JournalsjemsVol. 18, No. 11pp. 2627–2689

Legendrian knots and exact Lagrangian cobordisms

  • Tobias Ekholm

    Uppsala Universitet, Sweden
  • Ko Honda

    University of Southern California, Los Angeles, USA
  • Tamás Kálmán

    Tokyo Institute of Technology, Japan
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Abstract

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)(X,L) consisting of an exact symplectic manifold XX and an exact Lagrangian cobordism LXL\subset X which agrees with cylinders over Legendrian links Λ+\Lambda_+ and Λ\Lambda_- at the positive and negative ends induces a differential graded algebra (DGA) map from the Legendrian contact homology DGA of Λ+\Lambda_+ to that of Λ\Lambda_-. We give a gradient flow tree description of the DGA maps for certain pairs (X,L)(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.

Cite this article

Tobias Ekholm, Ko Honda, Tamás Kálmán, Legendrian knots and exact Lagrangian cobordisms. J. Eur. Math. Soc. 18 (2016), no. 11, pp. 2627–2689

DOI 10.4171/JEMS/650