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We obtain a set of generators and defining relations for these commutator subgroups. In particular, we prove that $VB_n'$ is finitely generated if and only if $n \\geq 4$, and $WB_n'$ is finitely generated for $n \\geq 3$. Also we prove that $VB_3'/VB_3'' =\\mathbb{Z}_3 \\oplus \\mathbb{Z}_3 \\oplus\\mathbb{Z}_3 \\oplus \\mathbb{Z}^{\\infty}$, $VB_4' / VB_4'' = \\mathbb{Z}_3 \\oplus \\mathbb{Z}_3 \\oplus \\mathbb{Z}_3$, $WB"},"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":"1802.01383","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-02-05T13:39:09Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"e52f5417e02ae033691f2bbf4d229d93d5afbbb7ea4b52cf231ea76e4acab31d","abstract_canon_sha256":"b54c8740ca67df2248ebfd7e67fe62355de6feb1980ab22c18153c282dfa9921"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:26.534648Z","signature_b64":"gyWp5C0sujQZHgHwC+DsTnUDxpDoTrDCqJ3LEX1dxhjxtZs8rPQFBtznwsEJgsXgdxNEfI8XwRy5BG1AZobjDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"27cac88ab08038921fb8e2e2ccd80d33ab0aa921952bdd75f491cc526214429e","last_reissued_at":"2026-05-18T00:24:26.534044Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:26.534044Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Commutator Subgroups of Virtual and Welded Braid Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Krishnendu Gongopadhyay, Mikhail V. Neshchadim, Valeriy G. Bardakov","submitted_at":"2018-02-05T13:39:09Z","abstract_excerpt":"Let $VB_n$, resp. $WB_n$ denote the virtual, resp. welded, braid group on $n$ strands. We study their commutator subgroups $VB_n' = [VB_n, VB_n]$ and, $WB_n' = [WB_n, WB_n]$ respectively. We obtain a set of generators and defining relations for these commutator subgroups. In particular, we prove that $VB_n'$ is finitely generated if and only if $n \\geq 4$, and $WB_n'$ is finitely generated for $n \\geq 3$. Also we prove that $VB_3'/VB_3'' =\\mathbb{Z}_3 \\oplus \\mathbb{Z}_3 \\oplus\\mathbb{Z}_3 \\oplus \\mathbb{Z}^{\\infty}$, $VB_4' / VB_4'' = \\mathbb{Z}_3 \\oplus \\mathbb{Z}_3 \\oplus \\mathbb{Z}_3$, $WB"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.01383","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":"1802.01383","created_at":"2026-05-18T00:24:26.534135+00:00"},{"alias_kind":"arxiv_version","alias_value":"1802.01383v1","created_at":"2026-05-18T00:24:26.534135+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.01383","created_at":"2026-05-18T00:24:26.534135+00:00"},{"alias_kind":"pith_short_12","alias_value":"E7FMRCVQQA4J","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_16","alias_value":"E7FMRCVQQA4JEH5Y","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_8","alias_value":"E7FMRCVQ","created_at":"2026-05-18T12:32:22.470017+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/E7FMRCVQQA4JEH5Y4LRMZWANGO","json":"https://pith.science/pith/E7FMRCVQQA4JEH5Y4LRMZWANGO.json","graph_json":"https://pith.science/api/pith-number/E7FMRCVQQA4JEH5Y4LRMZWANGO/graph.json","events_json":"https://pith.science/api/pith-number/E7FMRCVQQA4JEH5Y4LRMZWANGO/events.json","paper":"https://pith.science/paper/E7FMRCVQ"},"agent_actions":{"view_html":"https://pith.science/pith/E7FMRCVQQA4JEH5Y4LRMZWANGO","download_json":"https://pith.science/pith/E7FMRCVQQA4JEH5Y4LRMZWANGO.json","view_paper":"https://pith.science/paper/E7FMRCVQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1802.01383&json=true","fetch_graph":"https://pith.science/api/pith-number/E7FMRCVQQA4JEH5Y4LRMZWANGO/graph.json","fetch_events":"https://pith.science/api/pith-number/E7FMRCVQQA4JEH5Y4LRMZWANGO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/E7FMRCVQQA4JEH5Y4LRMZWANGO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/E7FMRCVQQA4JEH5Y4LRMZWANGO/action/storage_attestation","attest_author":"https://pith.science/pith/E7FMRCVQQA4JEH5Y4LRMZWANGO/action/author_attestation","sign_citation":"https://pith.science/pith/E7FMRCVQQA4JEH5Y4LRMZWANGO/action/citation_signature","submit_replication":"https://pith.science/pith/E7FMRCVQQA4JEH5Y4LRMZWANGO/action/replication_record"}},"created_at":"2026-05-18T00:24:26.534135+00:00","updated_at":"2026-05-18T00:24:26.534135+00:00"}