{"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 $6r-1$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.1259","kind":"arxiv","version":2},"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"}