{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:YV2PSTOHCNLVAZFVEEZKGZMW5E","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":"d3094e01fbb60e5d8752486ef61857b1718c23a6ccaa8203273e6790998ca2e7","cross_cats_sorted":[],"license":"","primary_cat":"math.GT","submitted_at":"2005-07-13T09:29:17Z","title_canon_sha256":"344afb8ebcc73b7679728da0853b08949770a1eb900093b2f859c4011da971d1"},"schema_version":"1.0","source":{"id":"math/0507266","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0507266","created_at":"2026-05-18T01:22:19Z"},{"alias_kind":"arxiv_version","alias_value":"math/0507266v2","created_at":"2026-05-18T01:22:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0507266","created_at":"2026-05-18T01:22:19Z"},{"alias_kind":"pith_short_12","alias_value":"YV2PSTOHCNLV","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"YV2PSTOHCNLVAZFV","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"YV2PSTOH","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:06c541bd31092f4184b1e18a6b348687a53438dbaedc7470c1d81fc08cddbd45","target":"graph","created_at":"2026-05-18T01:22:19Z","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 set of homology cobordisms from a surface to itself with markings of their boundaries has a natural monoid structure. To investigate the structure of this monoid, we define and study its Magnus representation and Reidemeister torsion invariants by generalizing Kirk-Livingston-Wang's argument over the Gassner representation of string links. Moreover, by applying Cochran and Harvey's framework of higher-order (non-commutative) Alexander invariants to them, we extract several pieces of information about the monoid and related objects.","authors_text":"Takuya Sakasai","cross_cats":[],"headline":"","license":"","primary_cat":"math.GT","submitted_at":"2005-07-13T09:29:17Z","title":"The Magnus representation and higher-order Alexander invariants for homology cobordisms of surfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0507266","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:d54df2344e89c117f579f884f6cd235d53750f94bb31ce4c22b836c13b2a79c6","target":"record","created_at":"2026-05-18T01:22:19Z","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":"d3094e01fbb60e5d8752486ef61857b1718c23a6ccaa8203273e6790998ca2e7","cross_cats_sorted":[],"license":"","primary_cat":"math.GT","submitted_at":"2005-07-13T09:29:17Z","title_canon_sha256":"344afb8ebcc73b7679728da0853b08949770a1eb900093b2f859c4011da971d1"},"schema_version":"1.0","source":{"id":"math/0507266","kind":"arxiv","version":2}},"canonical_sha256":"c574f94dc713575064b52132a36596e9008dd8ca0ab6aa44abb6a051d2daa38d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c574f94dc713575064b52132a36596e9008dd8ca0ab6aa44abb6a051d2daa38d","first_computed_at":"2026-05-18T01:22:19.350348Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:19.350348Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KgNw299u62wQEUlMsIcrZK5/oR1dRsZbFhGxJLJpABh0isrkklUjY0n0ncZ/0djiim7sd3a2+pj6XhSKb2MyAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:19.351247Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0507266","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d54df2344e89c117f579f884f6cd235d53750f94bb31ce4c22b836c13b2a79c6","sha256:06c541bd31092f4184b1e18a6b348687a53438dbaedc7470c1d81fc08cddbd45"],"state_sha256":"cba59e5ffc0596cde26ff7127a28c9a7e3c607f3ce935c5f0365c2a0f19d749b"}