Journal of the European Mathematical Society

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Volume 18, Issue 11, 2016, pp. 2627–2689
DOI: 10.4171/JEMS/650

Published online: 2016-10-12

Legendrian knots and exact Lagrangian cobordisms

Tobias Ekholm[1], Ko Honda[2] and Tamás Kálmán[3]

(1) Uppsala Universitet, Sweden
(2) University of Southern California, Los Angeles, USA
(3) Tokyo Institute of Technology, Japan

We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair $(X,L)$ consisting of an exact symplectic manifold $X$ and an exact Lagrangian cobordism $L\subset X$ which agrees with cylinders over Legendrian links $\Lambda_+$ and $\Lambda_-$ at the positive and negative ends induces a differential graded algebra (DGA) map from the Legendrian contact homology DGA of $\Lambda_+$ to that of $\Lambda_-$. We give a gradient flow tree description of the DGA maps for certain pairs $(X,L)$, which in turn yields a purely combinatorial description of the cobordism map for elementary cobordisms, i.e., cobordisms that correspond to certain local modifications of Legendrian knots. As an application, we find exact Lagrangian surfaces that fill a fixed Legendrian link and are not isotopic through exact Lagrangian surfaces.

Keywords: Contact structure, Legendrian knot, exact Lagrangian cobordism, contact homology

Ekholm Tobias, Honda Ko, Kálmán Tamás: Legendrian knots and exact Lagrangian cobordisms. J. Eur. Math. Soc. 18 (2016), 2627-2689. doi: 10.4171/JEMS/650