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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@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},
zbl = {1344.14022},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1645/}
}
TY - JOUR AU - Erwan Brugallé AU - Pierre-Vincent Koseleff AU - Daniel Pecker TI - The lexicographic degree of the first two-bridge knots JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2020 SP - 761 EP - 793 VL - 29 IS - 4 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1645/ DO - 10.5802/afst.1645 LA - en ID - AFST_2020_6_29_4_761_0 ER -
%0 Journal Article %A Erwan Brugallé %A Pierre-Vincent Koseleff %A Daniel Pecker %T The lexicographic degree of the first two-bridge knots %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2020 %P 761-793 %V 29 %N 4 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1645/ %R 10.5802/afst.1645 %G en %F AFST_2020_6_29_4_761_0
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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