Augmentations and exact Lagrangian cobordisms

Abstract

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