{"paper":{"title":"Closed Formulas for $\\eta$-Corrections in the Once Punctured Torus","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.GT","authors_text":"Nelson A. Colon Vargas","submitted_at":"2025-08-25T11:57:49Z","abstract_excerpt":"We study $\\eta$-correction terms in the Kauffman bracket skein algebra of the once-punctured torus $K_t(\\Sigma_{1,1})$. While the Frohman--Gelca product-to-sum rule gives an explicit multiplication formula on the closed torus, the once-punctured torus introduces correction terms in the ideal $(\\eta)$. We give a closed formula for the Chebyshev-threaded family generated by the primitive determinant-two pair \\[ P_n=T_n((1,2))\\cdot(1,0). \\] The correction $\\epsilon_n$ has an explicit Chebyshev expansion whose coefficients factor as geometric sums in $t^{\\pm4}$ and whose terms are governed by a pa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.18334","kind":"arxiv","version":3},"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"}