{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:D64XVZNJNTMPV7FJ3JLL7JUS4U","short_pith_number":"pith:D64XVZNJ","canonical_record":{"source":{"id":"1812.04930","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-12T13:13:24Z","cross_cats_sorted":["math.AT","math.NA"],"title_canon_sha256":"2403706d06d2a755752840a1132c18a4dd615ddbbee8f9dc66224a836886c4d3","abstract_canon_sha256":"ce4a411803ed1b24c13006b791e8079273e8194b7a167ee2792897cf354f9b73"},"schema_version":"1.0"},"canonical_sha256":"1fb97ae5a96cd8fafca9da56bfa692e5387682f6244f0741a321cce513b9172c","source":{"kind":"arxiv","id":"1812.04930","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.04930","created_at":"2026-05-17T23:58:20Z"},{"alias_kind":"arxiv_version","alias_value":"1812.04930v2","created_at":"2026-05-17T23:58:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.04930","created_at":"2026-05-17T23:58:20Z"},{"alias_kind":"pith_short_12","alias_value":"D64XVZNJNTMP","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"D64XVZNJNTMPV7FJ","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"D64XVZNJ","created_at":"2026-05-18T12:32:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:D64XVZNJNTMPV7FJ3JLL7JUS4U","target":"record","payload":{"canonical_record":{"source":{"id":"1812.04930","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-12T13:13:24Z","cross_cats_sorted":["math.AT","math.NA"],"title_canon_sha256":"2403706d06d2a755752840a1132c18a4dd615ddbbee8f9dc66224a836886c4d3","abstract_canon_sha256":"ce4a411803ed1b24c13006b791e8079273e8194b7a167ee2792897cf354f9b73"},"schema_version":"1.0"},"canonical_sha256":"1fb97ae5a96cd8fafca9da56bfa692e5387682f6244f0741a321cce513b9172c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:20.476803Z","signature_b64":"jQadN4wjGxrS4ylNeHNWfFGjjeLxU5E6kfONTzvW418OPqxiUtWO0SjCZtlD1T3kiat63udimHgADf7cghsWDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1fb97ae5a96cd8fafca9da56bfa692e5387682f6244f0741a321cce513b9172c","last_reissued_at":"2026-05-17T23:58:20.476182Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:20.476182Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1812.04930","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":"2026-05-17T23:58:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H1ZGu5RPemg1mscyX0IqNJ/psCyoVFspN+EStAEP2oQhyujzn6rHxruUCn5rB5L0WogUWFcYnDFPaawUp4PKBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T10:19:55.417679Z"},"content_sha256":"1484d002c52794128b41859035c069cec2294ac99ddf8420dcd5602fe15fc290","schema_version":"1.0","event_id":"sha256:1484d002c52794128b41859035c069cec2294ac99ddf8420dcd5602fe15fc290"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:D64XVZNJNTMPV7FJ3JLL7JUS4U","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Winding number and Cutting number of Harmonic cycle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.NA"],"primary_cat":"math.CO","authors_text":"Woong Kook, Younng-Jin Kim","submitted_at":"2018-12-12T13:13:24Z","abstract_excerpt":"A harmonic cycle $\\lambda$, also called a discrete harmonic form, is a solution of the Laplace's equation with the combinatorial Laplace operator obtained from the boundary operators of a chain complex. By the combinatorial Hodge theory, harmonic spaces are isomorphic to the homology groups with real coefficients. In particular, if a cell complex has a one dimensional reduced homology, it has a unique harmonic cycle up to scalar, which we call the \\emph{standard harmonic cycle}. In this paper, we will present a formula for the standard harmonic cycle $\\lambda$ of a cell complex based on a high"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.04930","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":"2026-05-17T23:58:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qHp13x2VposOUaixlQpc8bBnw/Vr5myr1wyWtqyi9CSJ67+GaI7F/CQ3w0GBySU55z1FNLpSVhaPuNQyBe6gDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T10:19:55.418019Z"},"content_sha256":"8d9f6be160c7d6f39a6872f5bfd743b14aecb9632a005b19a5a7401e7152e004","schema_version":"1.0","event_id":"sha256:8d9f6be160c7d6f39a6872f5bfd743b14aecb9632a005b19a5a7401e7152e004"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/D64XVZNJNTMPV7FJ3JLL7JUS4U/bundle.json","state_url":"https://pith.science/pith/D64XVZNJNTMPV7FJ3JLL7JUS4U/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/D64XVZNJNTMPV7FJ3JLL7JUS4U/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":"2026-06-05T10:19:55Z","links":{"resolver":"https://pith.science/pith/D64XVZNJNTMPV7FJ3JLL7JUS4U","bundle":"https://pith.science/pith/D64XVZNJNTMPV7FJ3JLL7JUS4U/bundle.json","state":"https://pith.science/pith/D64XVZNJNTMPV7FJ3JLL7JUS4U/state.json","well_known_bundle":"https://pith.science/.well-known/pith/D64XVZNJNTMPV7FJ3JLL7JUS4U/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:D64XVZNJNTMPV7FJ3JLL7JUS4U","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":"ce4a411803ed1b24c13006b791e8079273e8194b7a167ee2792897cf354f9b73","cross_cats_sorted":["math.AT","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-12T13:13:24Z","title_canon_sha256":"2403706d06d2a755752840a1132c18a4dd615ddbbee8f9dc66224a836886c4d3"},"schema_version":"1.0","source":{"id":"1812.04930","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.04930","created_at":"2026-05-17T23:58:20Z"},{"alias_kind":"arxiv_version","alias_value":"1812.04930v2","created_at":"2026-05-17T23:58:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.04930","created_at":"2026-05-17T23:58:20Z"},{"alias_kind":"pith_short_12","alias_value":"D64XVZNJNTMP","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"D64XVZNJNTMPV7FJ","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"D64XVZNJ","created_at":"2026-05-18T12:32:19Z"}],"graph_snapshots":[{"event_id":"sha256:8d9f6be160c7d6f39a6872f5bfd743b14aecb9632a005b19a5a7401e7152e004","target":"graph","created_at":"2026-05-17T23:58:20Z","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":"A harmonic cycle $\\lambda$, also called a discrete harmonic form, is a solution of the Laplace's equation with the combinatorial Laplace operator obtained from the boundary operators of a chain complex. By the combinatorial Hodge theory, harmonic spaces are isomorphic to the homology groups with real coefficients. In particular, if a cell complex has a one dimensional reduced homology, it has a unique harmonic cycle up to scalar, which we call the \\emph{standard harmonic cycle}. In this paper, we will present a formula for the standard harmonic cycle $\\lambda$ of a cell complex based on a high","authors_text":"Woong Kook, Younng-Jin Kim","cross_cats":["math.AT","math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-12T13:13:24Z","title":"Winding number and Cutting number of Harmonic cycle"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.04930","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":"sha256:1484d002c52794128b41859035c069cec2294ac99ddf8420dcd5602fe15fc290","target":"record","created_at":"2026-05-17T23:58:20Z","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":"ce4a411803ed1b24c13006b791e8079273e8194b7a167ee2792897cf354f9b73","cross_cats_sorted":["math.AT","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-12T13:13:24Z","title_canon_sha256":"2403706d06d2a755752840a1132c18a4dd615ddbbee8f9dc66224a836886c4d3"},"schema_version":"1.0","source":{"id":"1812.04930","kind":"arxiv","version":2}},"canonical_sha256":"1fb97ae5a96cd8fafca9da56bfa692e5387682f6244f0741a321cce513b9172c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1fb97ae5a96cd8fafca9da56bfa692e5387682f6244f0741a321cce513b9172c","first_computed_at":"2026-05-17T23:58:20.476182Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:58:20.476182Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jQadN4wjGxrS4ylNeHNWfFGjjeLxU5E6kfONTzvW418OPqxiUtWO0SjCZtlD1T3kiat63udimHgADf7cghsWDA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:58:20.476803Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.04930","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1484d002c52794128b41859035c069cec2294ac99ddf8420dcd5602fe15fc290","sha256:8d9f6be160c7d6f39a6872f5bfd743b14aecb9632a005b19a5a7401e7152e004"],"state_sha256":"3407a0b2c3a1200b7fda13f2f2df843a2a70c788c98ff887724a63ff9859de9d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a8PG44/DXeYJBHhUHM527O9Sh2cJiNiAZXFTQs1OIoBJWTpTx3rTOH47YUY/2OCizSZPdOfYMocxleAXqqt+Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T10:19:55.419858Z","bundle_sha256":"27e75eb14492b579b21dbf5d7e78fbd5fb15037cd50bdea47a88b095f7d2bd23"}}