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 when to the unknot. Those examples are of interest because the Legendrian contact homology algebra cannot be used to obstruct such a concordance.
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 de concordances du noeud pretzel 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.
Revised:
Accepted:
Published online:
Baptiste Chantraine 1; Noémie Legout 2
@article{CRMATH_2023__361_G10_1605_0, 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}, }
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/
[1] Ribbon concordance of knots is a partial ordering, Comm. Amer. Math. Soc., Volume 2, pp. 374-379 | DOI | MR | Zbl
[2] Quasi-positivité d’une courbe analytique dans une boule pseudo-convexe, C. R. Math. Acad. Sci. Paris, Volume 332 (2001) no. 9, pp. 825-830 | DOI | MR | Zbl
[3] Lagrangian concordance is not a symmetric relation, Quantum Topol., Volume 6 (2015) no. 3, pp. 451-474 | DOI | MR | Zbl
[4] Obstructions to Lagrangian concordance, Algebr. Geom. Topol., Volume 16 (2016) no. 2, pp. 797-824 | DOI | MR | Zbl
[5] Local Lagrangian -knots are trivial, Ann. Math., Volume 144 (1996) no. 1, pp. 61-76 | DOI | MR | Zbl
[6] Legendrian contact homology in , Surveys in differential geometry 2020. Surveys in 3-manifold topology and geometry (Surveys in Differential Geometry), Volume 25, International Press, Boston, MA, 2022, pp. 103-161 | MR | Zbl
[7] Smoothly embedding Seifert fibered spaces in , Trans. Am. Math. Soc., Volume 373 (2020) no. 7, pp. 4933-4974 | DOI | MR | Zbl
[8] Rational Witt classes of pretzel knots, Osaka J. Math., Volume 47 (2010) no. 4, pp. 977-1027 | MR | Zbl
[9] Doubly slice knots with low crossing number, New York J. Math., Volume 21 (2015), pp. 1007-1026 | MR | Zbl
[10] Doubly slice odd pretzel knots, Proc. Am. Math. Soc., Volume 148 (2020) no. 12, pp. 5413-5420 | DOI | MR | Zbl
[11] The cardinality of the augmentation category of a Legendrian link, Math. Res. Lett., Volume 24 (2017) no. 6, pp. 1845-1874 | DOI | MR | Zbl
[12] Algebraic functions and closed braids, Topology, Volume 22 (1983), pp. 191-202 | DOI | MR | Zbl
[13] Invertible knot cobordisms, Comment. Math. Helv., Volume 46 (1971), pp. 240-256 | DOI | MR | Zbl
[14] Obstructing Lagrangian concordance for closures of 3-braids (2022) | arXiv
[15] Knot Floer homology obstructs ribbon concordance, Ann. Math., Volume 190 (2019) no. 3, pp. 931-947 | DOI | MR | Zbl
Cited by Sources:
Comments - Policy