{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:VLTJLQRGOR3PWTDEWWXCOURBJA","short_pith_number":"pith:VLTJLQRG","canonical_record":{"source":{"id":"1603.04877","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-03-15T20:31:43Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"49994747deecb87cd2eabb189857aa8b9382ff0b9f6c23dddf5c9635b4f4f4ec","abstract_canon_sha256":"8f801b1ee4d1dfcfdc196159daecefb5c64c1fe6d23e32bacfb69fa6ca381308"},"schema_version":"1.0"},"canonical_sha256":"aae695c2267476fb4c64b5ae275221482c828f680f82c48c6072ff282f6528b7","source":{"kind":"arxiv","id":"1603.04877","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.04877","created_at":"2026-05-18T01:18:59Z"},{"alias_kind":"arxiv_version","alias_value":"1603.04877v1","created_at":"2026-05-18T01:18:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.04877","created_at":"2026-05-18T01:18:59Z"},{"alias_kind":"pith_short_12","alias_value":"VLTJLQRGOR3P","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"VLTJLQRGOR3PWTDE","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"VLTJLQRG","created_at":"2026-05-18T12:30:48Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:VLTJLQRGOR3PWTDEWWXCOURBJA","target":"record","payload":{"canonical_record":{"source":{"id":"1603.04877","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-03-15T20:31:43Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"49994747deecb87cd2eabb189857aa8b9382ff0b9f6c23dddf5c9635b4f4f4ec","abstract_canon_sha256":"8f801b1ee4d1dfcfdc196159daecefb5c64c1fe6d23e32bacfb69fa6ca381308"},"schema_version":"1.0"},"canonical_sha256":"aae695c2267476fb4c64b5ae275221482c828f680f82c48c6072ff282f6528b7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:59.773782Z","signature_b64":"tbGSHnf6COoXkr6rVXpK4XvHGR6X3XOC8BZS8YDGcBtnsUDktVr6F22WC/Zzrte+PYv37nciKanSSGnYugeABQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aae695c2267476fb4c64b5ae275221482c828f680f82c48c6072ff282f6528b7","last_reissued_at":"2026-05-18T01:18:59.773287Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:59.773287Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.04877","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":"2026-05-18T01:18:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v+RjS6rvK5iPtb73tr7zExggymwh9Zd/5BdyVtPtqi0vfqy6B02GdOxwdw63SWbInusS1qrUck94J9iSW/d/CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T11:40:45.383848Z"},"content_sha256":"9a13381a51a8d2da1264beda7b55c0227ed2ee47fd00981cf975d48cf0fa85bd","schema_version":"1.0","event_id":"sha256:9a13381a51a8d2da1264beda7b55c0227ed2ee47fd00981cf975d48cf0fa85bd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:VLTJLQRGOR3PWTDEWWXCOURBJA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Polyhedral Embeddings and Immersions of Many Triangulated 2-Manifolds with Few Vertices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.MG","authors_text":"Ulrich Brehm, Undine Leopold","submitted_at":"2016-03-15T20:31:43Z","abstract_excerpt":"This article presents an improvement and extension of the heuristic first presented by Hougardy, Lutz, and Zelke in 2010 for realizing triangulated orientable surfaces with few vertices by a simplex-wise linear embedding. The improvement consists in the applicability to non-orientable surfaces (simplex-wise linear immersions) as well as symmetric realizations. With the help of our algorithm, numerous - often symmetric - realizations of non-orientable surfaces with the minimal number or few vertices have been obtained for the first time. These examples include the the Projective Plane with one "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.04877","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":""},"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":"2026-05-18T01:18:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ysda1jLtoT4s2jJ8QLNb2Q/sLA4HsS2wqSe0GiCGf1ElXiq4Wa8okyx+AQ8hk0Upksd/bUAg983b5Pv0g8ygCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T11:40:45.384180Z"},"content_sha256":"153d24d3d56d4c70669d9306665eff8d01771985076cb12e1e3a6785152319ec","schema_version":"1.0","event_id":"sha256:153d24d3d56d4c70669d9306665eff8d01771985076cb12e1e3a6785152319ec"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VLTJLQRGOR3PWTDEWWXCOURBJA/bundle.json","state_url":"https://pith.science/pith/VLTJLQRGOR3PWTDEWWXCOURBJA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VLTJLQRGOR3PWTDEWWXCOURBJA/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":"2026-06-29T11:40:45Z","links":{"resolver":"https://pith.science/pith/VLTJLQRGOR3PWTDEWWXCOURBJA","bundle":"https://pith.science/pith/VLTJLQRGOR3PWTDEWWXCOURBJA/bundle.json","state":"https://pith.science/pith/VLTJLQRGOR3PWTDEWWXCOURBJA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VLTJLQRGOR3PWTDEWWXCOURBJA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:VLTJLQRGOR3PWTDEWWXCOURBJA","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":"8f801b1ee4d1dfcfdc196159daecefb5c64c1fe6d23e32bacfb69fa6ca381308","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-03-15T20:31:43Z","title_canon_sha256":"49994747deecb87cd2eabb189857aa8b9382ff0b9f6c23dddf5c9635b4f4f4ec"},"schema_version":"1.0","source":{"id":"1603.04877","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.04877","created_at":"2026-05-18T01:18:59Z"},{"alias_kind":"arxiv_version","alias_value":"1603.04877v1","created_at":"2026-05-18T01:18:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.04877","created_at":"2026-05-18T01:18:59Z"},{"alias_kind":"pith_short_12","alias_value":"VLTJLQRGOR3P","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"VLTJLQRGOR3PWTDE","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"VLTJLQRG","created_at":"2026-05-18T12:30:48Z"}],"graph_snapshots":[{"event_id":"sha256:153d24d3d56d4c70669d9306665eff8d01771985076cb12e1e3a6785152319ec","target":"graph","created_at":"2026-05-18T01:18:59Z","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":"This article presents an improvement and extension of the heuristic first presented by Hougardy, Lutz, and Zelke in 2010 for realizing triangulated orientable surfaces with few vertices by a simplex-wise linear embedding. The improvement consists in the applicability to non-orientable surfaces (simplex-wise linear immersions) as well as symmetric realizations. With the help of our algorithm, numerous - often symmetric - realizations of non-orientable surfaces with the minimal number or few vertices have been obtained for the first time. These examples include the the Projective Plane with one ","authors_text":"Ulrich Brehm, Undine Leopold","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-03-15T20:31:43Z","title":"Polyhedral Embeddings and Immersions of Many Triangulated 2-Manifolds with Few Vertices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.04877","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":"sha256:9a13381a51a8d2da1264beda7b55c0227ed2ee47fd00981cf975d48cf0fa85bd","target":"record","created_at":"2026-05-18T01:18:59Z","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":"8f801b1ee4d1dfcfdc196159daecefb5c64c1fe6d23e32bacfb69fa6ca381308","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-03-15T20:31:43Z","title_canon_sha256":"49994747deecb87cd2eabb189857aa8b9382ff0b9f6c23dddf5c9635b4f4f4ec"},"schema_version":"1.0","source":{"id":"1603.04877","kind":"arxiv","version":1}},"canonical_sha256":"aae695c2267476fb4c64b5ae275221482c828f680f82c48c6072ff282f6528b7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aae695c2267476fb4c64b5ae275221482c828f680f82c48c6072ff282f6528b7","first_computed_at":"2026-05-18T01:18:59.773287Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:18:59.773287Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tbGSHnf6COoXkr6rVXpK4XvHGR6X3XOC8BZS8YDGcBtnsUDktVr6F22WC/Zzrte+PYv37nciKanSSGnYugeABQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:18:59.773782Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.04877","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9a13381a51a8d2da1264beda7b55c0227ed2ee47fd00981cf975d48cf0fa85bd","sha256:153d24d3d56d4c70669d9306665eff8d01771985076cb12e1e3a6785152319ec"],"state_sha256":"720b66cf3f32a75b8ea2e2142ac3deac9e0dea782946b6539d3e7312e597e44b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l7eY4isZKKNKJwo0U5yhFJGBkl8+jE/gMcxpMhPp6AIzONkPJXmIGZF90bX/XCUswVgh3OFFAHary499ONmSAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T11:40:45.386466Z","bundle_sha256":"f2c85f93e1edd01bf7df7fe70e0a25c5013c3814a9038995ac6268c9310350e5"}}