Koseleff, P.-V.; Pecker, D.; Rouillier, F.; Tran, C. Computing Chebyshev knot diagrams. (English) Zbl 1383.57010 J. Symb. Comput. 86, 120-141 (2018). Summary: A Chebyshev curve \(C(a,b,c,\varphi)\) has a parametrization of the form \(x(t)=T_a(t)\); \(y(t)=T_b(t)\); \(z(t)=T_c(t+\varphi)\), where \(a\),\(b\),\(c\) are integers, \(T_n(t)\) is the Chebyshev polynomial of degree \(n\) and \(\varphi\in\mathbb{R}\). When \(C(a,b,c,\varphi)\) is nonsingular, it defines a polynomial knot. We determine all possible knot diagrams when \(\varphi\) varies. When \(a\),\(b\),\(c\) are integers, \((a,b)=1\), we show that one can list all possible knots \(C(a,b,c,\varphi)\) in \(\tilde{\mathcal{O}}(n^2)\) bit operations, with \(n=abc\). We give the parameterizations of minimal degree for all two-bridge knots with 10 crossings and fewer. Cited in 1 Document MSC: 57M25 Knots and links in the \(3\)-sphere (MSC2010) 68W30 Symbolic computation and algebraic computation 14H50 Plane and space curves Keywords:zero dimensional systems; Chebyshev curves; Lissajous curves; Lissajous knots; polynomial knots; Chebyshev polynomials; minimal polynomial; Chebyshev forms Software:ISOLATE PDFBibTeX XMLCite \textit{P. V. Koseleff} et al., J. Symb. Comput. 86, 120--141 (2018; Zbl 1383.57010) Full Text: DOI arXiv References: [1] Bogle, M. G.V.; Hearst, J. E.; Jones, V. F. R.; Stoilov, L., Lissajous knots, J. Knot Theory Ramif., 3, 2, 121-140 (1994) · Zbl 0834.57005 [2] Boocher, A.; Daigle, J.; Hoste, J.; Zheng, W., Sampling Lissajous and Fourier knots, Exp. Math., 18, 4, 481-497 (2009) · Zbl 1180.57007 [3] Brent, R., Multipleprecision zero-finding methods and the complexity of elementary function evaluation, (Traub, J. F., Analytic Computational Complexity (1975), Academic Press: Academic Press New York), 151-176 [4] Brent, R., Fast multiple precision evaluation of elementary functions, J. ACM, 23, 2, 242-251 (1976) · Zbl 0324.65018 [5] Brugallé, E.; Koseleff, P. V.; Pecker, D., On the lexicographic degree of two-bridge knots, J. Knot Theory Ramif., 26 (2016), 17 p · Zbl 1344.14022 [6] Cohen, M.; Krishnan, S. R., Random knots using Chebyshev billiard table diagrams, Topol. Appl., 194, 4-21 (2015) · Zbl 1328.57005 [7] Conway, J. H.; Jones, A. J., Trigonometric Diophantine equations (on vanishing sums of roots of unity), Acta Arith., 30, 3, 229-240 (1976) · Zbl 0349.10014 [8] Dimca, A.; Sticlaru, G., Chebyshev curves, free resolutions and rational curve arrangements, Math. Proc. Camb. Philos. Soc., 153, 3, 385-397 (2012) · Zbl 1253.14032 [9] Fischer, G., Plane Algebraic Curves, Student Mathematical Library, vol. 15 (June 2001), American Mathematical Society [10] Koseleff, P.-V.; Pecker, D., Chebyshev knots, J. Knot Theory Ramif., 20, 4, 575-593 (2011) · Zbl 1218.57009 [11] Koseleff, P.-V.; Pecker, D.; Rouillier, F., The first rational Chebyshev knots, Mega Conference, Barcelona. Mega Conference, Barcelona, J. Symb. Comput., 45, 12, 1341-1358 (2010) · Zbl 1202.57007 [12] Koseleff, P.-V.; Rouillier, F.; Tran, C., On the sign of a trigonometric expression, (ISSAC ’15: Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation (2015), ACM: ACM New York, NY, USA), 259-266 · Zbl 1345.68289 [13] Lissajous, J. A., Sur l’étude optique des mouvements vibratoires, Ann. Chim. Phys., LI (1857) [14] Mehlhorn, K.; Sagraloff, M., Computing real roots of real polynomials, J. Symb. Comput., 73, 46-86 (2016) · Zbl 1330.65072 [15] Murasugi, K., Knot Theory and Its Applications (2007), Birkhäuser [16] Myerson, G., Rational products of sines of rational angles, Aequ. Math., 1, 70-82 (1993) · Zbl 0769.11019 [17] Rouillier, F., Solving zero-dimensional systems through the rational univariate representation, Appl. Algebra Eng. Commun. Comput., 9, 5, 433-461 (1999) · Zbl 0932.12008 [18] Rouillier, F.; Zimmermann, P., Efficient isolation of polynomial real roots, J. Comput. Appl. Math., 162, 1, 33-50 (2003) · Zbl 1040.65041 [19] Soret, M.; Ville, M., Lissajous and Fourier knots, J. Knot Theory Ramif., 26 (2016), 27 p · Zbl 1337.57033 [20] Tran, C., Calcul formel dans la base des polynômes unitaires de Chebyshev (2015), Université Pierre et Marie Curie, UPMC, PhD thesis [21] Vassiliev, V. A., Cohomology of knot spaces, (Theory of Singularities and Its Applications. Theory of Singularities and Its Applications, Advances in Soviet Mathematics, vol. 1 (1990)) · Zbl 1015.57003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.