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For fixed $r$, the sets $\\mathcal{U}_{r, w}$ form an open cover of $X(\\beta)$. We conjecture that $\\mathcal{U}_{r,w}$ is given by the nonvanishing of some cluster variables in a single cluster for the cluster structure on $\\mathbb{C}[X(\\beta)]$ and that $\\mathcal{U}_{r,w}$ admits a cluster structure given by freezing these variabl"},"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":"2505.08211","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2025-05-13T03:57:38Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"196422a6de735d18098ee304e6655674638cf94d6aaadad0ee065a4e74b562a2","abstract_canon_sha256":"40fd8d54755641b0f12c3ec58b745fc0248ab62612232ac4532d7701f59bd1ee"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:02:26.661396Z","signature_b64":"Lp//8JnxhItnlVz70Ymgk+nBzdAxV0RFwjSpBGVhlDoyNaj2F/EGHlXTlsrWGxpAQ4iOD0KkO/v+yVam3pvsAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5076d8d92ecd407f0f543d52f3560e1305d67ee7130d15fca4fe22c86c32e588","last_reissued_at":"2026-07-05T11:02:26.660877Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:02:26.660877Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Splicing braid varieties","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AG","authors_text":"Eugene Gorsky, Jos\\'e Simental, Soyeon Kim, Tonie Scroggin","submitted_at":"2025-05-13T03:57:38Z","abstract_excerpt":"For a positive braid $\\beta \\in \\mathrm{Br}^{+}_{k}$, we consider the braid variety $X(\\beta)$. 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