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From the obtained presentations, we show that the Kauffman bracket skein modules of $\\Sigma_{0,1}((k_1,1),(k_2,1))$ are free with infinitely many generators when $k_1,k_2\\ge 1$ and that of $\\Sigma_{0,0}((k_1,1),(k_2,1),(k_3,1))$ are finitely generated when $k_1,k_2,k_3 \\ge 2$. We also show that the empty link in either case is not trivial."},"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":"2409.09438","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2024-09-14T13:39:30Z","cross_cats_sorted":[],"title_canon_sha256":"0616b4574c05a9a3ac2ee67e079793905c1e6f9f51d1da8853cf8489c0880775","abstract_canon_sha256":"60130665968e394f9be1839f8bb53c2d49f678771881d28f7f56209539a7f094"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:00:39.072091Z","signature_b64":"t5KUE4CdI92LRv5x3XBvTO8u5uXqqMQ2N3/v+lSzy2s+yQfkcCXrxc2+0ecCKNyiDOtN/Jfq7ezjn7udj/CbAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e362333ca399c153a74090dc3236a7e78aa25b5ff89a5a9397368822b8a267e0","last_reissued_at":"2026-07-05T11:00:39.071610Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:00:39.071610Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Kauffman bracket skein module of two families of Seifert manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Minyi Liang, Shangjun Shi, Xiao Wang","submitted_at":"2024-09-14T13:39:30Z","abstract_excerpt":"We compute the Kauffman bracket skein modules of Seifert manifolds $\\Sigma_{0,1}((k_1,1),(k_2,1))$ and $\\Sigma_{0,0}((k_1,1),(k_2,1),(k_3,1))$ by providing presentations of them. From the obtained presentations, we show that the Kauffman bracket skein modules of $\\Sigma_{0,1}((k_1,1),(k_2,1))$ are free with infinitely many generators when $k_1,k_2\\ge 1$ and that of $\\Sigma_{0,0}((k_1,1),(k_2,1),(k_3,1))$ are finitely generated when $k_1,k_2,k_3 \\ge 2$. 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