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Augmentations and exact Lagrangian cobordisms

 dc.contributor.advisor Ng, Lenhard dc.contributor.author Pan, YU dc.date.accessioned 2017-05-16T17:27:35Z dc.date.available 2017-05-16T17:27:35Z dc.date.issued 2017 dc.identifier.uri https://hdl.handle.net/10161/14398 dc.description.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.

dc.subject Mathematics dc.subject Augmentations dc.subject Contact Topology dc.subject Lagrangian cobordisms dc.subject Lengendrian knots dc.title Augmentations and exact Lagrangian cobordisms dc.type Dissertation dc.department Mathematics
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