Legendrian knots and exact Lagrangian cobordisms

By T. Ekholm, K. Honda, Tam'as K'alm'an. 2012

174 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e8a21a3225ec3b564a404ffc655fcbd075c83774

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The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Lenhard L. Ng
2. Georgios Dimitroglou Rizell
3. Yankı Lekili
4. Roger Casals
5. Dan Rutherford
6. T. Ekholm
7. Steven Sivek
8. Vivek V. Shende

In Geometry and Topology

1. A. Oancea
2. T. Ekholm
3. Georgios Dimitroglou Rizell
4. Kai Cieliebak
5. Lenhard L. Ng
6. Vivek V. Shende
7. Roger Casals
8. M. Abouzaid

In Mathematical Physics

1. A. Oancea
2. T. Ekholm
3. Georgios Dimitroglou Rizell
4. Kai Cieliebak
5. Lenhard L. Ng
6. Vivek V. Shende
7. Yankı Lekili
8. Frederick J. Bourgeois III

Citations in the corpus

• Duality between Lagrangian and Legendrian invariants (2017, background): “A well known argument [29] shows that that CW∗(L) with differential m1 is quasiisomorphic to the wrapped Floer cohomology by a geometrically defined chain map.”
• Symplectic homology and the Eilenberg-Steenrod axioms (2015, background): “The Lagrangian symplectic homology group SH ∗ (L) is isomorphic to the linearized Legendrian contact homology group of ∂L [36, 39].”
• Infinitely many Lagrangian fillings (2020, background): “The classification of Legendrian knots and their Lagrangian fillings has been one of the central areas of research in low-dimensional contact topology [20, 21, 41, 53, 55, 56, 57].”
• Legendrian ambient surgery and Legendrian contact homology (2012, background): “ll hasthe same expected dimension as the modulispace without any boundarypoint constraints. Remark 1.7. In the case when LS ⊂ J1M is obtained from an ambient surgeryon a Legendrian knot, theresult in [14] instead enables usto compute the DGA morphism ΦV S , and consequently also the Chekanov-Eliashb”
• Lagrangian cobordisms via generating families: Construction and geography (2014, unknown): “We note that Ekholm, Honda, and Kálmán’s construction of Lagrangian cobordisms in [17] (see also Dimitriglou Rizell [12]) would work equally well for these applications.”
• Augmentations are sheaves (2015, methodology): “th trivial Maslov number, and let L 1 ; L 2 be the corresponding augmentations of the DGA of . If L 1;L 2 are isotopic through exact Lagrangian fillings, then L 1 ˘= L 2 in Aug +() . Proof. From [EHK], given these hypotheses, L 1 ; L 2 are DGA homotopic. Remark 5.19. As before, we can generalize C”