Doubly slice knots and obstruction to Lagrangian concordance

Baptiste Chantraine; Noémie Legout

doi:10.5802/crmath.478

Géométrie et Topologie

Doubly slice knots and obstruction to Lagrangian concordance
[Noeuds doublement bordant et obstruction aux concordances lagrangiennes]

Baptiste Chantraine ¹ ; Noémie Legout ²

¹ Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, LMJL, UMR 6629, F-44000 Nantes, France
² Uppsala University, Department of Mathematics, Box 480, 751 06 Uppsala, Sweden

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1605-1609.

Résumé
Abstract

Dans cette note, nous remarquons qu’un résultat d’Eliashberg et Polterovitch permet d’utiliser la notion de nœuds doublement bordant afin d’obstruer la possibilité pour un noeud legendrien d’apparaitre comme une tranche dans une concordance lagrangienne du noeud legendrien trivial d’invariant de Thurston–Bennequin maximal vers lui-même. Cela permet d’obstruer l’existence pour $m \geq 4$ de concordances du noeud pretzel $P (3, - 3, - m)$ vers le noeud trivial. Ces exemples s’avèrent particulièrement intéressants car l’algèbre d’homologie de contact legendrienne ne permet pas d’obstruer une telle concordance.

In this short note we observe that a result of Eliashberg and Polterovitch allows to use the doubly slice genus as an obstruction for a Legendrian knot to be a slice of a Lagrangian concordance from the trivial Legendrian knot with maximal Thurston–Bennequin invariant to itself. This allows to obstruct concordances from the Pretzel knot $P (3, - 3, - m)$ when $m \geq 4$ to the unknot. Those examples are of interest because the Legendrian contact homology algebra cannot be used to obstruct such a concordance.

Reçu le : 2022-07-15
Révisé le : 2023-01-10
Accepté le : 2023-02-16
Publié le : 2023-11-17

DOI : 10.5802/crmath.478

Classification : 57K33, 57K10

Affiliations des auteurs :

Baptiste Chantraine ¹ ; Noémie Legout ²

¹ Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, LMJL, UMR 6629, F-44000 Nantes, France
² Uppsala University, Department of Mathematics, Box 480, 751 06 Uppsala, Sweden

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Baptiste Chantraine and No\'emie Legout},
     title = {Doubly slice knots and obstruction to {Lagrangian} concordance},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1605--1609},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.478},
     language = {en},
}

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Baptiste Chantraine; Noémie Legout. Doubly slice knots and obstruction to Lagrangian concordance. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1605-1609. doi : 10.5802/crmath.478. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.478/

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