kind":"arxiv_version","alias_value":"0907.4639v2","created_at":"2026-05-18T01:22:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.4639","created_at":"2026-05-18T01:22:31Z"},{"alias_kind":"pith_short_12","alias_value":"EKHNNSP2R47J","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"EKHNNSP2R47JZSJJ","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"EKHNNSP2","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:37b75c4aef870a88f839ebd087b4208e2dccc34a1be60b44f44046561dae35ba","target":"graph","created_at":"2026-05-18T01:22:31Z","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":"The instanton Floer homology of a knot in the three-sphere is a vector space with a canonical mod 2 grading. It carries a distinguished endomorphism of even degree,arising from the 2-dimensional homology class represented by a Seifert surface. The Floer homology decomposes as a direct sum of the generalized eigenspaces of this endomorphism. We show that the Euler characteristics of these generalized eigenspaces are the coefficients of the Alexander polynomial of the knot. Among other applications, we deduce that instanton homology detects fibered knots.","authors_text":"P. B. Kronheimer, T. S. Mrowka","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2009-07-27T14:27:03Z","title":"Instanton Floer homology and the Alexander polynomial"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.4639","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":"sha256:7f47a996aa5ca9a10748b730aa0582c3c41ff884bd212f2c24ebd949da150c9d","target":"record","created_at":"2026-05-18T01:22:31Z","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":"121fbf4f5abae3bce45f273dd9d3ba4f7810714e2243a5f7f1c45e7a22d37609","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2009-07-27T14:27:03Z","title_canon_sha256":"fffa35cdd7a3a669c9c0cab3df2f6a5a8972d3d6cd850755f33391851a1c4bd4"},"schema_version":"1.0","source":{"id":"0907.4639","kind":"arxiv","version":2}},"canonical_sha256":"228ed6c9fa8f3e9cc9293689b800728f247320676ee930f531cd05a486ec771e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"228ed6c9fa8f3e9cc9293689b800728f247320676ee930f531cd05a486ec771e","first_computed_at":"2026-05-18T01:22:31.001719Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:31.001719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fKv7VdVOVnPZMxXpFn9dCUBHtA/8qxUrIUoBznbebVTyuVZRFedYG91IEWWaN8Ehg72N0UVfOinpN9CHHhcVCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:31.002366Z","signed_message":"canonical_sha256_bytes"},"source_id":"0907.4639","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7f47a996aa5ca9a10748b730aa0582c3c41ff884bd212f2c24ebd949da150c9d","sha256:37b75c4aef870a88f839ebd087b4208e2dccc34a1be60b44f44046561dae35ba"],"state_sha256":"40b4ea9a0a34b1f09ae147f58f492b42e6c588edc12a3164c607b21bef0d9fe8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LKHBGHqWv24P8v4fazQXLtR91pSB9qBJnBE3acFZDXw/Tt+tqtzf6tpVZX7OYDDAISt2FpF4t+Gws29r2l3PAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T18:48:07.423021Z","bundle_sha256":"7e48eb3ff6ddbeaddbf1db85f8d106e4ee7a5cc2866b2738bf6c4366b9ae7b51"}}