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For ordinary knots $K$ they are closely related to the bridge number $b(K)$ and the unknotting number $u(K)$ and we have $vu(K)\\leq u(K), vb(K)\\leq b(K).$\n  There are no ordinary knots $K$ with $b(K)=1.$ We show there are infinitely many homotopy classes of virtual knots each of which contains infinitely many isotopy classes of $K$ with $vb(K)=1.$ In fact for each $i\\in \\N$ there exists $K$ virtually homotopic (but not virtually isotopic) to the unknot with $vb(K)=1$ and $vu(K)=i"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0712.2347","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2007-12-14T13:29:07Z","cross_cats_sorted":[],"title_canon_sha256":"921ac5b53acfc495385ebd41fd4a39924fffa2495669838a6199551d3acdc013","abstract_canon_sha256":"811c332daf325efdc02487211391e9a70ab6cbb9c69ea783409a9ed42119514f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:53:30.729793Z","signature_b64":"xOea7gZz+zwIOEwEoL2GSdKDY5Op/OTY54dLetC0fl8CQ0ZWML25m4tGhRszpTK7q/jJw8IBrv5++MsjhvFODw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf3840f0022e39ef8e2e1bec2595c86efc84cd3a98ea7a7627db5312730ff567","last_reissued_at":"2026-05-18T02:53:30.729024Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:53:30.729024Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Virtual Bridge Number One Knots","license":"","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Evarist Byberi, Vladimir Chernov (Tchernov)","submitted_at":"2007-12-14T13:29:07Z","abstract_excerpt":"We define the virtual bridge number $vb(K)$ and the virtual unknotting number $vu(K)$ invariants for virtual knots. For ordinary knots $K$ they are closely related to the bridge number $b(K)$ and the unknotting number $u(K)$ and we have $vu(K)\\leq u(K), vb(K)\\leq b(K).$\n  There are no ordinary knots $K$ with $b(K)=1.$ We show there are infinitely many homotopy classes of virtual knots each of which contains infinitely many isotopy classes of $K$ with $vb(K)=1.$ In fact for each $i\\in \\N$ there exists $K$ virtually homotopic (but not virtually isotopic) to the unknot with $vb(K)=1$ and $vu(K)=i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.2347","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0712.2347","created_at":"2026-05-18T02:53:30.729139+00:00"},{"alias_kind":"arxiv_version","alias_value":"0712.2347v1","created_at":"2026-05-18T02:53:30.729139+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0712.2347","created_at":"2026-05-18T02:53:30.729139+00:00"},{"alias_kind":"pith_short_12","alias_value":"X44EB4ACFY46","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_16","alias_value":"X44EB4ACFY467DRO","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_8","alias_value":"X44EB4AC","created_at":"2026-05-18T12:25:56.245647+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/X44EB4ACFY467DRODPWCLFOIN3","json":"https://pith.science/pith/X44EB4ACFY467DRODPWCLFOIN3.json","graph_json":"https://pith.science/api/pith-number/X44EB4ACFY467DRODPWCLFOIN3/graph.json","events_json":"https://pith.science/api/pith-number/X44EB4ACFY467DRODPWCLFOIN3/events.json","paper":"https://pith.science/paper/X44EB4AC"},"agent_actions":{"view_html":"https://pith.science/pith/X44EB4ACFY467DRODPWCLFOIN3","download_json":"https://pith.science/pith/X44EB4ACFY467DRODPWCLFOIN3.json","view_paper":"https://pith.science/paper/X44EB4AC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0712.2347&json=true","fetch_graph":"https://pith.science/api/pith-number/X44EB4ACFY467DRODPWCLFOIN3/graph.json","fetch_events":"https://pith.science/api/pith-number/X44EB4ACFY467DRODPWCLFOIN3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/X44EB4ACFY467DRODPWCLFOIN3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/X44EB4ACFY467DRODPWCLFOIN3/action/storage_attestation","attest_author":"https://pith.science/pith/X44EB4ACFY467DRODPWCLFOIN3/action/author_attestation","sign_citation":"https://pith.science/pith/X44EB4ACFY467DRODPWCLFOIN3/action/citation_signature","submit_replication":"https://pith.science/pith/X44EB4ACFY467DRODPWCLFOIN3/action/replication_record"}},"created_at":"2026-05-18T02:53:30.729139+00:00","updated_at":"2026-05-18T02:53:30.729139+00:00"}