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The lexicographic degree of the first two-bridge knots
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 4, pp. 761-793.

Nous étudions les degrés des représentations polynomiales des nœuds. Nous donnons en particulier le degré lexicographique des nœuds à deux ponts à moins de 11 croisements. Nous estimons d’abord le degré total d’une paramétrisation polynomiale de degré lexicographique. Celà nous permet de nous ramener à un problème d’étude de courbes algébriques planes trigonales, et en particulier d’utiliser la méthode des tresses développée par Orevkov.

We study the degree of polynomial representations of knots. We give the lexicographic degree of all two-bridge knots with 11 or fewer crossings. First, we estimate the total degree of a lexicographic parametrisation of such a knot. This allows us to transform this problem into a study of real algebraic trigonal plane curves, and in particular to use the braid theoretical method developed by Orevkov.

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DOI : https://doi.org/10.5802/afst.1645
Classification : 14H50,  57M25,  11A55,  14P99
Mots clés : Real pseudoholomorphic curves, polynomial knots, two-bridge knots, Chebyshev curves
@article{AFST_2020_6_29_4_761_0,
     author = {Erwan Brugall\'e and Pierre-Vincent Koseleff and Daniel Pecker},
     title = {The lexicographic degree of the first two-bridge knots},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {761--793},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 29},
     number = {4},
     year = {2020},
     doi = {10.5802/afst.1645},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1645/}
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Erwan Brugallé; Pierre-Vincent Koseleff; Daniel Pecker. The lexicographic degree of the first two-bridge knots. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 4, pp. 761-793. doi : 10.5802/afst.1645. https://afst.centre-mersenne.org/articles/10.5802/afst.1645/

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