{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:F2RNYOAPHYDR2DKHOI52JISYZI","short_pith_number":"pith:F2RNYOAP","canonical_record":{"source":{"id":"2407.08158","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-07-11T03:25:38Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"3a52c08f80c985e98ef3a145232ff4e7e37e3dd18f2a17b1d1f92225d2166f02","abstract_canon_sha256":"0ab8e1f44ea86565d18c8bfc85e38ea10e5a04818bfed863b30834ad25059c31"},"schema_version":"1.0"},"canonical_sha256":"2ea2dc380f3e071d0d47723ba4a258ca19bdcfac9765c65f924f42baa534a2af","source":{"kind":"arxiv","id":"2407.08158","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.08158","created_at":"2026-07-05T12:06:00Z"},{"alias_kind":"arxiv_version","alias_value":"2407.08158v1","created_at":"2026-07-05T12:06:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.08158","created_at":"2026-07-05T12:06:00Z"},{"alias_kind":"pith_short_12","alias_value":"F2RNYOAPHYDR","created_at":"2026-07-05T12:06:00Z"},{"alias_kind":"pith_short_16","alias_value":"F2RNYOAPHYDR2DKH","created_at":"2026-07-05T12:06:00Z"},{"alias_kind":"pith_short_8","alias_value":"F2RNYOAP","created_at":"2026-07-05T12:06:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:F2RNYOAPHYDR2DKHOI52JISYZI","target":"record","payload":{"canonical_record":{"source":{"id":"2407.08158","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-07-11T03:25:38Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"3a52c08f80c985e98ef3a145232ff4e7e37e3dd18f2a17b1d1f92225d2166f02","abstract_canon_sha256":"0ab8e1f44ea86565d18c8bfc85e38ea10e5a04818bfed863b30834ad25059c31"},"schema_version":"1.0"},"canonical_sha256":"2ea2dc380f3e071d0d47723ba4a258ca19bdcfac9765c65f924f42baa534a2af","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:06:00.691973Z","signature_b64":"XCA/QBonos+Di1vqQnKQoN4SkAEtzKeASKpf57YLSgD7SYrXgSd4NJnkgiOsiTw0Q0WS0neHEGw16G8NiAKsDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ea2dc380f3e071d0d47723ba4a258ca19bdcfac9765c65f924f42baa534a2af","last_reissued_at":"2026-07-05T12:06:00.691406Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:06:00.691406Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2407.08158","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":"2026-07-05T12:06:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3Hlaxj9fNlZnN5ahkGN9umddIe94uxFGhgPWILpyyAhSZvvfLMzPyfUgo7hSRBgXW/WY+tuDIPuBTKrLYoRCBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T19:34:25.778591Z"},"content_sha256":"c618b461ad571a0990afbf45995c457f5710365e2cc5a91b4b8c6f6301bdfd50","schema_version":"1.0","event_id":"sha256:c618b461ad571a0990afbf45995c457f5710365e2cc5a91b4b8c6f6301bdfd50"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:F2RNYOAPHYDR2DKHOI52JISYZI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Topology of Cut Complexes II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CO","authors_text":"Lei Xue, Margaret Bayer, Marija Jeli\\'c Milutinovi\\'c, Mark Denker, Sheila Sundaram","submitted_at":"2024-07-11T03:25:38Z","abstract_excerpt":"We continue the study of the $k$-cut complex $\\Delta_k(G)$ of a graph $G$ initiated in the paper of Bayer, Denker, Jeli\\'c Milutinovi\\'c, Rowlands, Sundaram and Xue [Topology of cut complexes of graphs, SIAM J. on Discrete Math. 38(2): 1630--1675 (2024)].\n  We give explicit formulas for the $f$- and $h$-polynomials of the cut complex $\\Delta_k(G_1+G_2) $ of the disjoint union of two graphs $G_1$ and $G_2$, and for the homology representation of $\\Delta_k(K_m+K_n)$.\n  We also study the cut complex of the squared path and the grid graph. Our techniques include tools from combinatorial topology, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.08158","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/2407.08158/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":"2026-07-05T12:06:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sgb2jARPYjMfcsID6GDlLVcfizw5E7Z1arfTscu2wSzSOeLPAwvwzKq/NzXADHg6h37SWFkHKHGve5GdZfj6BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T19:34:25.778959Z"},"content_sha256":"c7cad120d0db305aa51ac4f14fe5fa0f1c772dbaab2da009b07b61cacb3d4794","schema_version":"1.0","event_id":"sha256:c7cad120d0db305aa51ac4f14fe5fa0f1c772dbaab2da009b07b61cacb3d4794"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F2RNYOAPHYDR2DKHOI52JISYZI/bundle.json","state_url":"https://pith.science/pith/F2RNYOAPHYDR2DKHOI52JISYZI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F2RNYOAPHYDR2DKHOI52JISYZI/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":"2026-07-06T19:34:25Z","links":{"resolver":"https://pith.science/pith/F2RNYOAPHYDR2DKHOI52JISYZI","bundle":"https://pith.science/pith/F2RNYOAPHYDR2DKHOI52JISYZI/bundle.json","state":"https://pith.science/pith/F2RNYOAPHYDR2DKHOI52JISYZI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F2RNYOAPHYDR2DKHOI52JISYZI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:F2RNYOAPHYDR2DKHOI52JISYZI","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":"0ab8e1f44ea86565d18c8bfc85e38ea10e5a04818bfed863b30834ad25059c31","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-07-11T03:25:38Z","title_canon_sha256":"3a52c08f80c985e98ef3a145232ff4e7e37e3dd18f2a17b1d1f92225d2166f02"},"schema_version":"1.0","source":{"id":"2407.08158","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.08158","created_at":"2026-07-05T12:06:00Z"},{"alias_kind":"arxiv_version","alias_value":"2407.08158v1","created_at":"2026-07-05T12:06:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.08158","created_at":"2026-07-05T12:06:00Z"},{"alias_kind":"pith_short_12","alias_value":"F2RNYOAPHYDR","created_at":"2026-07-05T12:06:00Z"},{"alias_kind":"pith_short_16","alias_value":"F2RNYOAPHYDR2DKH","created_at":"2026-07-05T12:06:00Z"},{"alias_kind":"pith_short_8","alias_value":"F2RNYOAP","created_at":"2026-07-05T12:06:00Z"}],"graph_snapshots":[{"event_id":"sha256:c7cad120d0db305aa51ac4f14fe5fa0f1c772dbaab2da009b07b61cacb3d4794","target":"graph","created_at":"2026-07-05T12:06:00Z","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/2407.08158/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We continue the study of the $k$-cut complex $\\Delta_k(G)$ of a graph $G$ initiated in the paper of Bayer, Denker, Jeli\\'c Milutinovi\\'c, Rowlands, Sundaram and Xue [Topology of cut complexes of graphs, SIAM J. on Discrete Math. 38(2): 1630--1675 (2024)].\n  We give explicit formulas for the $f$- and $h$-polynomials of the cut complex $\\Delta_k(G_1+G_2) $ of the disjoint union of two graphs $G_1$ and $G_2$, and for the homology representation of $\\Delta_k(K_m+K_n)$.\n  We also study the cut complex of the squared path and the grid graph. Our techniques include tools from combinatorial topology, ","authors_text":"Lei Xue, Margaret Bayer, Marija Jeli\\'c Milutinovi\\'c, Mark Denker, Sheila Sundaram","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-07-11T03:25:38Z","title":"Topology of Cut Complexes II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.08158","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:c618b461ad571a0990afbf45995c457f5710365e2cc5a91b4b8c6f6301bdfd50","target":"record","created_at":"2026-07-05T12:06:00Z","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":"0ab8e1f44ea86565d18c8bfc85e38ea10e5a04818bfed863b30834ad25059c31","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-07-11T03:25:38Z","title_canon_sha256":"3a52c08f80c985e98ef3a145232ff4e7e37e3dd18f2a17b1d1f92225d2166f02"},"schema_version":"1.0","source":{"id":"2407.08158","kind":"arxiv","version":1}},"canonical_sha256":"2ea2dc380f3e071d0d47723ba4a258ca19bdcfac9765c65f924f42baa534a2af","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2ea2dc380f3e071d0d47723ba4a258ca19bdcfac9765c65f924f42baa534a2af","first_computed_at":"2026-07-05T12:06:00.691406Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:06:00.691406Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XCA/QBonos+Di1vqQnKQoN4SkAEtzKeASKpf57YLSgD7SYrXgSd4NJnkgiOsiTw0Q0WS0neHEGw16G8NiAKsDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T12:06:00.691973Z","signed_message":"canonical_sha256_bytes"},"source_id":"2407.08158","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c618b461ad571a0990afbf45995c457f5710365e2cc5a91b4b8c6f6301bdfd50","sha256:c7cad120d0db305aa51ac4f14fe5fa0f1c772dbaab2da009b07b61cacb3d4794"],"state_sha256":"c1b77d34e40076f0fec8383824fdc2c7066eee6e78f0169e9597c20f574559a4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"slZYS+J69tzsGcEnwg/q+s/pi4BLqNUbTr7NS+sgZOgbg8ISuG4cq2mX5rKaNMJEUnEpOgWiHegC26sFEBhODQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T19:34:25.780944Z","bundle_sha256":"1bbc6028e60970c5a284e7758bb838d4973882483f6d43c01bb72ab1c3140b0a"}}