{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:ZKADUITLC3HAYYUYQZYPATY36U","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":"c197af9889f8df4718903d7d0fe8a4f7b48638c7b1f0e249f75d109457c4bb1f","cross_cats_sorted":["cs.CG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2017-08-25T13:38:17Z","title_canon_sha256":"5b22ba719061bd56a793f71073bd6e8fb29e2123fc8d747c2dde60e8a87fda03"},"schema_version":"1.0","source":{"id":"1708.07734","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.07734","created_at":"2026-05-18T00:07:33Z"},{"alias_kind":"arxiv_version","alias_value":"1708.07734v2","created_at":"2026-05-18T00:07:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.07734","created_at":"2026-05-18T00:07:33Z"},{"alias_kind":"pith_short_12","alias_value":"ZKADUITLC3HA","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_16","alias_value":"ZKADUITLC3HAYYUY","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_8","alias_value":"ZKADUITL","created_at":"2026-05-18T12:31:59Z"}],"graph_snapshots":[{"event_id":"sha256:f4b71d39699b749f612215c44bed727344df9bd64d3ebf92fc05c8abbc651703","target":"graph","created_at":"2026-05-18T00:07:33Z","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":"We prove that the problem of deciding whether a 2- or 3-dimensional simplicial complex embeds into $\\mathbb{R}^3$ is NP-hard. Our construction also shows that deciding whether a 3-manifold with boundary tori admits an $\\mathbb{S}^{3}$ filling is NP-hard. The former stands in contrast with the lower dimensional cases which can be solved in linear time,and the latter with a variety of computational problems in 3-manifold topology (for example, unknot or 3-sphere recognition, which are in NP and co-NP assuming the Generalized Riemann Hypothesis). Our reduction encodes a satisfiability instance in","authors_text":"Arnaud de Mesmay, Eric Sedgwick, Martin Tancer, Yo'av Rieck","cross_cats":["cs.CG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2017-08-25T13:38:17Z","title":"Embeddability in $\\mathbb{R}^3$ is NP-hard"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.07734","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":"sha256:7438717601447d6b5792d4ed453665bd711a7188a0c8a6fb6f31cdbe3134b8e9","target":"record","created_at":"2026-05-18T00:07:33Z","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":"c197af9889f8df4718903d7d0fe8a4f7b48638c7b1f0e249f75d109457c4bb1f","cross_cats_sorted":["cs.CG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2017-08-25T13:38:17Z","title_canon_sha256":"5b22ba719061bd56a793f71073bd6e8fb29e2123fc8d747c2dde60e8a87fda03"},"schema_version":"1.0","source":{"id":"1708.07734","kind":"arxiv","version":2}},"canonical_sha256":"ca803a226b16ce0c62988670f04f1bf51d4d9c81295c91eb1ce3a8f5a9ca6858","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ca803a226b16ce0c62988670f04f1bf51d4d9c81295c91eb1ce3a8f5a9ca6858","first_computed_at":"2026-05-18T00:07:33.052725Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:07:33.052725Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2spEgXqjz1k2/jIYRLwlsI2sqOyuWWH13RMhUroneL3SgpMat5ICp1B/F4DKuYCDhM4XNYbTnh61t76KLImWBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:07:33.053375Z","signed_message":"canonical_sha256_bytes"},"source_id":"1708.07734","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7438717601447d6b5792d4ed453665bd711a7188a0c8a6fb6f31cdbe3134b8e9","sha256:f4b71d39699b749f612215c44bed727344df9bd64d3ebf92fc05c8abbc651703"],"state_sha256":"970384be60d4391d03cb01ad0a9991b29326a11e52482c7562ce5aef98a2699d"}