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research-article

The first rational Chebyshev knots

Authors:

P.-V. Koseleff,

F. RouillierAuthors Info & Claims

Journal of Symbolic Computation, Volume 45, Issue 12

Pages 1341 - 1358

https://doi.org/10.1016/j.jsc.2010.06.014

Published: 01 December 2010 Publication History

Abstract

A Chebyshev knot C(a,b,c, ) is a knot which has a parametrization of the form x(t)=Ta(t);y(t)=Tb(t);z(t)=Tc(t+ ), where a,b,c are integers, Tn(t) is the Chebyshev polynomial of degree n and R. We show that any rational 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 C(a,b,c, ). We deduce the list of minimal Chebyshev representations of rational knots with 10 or fewer crossings.

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Cited By

Koseleff PPecker DRouillier FTran C(2018)Computing Chebyshev knot diagramsJournal of Symbolic Computation10.1016/j.jsc.2017.04.00186:C(120-141)Online publication date: 1-May-2018
https://dl.acm.org/doi/10.1016/j.jsc.2017.04.001
Koseleff PRouillier FTran CRobertz DLinton SYokoyama K(2015)On the Sign of a Trigonometric ExpressionProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756664(259-266)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756664

The first rational Chebyshev knots
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis

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Published In

cover image Journal of Symbolic Computation

Journal of Symbolic Computation Volume 45, Issue 12

December, 2010

221 pages

ISSN:0747-7171

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Publisher

Academic Press, Inc.

United States

Publication History

Published: 01 December 2010

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Cited By

Koseleff PPecker DRouillier FTran C(2018)Computing Chebyshev knot diagramsJournal of Symbolic Computation10.1016/j.jsc.2017.04.00186:C(120-141)Online publication date: 1-May-2018
https://dl.acm.org/doi/10.1016/j.jsc.2017.04.001
Koseleff PRouillier FTran CRobertz DLinton SYokoyama K(2015)On the Sign of a Trigonometric ExpressionProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756664(259-266)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756664

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