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

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.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1645
Classification: 14H50, 57M25, 11A55, 14P99
Keywords: Real pseudoholomorphic curves, polynomial knots, two-bridge knots, Chebyshev curves

Erwan Brugallé 1; Pierre-Vincent Koseleff 2; Daniel Pecker 3

1 Université de Nantes, Laboratoire de Mathématiques Jean Leray (CNRS LMJL UMR 6629) (France)
2 Sorbonne Université (UPMC–Paris 6), Institut de Mathématiques de Jussieu (CNRS IMJ-PRG UMR 7586) & Inria-Paris (France)
3 Sorbonne Université (UPMC–Paris 6), Institut de Mathématiques de Jussieu (CNRS IMJ-PRG UMR 7586) (France)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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, Serie 6, Volume 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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