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Journal Articles Journal of Symbolic Computation Year : 2010

The first rational Chebyshev knots

Abstract

A Chebyshev knot ${\cal C}(a,b,c,\phi)$ is a knot which 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 polynomial of degree $n$ and $\phi \in \R.$ We show that any two-bridge knot is a Chebyshev knot with $a=3$ and also with $a=4$. For every $a,b,c$ integers ($a=3, 4$ and $a$, $b$ coprime), we describe an algorithm that gives all Chebyshev knots $\cC(a,b,c,\phi)$. We deduce a list of minimal Chebyshev representations of two-bridge knots with small crossing number.
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Dates and versions

hal-00429510 , version 1 (03-11-2009)

Identifiers

Cite

Pierre-Vincent Koseleff, Daniel Pecker, Fabrice Rouillier. The first rational Chebyshev knots. Journal of Symbolic Computation, 2010, 45 (12), pp.1341-1358. ⟨10.1016/j.jsc.2010.06.014⟩. ⟨hal-00429510⟩
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