{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:U6XBMGVWTWWGEBQW32BCIQBDAC","short_pith_number":"pith:U6XBMGVW","canonical_record":{"source":{"id":"2504.02625","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2025-04-03T14:23:50Z","cross_cats_sorted":["math.CO","math.QA"],"title_canon_sha256":"7e375f8966311439fa8a04220efedd6a649fe9ad98f57b05fb7c4d08586f8f6b","abstract_canon_sha256":"01103b622613b1e62a848368b3b7185d4ef5a881652ef60720c8ba04c9d56680"},"schema_version":"1.0"},"canonical_sha256":"a7ae161ab69dac620616de822440230084e78d29fe9716a0f2c0a45600043c08","source":{"kind":"arxiv","id":"2504.02625","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.02625","created_at":"2026-07-05T10:44:00Z"},{"alias_kind":"arxiv_version","alias_value":"2504.02625v1","created_at":"2026-07-05T10:44:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.02625","created_at":"2026-07-05T10:44:00Z"},{"alias_kind":"pith_short_12","alias_value":"U6XBMGVWTWWG","created_at":"2026-07-05T10:44:00Z"},{"alias_kind":"pith_short_16","alias_value":"U6XBMGVWTWWGEBQW","created_at":"2026-07-05T10:44:00Z"},{"alias_kind":"pith_short_8","alias_value":"U6XBMGVW","created_at":"2026-07-05T10:44:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:U6XBMGVWTWWGEBQW32BCIQBDAC","target":"record","payload":{"canonical_record":{"source":{"id":"2504.02625","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2025-04-03T14:23:50Z","cross_cats_sorted":["math.CO","math.QA"],"title_canon_sha256":"7e375f8966311439fa8a04220efedd6a649fe9ad98f57b05fb7c4d08586f8f6b","abstract_canon_sha256":"01103b622613b1e62a848368b3b7185d4ef5a881652ef60720c8ba04c9d56680"},"schema_version":"1.0"},"canonical_sha256":"a7ae161ab69dac620616de822440230084e78d29fe9716a0f2c0a45600043c08","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:44:00.040761Z","signature_b64":"HsWZTUq5qwlmkcW7tkQSsYON/XWdBr++s6M/ZavXU1UpbwehIKCeUb29CAo8LE/kMKFIbeMKYT2eS9BARLW0CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7ae161ab69dac620616de822440230084e78d29fe9716a0f2c0a45600043c08","last_reissued_at":"2026-07-05T10:44:00.040257Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:44:00.040257Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2504.02625","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T10:44:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x4Z/Hc5AcR0xNVFglQl+NJHaqZPKr+7/WKaBDHIOqeugO6VZ2iQS/bFCBC5lcAhEShwBB8bdIm3vtdcC0ZpMAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T14:57:38.472866Z"},"content_sha256":"38a68133440a5f3264601d5b9555ca3684ed236d9af648e1d6800e5f7955099e","schema_version":"1.0","event_id":"sha256:38a68133440a5f3264601d5b9555ca3684ed236d9af648e1d6800e5f7955099e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:U6XBMGVWTWWGEBQW32BCIQBDAC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A spanning tree model for Khovanov homology, Rasmussen's s-invariant and exotic discs in the $4$-ball","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.QA"],"primary_cat":"math.GT","authors_text":"Aninda Banerjee, Apratim Chakraborty, Swarup Kumar Das","submitted_at":"2025-04-03T14:23:50Z","abstract_excerpt":"The checkerboard coloring of knot diagrams offers a graph-theoretical approach to address topological questions. Champanerkar and Kofman defined a complex generated by the spanning trees of a graph obtained from the checkerboard coloring whose homology is the reduced Khovanov homology. Notably, the differential in their chain complex was not explicitly defined. We explicitly define the combinatorial form of the differential within the spanning tree complex. We additionally provide a description of Rasmussen's $s$-invariant within the context of the spanning tree complex. Applying our technique"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.02625","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2504.02625/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T10:44:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ur7fVi73A69cJgeSKgSLNc13kZ/yKQNNafsZY/zOXRmcmXcI/3IPGV1STfSpgwyOiBe33eF4tX8Docta0pgRBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T14:57:38.473804Z"},"content_sha256":"9cb39946410e9c51160812afa4929c572446e0c930aac95e7d3bc4dceb36b0e2","schema_version":"1.0","event_id":"sha256:9cb39946410e9c51160812afa4929c572446e0c930aac95e7d3bc4dceb36b0e2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U6XBMGVWTWWGEBQW32BCIQBDAC/bundle.json","state_url":"https://pith.science/pith/U6XBMGVWTWWGEBQW32BCIQBDAC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U6XBMGVWTWWGEBQW32BCIQBDAC/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":"2026-08-10T14:57:38Z","links":{"resolver":"https://pith.science/pith/U6XBMGVWTWWGEBQW32BCIQBDAC","bundle":"https://pith.science/pith/U6XBMGVWTWWGEBQW32BCIQBDAC/bundle.json","state":"https://pith.science/pith/U6XBMGVWTWWGEBQW32BCIQBDAC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U6XBMGVWTWWGEBQW32BCIQBDAC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:U6XBMGVWTWWGEBQW32BCIQBDAC","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":"01103b622613b1e62a848368b3b7185d4ef5a881652ef60720c8ba04c9d56680","cross_cats_sorted":["math.CO","math.QA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2025-04-03T14:23:50Z","title_canon_sha256":"7e375f8966311439fa8a04220efedd6a649fe9ad98f57b05fb7c4d08586f8f6b"},"schema_version":"1.0","source":{"id":"2504.02625","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.02625","created_at":"2026-07-05T10:44:00Z"},{"alias_kind":"arxiv_version","alias_value":"2504.02625v1","created_at":"2026-07-05T10:44:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.02625","created_at":"2026-07-05T10:44:00Z"},{"alias_kind":"pith_short_12","alias_value":"U6XBMGVWTWWG","created_at":"2026-07-05T10:44:00Z"},{"alias_kind":"pith_short_16","alias_value":"U6XBMGVWTWWGEBQW","created_at":"2026-07-05T10:44:00Z"},{"alias_kind":"pith_short_8","alias_value":"U6XBMGVW","created_at":"2026-07-05T10:44:00Z"}],"graph_snapshots":[{"event_id":"sha256:9cb39946410e9c51160812afa4929c572446e0c930aac95e7d3bc4dceb36b0e2","target":"graph","created_at":"2026-07-05T10:44:00Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2504.02625/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The checkerboard coloring of knot diagrams offers a graph-theoretical approach to address topological questions. Champanerkar and Kofman defined a complex generated by the spanning trees of a graph obtained from the checkerboard coloring whose homology is the reduced Khovanov homology. Notably, the differential in their chain complex was not explicitly defined. We explicitly define the combinatorial form of the differential within the spanning tree complex. We additionally provide a description of Rasmussen's $s$-invariant within the context of the spanning tree complex. Applying our technique","authors_text":"Aninda Banerjee, Apratim Chakraborty, Swarup Kumar Das","cross_cats":["math.CO","math.QA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2025-04-03T14:23:50Z","title":"A spanning tree model for Khovanov homology, Rasmussen's s-invariant and exotic discs in the $4$-ball"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.02625","kind":"arxiv","version":1},"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":"sha256:38a68133440a5f3264601d5b9555ca3684ed236d9af648e1d6800e5f7955099e","target":"record","created_at":"2026-07-05T10:44:00Z","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":"01103b622613b1e62a848368b3b7185d4ef5a881652ef60720c8ba04c9d56680","cross_cats_sorted":["math.CO","math.QA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2025-04-03T14:23:50Z","title_canon_sha256":"7e375f8966311439fa8a04220efedd6a649fe9ad98f57b05fb7c4d08586f8f6b"},"schema_version":"1.0","source":{"id":"2504.02625","kind":"arxiv","version":1}},"canonical_sha256":"a7ae161ab69dac620616de822440230084e78d29fe9716a0f2c0a45600043c08","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a7ae161ab69dac620616de822440230084e78d29fe9716a0f2c0a45600043c08","first_computed_at":"2026-07-05T10:44:00.040257Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:44:00.040257Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HsWZTUq5qwlmkcW7tkQSsYON/XWdBr++s6M/ZavXU1UpbwehIKCeUb29CAo8LE/kMKFIbeMKYT2eS9BARLW0CQ==","signature_status":"signed_v1","signed_at":"2026-07-05T10:44:00.040761Z","signed_message":"canonical_sha256_bytes"},"source_id":"2504.02625","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:38a68133440a5f3264601d5b9555ca3684ed236d9af648e1d6800e5f7955099e","sha256:9cb39946410e9c51160812afa4929c572446e0c930aac95e7d3bc4dceb36b0e2"],"state_sha256":"b311bf1f5b32e8c5095943b6ff56977315cd412b761cc72bf037a3c7843b4370"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pt3See/bPXzpOT+VQJSvhooUI5xXYag4LOcf+EKFbHt3vkgqF9mDniSDGZiwbXZ1bFtqed7eeGpBj4pkz838Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T14:57:38.479436Z","bundle_sha256":"a651e876e14e0caa1d300abfa48199212e17d5f411985f2f23aa0cf1c1c49e34"}}