{"paper":{"title":"Computing Chebyshev knot diagrams","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.SC","authors_text":"C Tran (UPMC, D Pecker (IMJ-PRG, Fabrice Rouillier (OURAGAN, IMJ-PRG, IMJ-PRG), P.-V Koseleff (OURAGAN, UPMC)","submitted_at":"2015-12-24T09:23:25Z","abstract_excerpt":"A Chebyshev curve $\\mathcal{C}(a,b,c,\\phi)$ has a parametrization of the form$ x(t)=T\\_a(t)$; \\ $y(t)=T\\_b(t)$; $z(t)= T\\_c(t + \\phi)$, where $a,b,c$are integers, $T\\_n(t)$ is the Chebyshev polynomialof degree $n$ and $\\phi \\in \\mathbb{R}$. When $\\mathcal{C}(a,b,c,\\phi)$ is nonsingular,it defines a polynomial knot. We determine all possible knot diagrams when $\\phi$ varies. Let $a,b,c$ be integers, $a$ is odd, $(a,b)=1$, we show that one can list all possible knots $\\mathcal{C}(a,b,c,\\phi)$ in$\\tilde{\\mathcal{O}}(n^2)$ bit operations, with $n=abc$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.07766","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"}