{"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 $X(\\beta)$. We define a family of open sets $\\mathcal{U}_{r, w}$ in $X(\\beta)$, where $w \\in S_k$ is a permutation and $r$ is a positive integer no greater than the length of $\\beta$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.08211","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.08211/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}