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Title Augmentations and exact Lagrangian cobordisms
Publication Date
Date Accessioned
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
Date Indexed 2020-04-26
Issued Date 2017-01-01 00:00:00

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…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…

…a pair of exact Lagrangian cobordisms a chain complex, called the Cthulhu chain complex, whose differential counts holomorphic disks with boundary on the union of cobordisms (see Chapter 4). We construct a special pair of exact Lagrangian

…Isomorphism [Sei08, DR16]. This gives a strong obstruction to the existence of exact Lagrangian cobordisms that recovers Chantraine’s obstruction ` ˘ the relation (1.1) . Theorem 1.1.3 ([Pan16a]). If there is an exact…