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As an application, we give a family $\\mathcal K^3$ of alternating knots, including $(2,n)$ torus knots, 2-bridge knots and alternating pretzel knots as its subfamilies, such that the minimal crossing number of any alternating knot in $\\mathcal"},"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":"1106.1259","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-06-07T05:46:43Z","cross_cats_sorted":[],"title_canon_sha256":"2bc5685a8ad921cd31c3b2851d10fd04fdaa3dee222d9f17441fe731bb469d13","abstract_canon_sha256":"ecf5be7511fb3225d646027b7612eec8d69b39560e82a21be006e46f224006eb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:20:23.003361Z","signature_b64":"0x5cT1JybuF3CoTSD4y85vzbX9FIuqP9a3Iwi3ZyJstcjy7NEwJ/mRhrNU7BsXobYILiKeXxY9CyXSbvW2TXBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ec86164c2cbd40c9f746c74971f6f9a19af7239e0ee77b94feb5e1973b1c367","last_reissued_at":"2026-05-18T04:20:23.002725Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:20:23.002725Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The canonical genus for Whitehead doubles of a family of alternating knots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Hee Jeong Jang, Sang Youl Lee","submitted_at":"2011-06-07T05:46:43Z","abstract_excerpt":"For any given integer $r \\geq 1$ and a quasitoric braid $\\beta_r=(\\sigma_r^{-\\epsilon} \\sigma_{r-1}^{\\epsilon}...$ $ \\sigma_{1}^{(-1)^{r}\\epsilon})^3$ with $\\epsilon=\\pm 1$, we prove that the maximum degree in $z$ of the HOMFLYPT polynomial $P_{W_2(\\hat\\beta_r)}(v,z)$ of the doubled link $W_2(\\hat\\beta_r)$ of the closure $\\hat\\beta_r$ is equal to $6r-1$. 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