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2018 Nonfillable Legendrian knots in the $3$–sphere
Tolga Etgü
Algebr. Geom. Topol. 18(2): 1077-1088 (2018). DOI: 10.2140/agt.2018.18.1077

Abstract

If $\Lambda$ is a Legendrian knot in the standard contact 3 –sphere that bounds an orientable exact Lagrangian surface $\Sigma$ in the standard symplectic 4 –ball, then the genus of $\Sigma$ is equal to the slice genus of (the smooth knot underlying) $\Lambda$, the rotation number of $\Lambda$ is zero as well as the sum of the Thurston–Bennequin number of $\Lambda$ and the Euler characteristic of $\Sigma$, and moreover, the linearized contact homology of $\Lambda$ with respect to the augmentation induced by $\Sigma$ is isomorphic to the (singular) homology of $\Sigma$. It was asked by Ekholm, Honda and Kálmán (2016) whether the converse of this statement holds. We give a negative answer, providing a family of Legendrian knots with augmentations which are not induced by any exact Lagrangian filling although the associated linearized contact homology is isomorphic to the homology of the smooth surface of minimal genus in the 4 –ball bounding the knot.

Citation

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Tolga Etgü. "Nonfillable Legendrian knots in the $3$–sphere." Algebr. Geom. Topol. 18 (2) 1077 - 1088, 2018. https://doi.org/10.2140/agt.2018.18.1077

Information

Received: 19 April 2017; Revised: 29 August 2017; Accepted: 10 September 2017; Published: 2018
First available in Project Euclid: 22 March 2018

zbMATH: 06859614
MathSciNet: MR3773748
Digital Object Identifier: 10.2140/agt.2018.18.1077

Subjects:
Primary: 57R17
Secondary: 16E45, 53D42

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.18 • No. 2 • 2018
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