Full Record

Author | Pan, YU |

Title | Augmentations and exact Lagrangian cobordisms |

URL | http://hdl.handle.net/10161/14398 |

Publication Date | 2017 |

Date Accessioned | 2017-05-16 17:27:35 |

University/Publisher | Duke University |

Abstract | To a Legendrian knot, one can associate an $A_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials. As a related project, we study exact Lagrangian fillings of Legendrian $(2,n)$ links. For a Legendrian $(2,n)$ torus knot or link with maximal Thurston – Bennequin number, Ekholm, Honda, and K{\'a}lm{\'a}n constructed $C_n$ exact Lagrangian fillings, where $C_n$ is the $n$ – th Catalan number. We show that these exact Lagrangian fillings are pairwise non – isotopic through exact Lagrangian isotopy. To do that, we compute the augmentations induced by the exact Lagrangian fillings $L$ to $\mathbbZ_2[H_1(L)]$ and distinguish the resulting augmentations. |

Subjects/Keywords | Mathematics; Augmentations; Contact Topology; Lagrangian cobordisms; Lengendrian knots |

Contributors | Ng, Lenhard (advisor) |

Country of Publication | us |

Record ID | handle:10161/14398 |

Repository | duke |

Date Retrieved | 2020-04-23 |

Date Indexed | 2020-04-26 |

Issued Date | 2017-01-01 00:00:00 |

Sample Search Hits | Sample Images | Cited Works

…cobordism. . . . . . . . . . . . . . . . . . . . . . . . . .
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2.10 The relation among *cobordisms* Σ` , Σ´ , and Σ. . . . . . . . . . . . .
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A neighborhood of the base point ˚ on Λ. The arrow indicates the…

…*cobordisms* in R ˆ R3 , dpet αq . . . . . . . . . . .
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A sketch of the J-holomorphic disks in the differential d. Here a˘ are
Reeb chords to Λ1˘ from Λ2˘ , respectively, and x, x1 , x2 are double
points of Σ1 Y Σ2 . In these examples, p´ is a word of one…

…exact Lagrangian condition and consider topological *cobordisms*,
this question is easy. Any pair of knots can be connected by a cobordism. Because
of the additional geometric structure exact Lagrangian cobordism has, the question above is hard to answer…

…Lagrangian cobordism between them. In this way, we can give
obstructions to the existence of exact Lagrangian *cobordisms*.
Several obstructions have been made. Chantraine [Cha10] first gave an obstruction in terms of the Thurston–Bennequin number…

…obstruction is given by Cornwell, Ng and Sivek [CNS16] based on a key
property of exact Lagrangian *cobordisms* from the work of [EHK16]. Chekanov–
Eliashberg differential graded algebra (DGA) is one of the most useful structural…

…from the category whose objects
are Legendrian knots and morphisms are exact Lagrangian *cobordisms* to a category
whose objects are DGAs and morphisms are DGA maps.
CpLegendrian knots, Exact Lagrangian cobordismsq Ñ CpDGAs, DGA mapsq
When Λ´ is empty, Σ…

…cobordism from Λ´ to Λ` and Λ´ has an augmentation ´ . We can
compose the DGA map DGApΛ` q Ñ DGApΛ´ q induced by Σ and ´ : DGApΛ´ q Ñ
pF, 0q to get an augmentation ` of Λ` . This gives an obstruction of the existence
of exact Lagrangian *cobordisms* as follows…

…wrapped Floer homology of the 2–copy of Σ. The wrapped Floer homology for
Lagrangian *cobordisms* was recently introduced by Chantraine, Dimitroglou Rizell,
Ghiggini and Golovko [CDRGG15] in the spirit of Symplectic Field Theory. They
associated to…