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Nonfillable Legendrian knots in the $3$–sphere

Tolga Etgü

Algebraic & Geometric Topology 18 (2018) 1077–1088

If Λ is a Legendrian knot in the standard contact 3–sphere that bounds an orientable exact Lagrangian surface Σ in the standard symplectic 4–ball, then the genus of Σ is equal to the slice genus of (the smooth knot underlying) Λ, the rotation number of Λ is zero as well as the sum of the Thurston–Bennequin number of Λ and the Euler characteristic of Σ, and moreover, the linearized contact homology of Λ with respect to the augmentation induced by Σ is isomorphic to the (singular) homology of Σ. 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.

Legendrian knot, Lagrangian filling, augmentation, linearized contact homology
Mathematical Subject Classification 2010
Primary: 57R17
Secondary: 16E45, 53D42
Received: 19 April 2017
Revised: 29 August 2017
Accepted: 10 September 2017
Published: 12 March 2018
Tolga Etgü
Department of Mathematics
Koç University