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One can see from the definitions that for an classical knot $K$ the values of these invariants are less or equal than the classical canonical genus $g_c(K)$ and the bridge number $b(K)$ respectively. We use Manturov's projection from the category of virtual knot diagrams to the category of classical knot diagrams, to show that for every classical knot type $K$ we have $g_{vc}(K)=g_c(K)$ and $vb(K)=b(K)$."},"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":"1209.0178","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2012-09-02T14:15:41Z","cross_cats_sorted":[],"title_canon_sha256":"400729323f1c919b48fa0ca6ab1f884fcb5893e6a9550cfa68d0548222a9ec34","abstract_canon_sha256":"7f1b341a325530b93e7a64e7bb843f24cbb5813ed679d4b6d6bb2f9677b5f8f4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:53:30.486237Z","signature_b64":"vXEFW8honEG0tkgD21FdkiMtTbD7NlUdxj0p5wym02OJsUTOxrv6fwBFIbCJl24RpSxY424R950F0BgCbN4VDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb48e7d6f355bc86b576a3e7b78b9b9208946ae1a904683add8bb2133d5ff0e0","last_reissued_at":"2026-05-18T02:53:30.485611Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:53:30.485611Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some Corollaries of Manturov's projection Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Vladimir Chernov","submitted_at":"2012-09-02T14:15:41Z","abstract_excerpt":"In our works with Stoimenow, Vdovina and with Byberi, we introduced the virtual canonical genus $g_{vc}(K)$ and the virtual bridge number $vb(K)$ invariants of virtual knots. 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