{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:26HGDRQXVNJSFX6IKXUPFMYDH7","short_pith_number":"pith:26HGDRQX","canonical_record":{"source":{"id":"math/0606231","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2006-06-09T21:28:00Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"1655f446065be8e38f4dc4913f9447c8366ffae732b020814a5ac0c0e0a108d0","abstract_canon_sha256":"51ab46a4a7480849691bb560e6d6fd1a15e404002759c95cc99c29b524912258"},"schema_version":"1.0"},"canonical_sha256":"d78e61c617ab5322dfc855e8f2b3033fe3f6abaa07a8e5287b01013bd5491da1","source":{"kind":"arxiv","id":"math/0606231","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0606231","created_at":"2026-05-18T02:41:31Z"},{"alias_kind":"arxiv_version","alias_value":"math/0606231v1","created_at":"2026-05-18T02:41:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0606231","created_at":"2026-05-18T02:41:31Z"},{"alias_kind":"pith_short_12","alias_value":"26HGDRQXVNJS","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"26HGDRQXVNJSFX6I","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"26HGDRQX","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:26HGDRQXVNJSFX6IKXUPFMYDH7","target":"record","payload":{"canonical_record":{"source":{"id":"math/0606231","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2006-06-09T21:28:00Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"1655f446065be8e38f4dc4913f9447c8366ffae732b020814a5ac0c0e0a108d0","abstract_canon_sha256":"51ab46a4a7480849691bb560e6d6fd1a15e404002759c95cc99c29b524912258"},"schema_version":"1.0"},"canonical_sha256":"d78e61c617ab5322dfc855e8f2b3033fe3f6abaa07a8e5287b01013bd5491da1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:31.851452Z","signature_b64":"llYREcLY21oSJowlCl/54FBjllOR+HyT/GvxQdXUjOCoD4Xe+KrJyv0I0w6KWJGXljmZ8S52nr/PjBM0R0DfAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d78e61c617ab5322dfc855e8f2b3033fe3f6abaa07a8e5287b01013bd5491da1","last_reissued_at":"2026-05-18T02:41:31.851093Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:31.851093Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0606231","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":"2026-05-18T02:41:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kRVcNLkZD7DTSR0U0kQ4zQG8Uta7sIbt+qIrMBwYoXNhQEuBVKijmYvocPkp9w+bLiQH1Yzdflz5EfDUdP7hBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T08:57:25.508801Z"},"content_sha256":"67ecacd989e2e655c17104da0fa102c97cebef851c7f2fc57db82111165aab6c","schema_version":"1.0","event_id":"sha256:67ecacd989e2e655c17104da0fa102c97cebef851c7f2fc57db82111165aab6c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:26HGDRQXVNJSFX6IKXUPFMYDH7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Intrinsic Linking and Knotting in Virtual Spatial Graphs","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GT","authors_text":"Blake Mellor, Thomas Fleming","submitted_at":"2006-06-09T21:28:00Z","abstract_excerpt":"We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that these filtrations are descending and non-terminating. We also provide several examples of intrinsically virtually linked and knotted graphs. As a byproduct, we introduce the {\\it virtual unknotting number} of a knot, and show that any knot with non-trivial Jones polynomial has virtual unknotting number at least 2."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0606231","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":"2026-05-18T02:41:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HqJMrhcs97qSbBvSoP5C7hl7roN3T/XwR/Uw4uWchMx9GPcBfcmls3Rwoh4gxXlb1FIlcV87f/qrclDxszy7Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T08:57:25.509141Z"},"content_sha256":"e50173cc8a1166be4c97464f71c1d4b1be25b79198857c9974796a81b3060f6b","schema_version":"1.0","event_id":"sha256:e50173cc8a1166be4c97464f71c1d4b1be25b79198857c9974796a81b3060f6b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/26HGDRQXVNJSFX6IKXUPFMYDH7/bundle.json","state_url":"https://pith.science/pith/26HGDRQXVNJSFX6IKXUPFMYDH7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/26HGDRQXVNJSFX6IKXUPFMYDH7/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":"2026-06-23T08:57:25Z","links":{"resolver":"https://pith.science/pith/26HGDRQXVNJSFX6IKXUPFMYDH7","bundle":"https://pith.science/pith/26HGDRQXVNJSFX6IKXUPFMYDH7/bundle.json","state":"https://pith.science/pith/26HGDRQXVNJSFX6IKXUPFMYDH7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/26HGDRQXVNJSFX6IKXUPFMYDH7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:26HGDRQXVNJSFX6IKXUPFMYDH7","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":"51ab46a4a7480849691bb560e6d6fd1a15e404002759c95cc99c29b524912258","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.GT","submitted_at":"2006-06-09T21:28:00Z","title_canon_sha256":"1655f446065be8e38f4dc4913f9447c8366ffae732b020814a5ac0c0e0a108d0"},"schema_version":"1.0","source":{"id":"math/0606231","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0606231","created_at":"2026-05-18T02:41:31Z"},{"alias_kind":"arxiv_version","alias_value":"math/0606231v1","created_at":"2026-05-18T02:41:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0606231","created_at":"2026-05-18T02:41:31Z"},{"alias_kind":"pith_short_12","alias_value":"26HGDRQXVNJS","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"26HGDRQXVNJSFX6I","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"26HGDRQX","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:e50173cc8a1166be4c97464f71c1d4b1be25b79198857c9974796a81b3060f6b","target":"graph","created_at":"2026-05-18T02:41:31Z","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 introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that these filtrations are descending and non-terminating. We also provide several examples of intrinsically virtually linked and knotted graphs. As a byproduct, we introduce the {\\it virtual unknotting number} of a knot, and show that any knot with non-trivial Jones polynomial has virtual unknotting number at least 2.","authors_text":"Blake Mellor, Thomas Fleming","cross_cats":["math.CO"],"headline":"","license":"","primary_cat":"math.GT","submitted_at":"2006-06-09T21:28:00Z","title":"Intrinsic Linking and Knotting in Virtual Spatial Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0606231","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":"sha256:67ecacd989e2e655c17104da0fa102c97cebef851c7f2fc57db82111165aab6c","target":"record","created_at":"2026-05-18T02:41:31Z","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":"51ab46a4a7480849691bb560e6d6fd1a15e404002759c95cc99c29b524912258","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.GT","submitted_at":"2006-06-09T21:28:00Z","title_canon_sha256":"1655f446065be8e38f4dc4913f9447c8366ffae732b020814a5ac0c0e0a108d0"},"schema_version":"1.0","source":{"id":"math/0606231","kind":"arxiv","version":1}},"canonical_sha256":"d78e61c617ab5322dfc855e8f2b3033fe3f6abaa07a8e5287b01013bd5491da1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d78e61c617ab5322dfc855e8f2b3033fe3f6abaa07a8e5287b01013bd5491da1","first_computed_at":"2026-05-18T02:41:31.851093Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:41:31.851093Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"llYREcLY21oSJowlCl/54FBjllOR+HyT/GvxQdXUjOCoD4Xe+KrJyv0I0w6KWJGXljmZ8S52nr/PjBM0R0DfAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:41:31.851452Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0606231","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:67ecacd989e2e655c17104da0fa102c97cebef851c7f2fc57db82111165aab6c","sha256:e50173cc8a1166be4c97464f71c1d4b1be25b79198857c9974796a81b3060f6b"],"state_sha256":"0deb271dd3993af889d67edacf7bbd4017212fc4453cbbd25d2f539570412890"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LMcyVDPghjnRbUof99n3K6Us94ZcmkqcUblCt55Rr8/l7UoSm0iOnhuzD0+0ss3si4hCjj644NWtolMARA5QDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T08:57:25.511026Z","bundle_sha256":"1aedfe1ad28c16dcc60f05dbf3f8cc0fab4ff74fe6670773dce33582824e0d9c"}}