{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:NYL2Y7FFFJF36ECKOTCE7BOXPG","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":"ed6c05063170d5a5e185fd91ffaf4eaf2c5ee4d9843bb44899de319eae97efe9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2019-08-14T23:25:57Z","title_canon_sha256":"f199523ae690ac3f35d4e3b963363229ca7d9e5d04d8b7d1921e33bd61510577"},"schema_version":"1.0","source":{"id":"1908.05371","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.05371","created_at":"2026-07-05T03:49:06Z"},{"alias_kind":"arxiv_version","alias_value":"1908.05371v2","created_at":"2026-07-05T03:49:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.05371","created_at":"2026-07-05T03:49:06Z"},{"alias_kind":"pith_short_12","alias_value":"NYL2Y7FFFJF3","created_at":"2026-07-05T03:49:06Z"},{"alias_kind":"pith_short_16","alias_value":"NYL2Y7FFFJF36ECK","created_at":"2026-07-05T03:49:06Z"},{"alias_kind":"pith_short_8","alias_value":"NYL2Y7FF","created_at":"2026-07-05T03:49:06Z"}],"graph_snapshots":[{"event_id":"sha256:665e442113b44dfc210d169637ca015184458a1a4653b40c2f1b7b36ef063043","target":"graph","created_at":"2026-07-05T03:49:06Z","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/1908.05371/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we introduce the relative $\\mathcal{L}$-invariant $r\\mathcal{L}(X)$ of a smooth, orientable, compact 4-manifold $X$ with boundary. This invariant is defined by measuring the lengths of certain paths in the cut complex of a trisection surface for $X$. This is motivated by the definition of the $\\mathcal{L}$-invariant for smooth, orientable, closed 4-manifolds by Kirby and Thompson. We show that if $X$ is a rational homology ball, then $r\\mathcal{L}(X)=0$ if and only if $X\\cong B^4$.\n  In order to better understand relative trisections, we also produce an algorithm to glue two rel","authors_text":"Gabriel Islambouli, Maggie Miller, Maggy Tomova, Nickolas A. Castro","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2019-08-14T23:25:57Z","title":"The relative $\\mathcal{L}$-invariant of a compact $4$-manifold"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.05371","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:ec50f3ba97403d7f38898f98777894b180bad833a135fd8c8a44994884ba9e9f","target":"record","created_at":"2026-07-05T03:49:06Z","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":"ed6c05063170d5a5e185fd91ffaf4eaf2c5ee4d9843bb44899de319eae97efe9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2019-08-14T23:25:57Z","title_canon_sha256":"f199523ae690ac3f35d4e3b963363229ca7d9e5d04d8b7d1921e33bd61510577"},"schema_version":"1.0","source":{"id":"1908.05371","kind":"arxiv","version":2}},"canonical_sha256":"6e17ac7ca52a4bbf104a74c44f85d779920494f62babcc9af9b3b062297b2fe5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6e17ac7ca52a4bbf104a74c44f85d779920494f62babcc9af9b3b062297b2fe5","first_computed_at":"2026-07-05T03:49:06.723177Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:49:06.723177Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"T5RGjMiFnQhwkpFCWRLLEXJIdYKM4CWPA5ncui4J6g08TpyOZZthm4JNteTpNB0t0KhNJbDy1jV32oShOW57Cw==","signature_status":"signed_v1","signed_at":"2026-07-05T03:49:06.723579Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.05371","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ec50f3ba97403d7f38898f98777894b180bad833a135fd8c8a44994884ba9e9f","sha256:665e442113b44dfc210d169637ca015184458a1a4653b40c2f1b7b36ef063043"],"state_sha256":"3bdb53fab7f17e57d6a317047b491f90852d632da9cc61c1fce41fca84b05d0a"}