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We localize by inverting the nonzero characters to get an algebra $S^{-1}K_N(F)$ over the function field of the character variety. We prove that if $F$ is noncompact, the algebra $S^{-1}K_N(F)$ is a symmetric Frobenius algebra. Along the way we prove $K(F)$ is finitely generated, $K_N(F)$ is a finite rank module over $"},"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":"1501.02631","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-01-12T13:02:00Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"02b5876ac6df853670fe7a39c6ec2f0b976fab19606ea78a7f7eb41e5795659e","abstract_canon_sha256":"cfb6b82d5c111c88e805ef16d8340c6243c73204e83f880006dd9dbfe62df593"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:00.646280Z","signature_b64":"L4Eio18S6CwdoOOSQ2IL+Q0BIuPE4zTx4xD1MpMYnxHi/zms9ESUqkhY+lCd3UgE4E+w+y4iswNbTdzl/HpBAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e2859aee510a1b132c8f1d87292e894f537b86598b29d2b5bd2b32ae43d180a7","last_reissued_at":"2026-05-18T00:21:00.645782Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:00.645782Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Localized Skein Algebra is Frobenius","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.GT","authors_text":"Charles Frohman, Nel Abdiel","submitted_at":"2015-01-12T13:02:00Z","abstract_excerpt":"When $A$ in the Kauffman bracket skein relation is a primitive $2N$th root of unity, where $N\\geq 3$ is odd, the Kauffman bracket skein algebra $K_N(F)$ of a finite type surface $F$ is a ring extension of the $SL_2\\mathbb{C}$-characters $\\chi(F)$ of the fundamental group of $F$. We localize by inverting the nonzero characters to get an algebra $S^{-1}K_N(F)$ over the function field of the character variety. We prove that if $F$ is noncompact, the algebra $S^{-1}K_N(F)$ is a symmetric Frobenius algebra. 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