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Given an augmentation for a Legendrian surface in a $1$-jet space, $Lambda subset J^1(M)$, we explicitly construct an object, $mathcal{F} in Sh_{Lambda}$, of the (derived) category from arXiv:1402.0490 of constructible sheaves on $Mtimes R$ with singular support determined by $Lambda$. In the construction, we introduce a simplicial Legendrian DGA (differential graded algebra) for Legendrian submanifolds in $1$-jet spaces that, based on arXiv:1608.02984 and arXiv:1608.03011, is equivalent to the Legendrian contact homology DGA in the case of Legendrian surfaces. In addition, we extend the approach of arXiv:1402.0490 for $1$-dimensional Legendrian knots to obtain a combinatorial model for sheaves in $Sh_Lambda$ in the $2$-dimensional case.
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The augmentation variety of a knot is the locus, in the 3-dimensional coefficient space of the knot contact homology dg-algebra, where the algebra admits a unital chain map to the complex numbers. We explain how to express the Alexander polynomial of a knot in terms of the augmentation variety: it is the exponential of the integral of a ratio of two partial derivatives. The expression is derived from a description of the Alexander polynomial as a count of Floer strips and holomorphic annuli, in the cotangent bundle of Euclidean 3-space, stretching between a Lagrangian with the topology of the knot complement and the zero-section, and from a description of the boundary of the moduli space of such annuli with one positive puncture.
In this article, we prove a Legendrian Whitney trick which allows for the removal of intersections between codimension-two contact submanifolds and Legendrian submanifolds, assuming such a smooth cancellation is possible. This technique is applied to show the existence h-principle for codimension-two contact embeddings with a prescribed contact structure.
Let $X$ be a Weinstein manifold with ideal contact boundary $Y$. If $Lambdasubset Y$ is a link of Legendrian spheres in $Y$ then by attaching Weinstein handles to $X$ along $Lambda$ we get a Weinstein cobordism $X_{Lambda}$ with a collection of Lagrangian co-core disks $C$ corresponding to $Lambda$. In cite{BEE, EL} it was shown that the wrapped Floer cohomology $CW^{ast}(C)$ of $C$ in the Weinstein manifold $X_{Lambda}=Xcup X_{Lambda}$is naturally isomorphic to the Legendrian differential graded algebra $CE^{ast}(Lambda)$ of $Lambda$ in $Y$. The argument uses properties of moduli spaces of holomorphic curves, the proofs of which were only sketched. The purpose of this paper is to provide proofs of these properties.
Sivek proves a van Kampen decomposition theorem for the combinatorial Legendrian contact algebra (also known as the Chekanov-Eliashberg algebra) of knots in standard contact $R^3$ . We prove an analogous result for the holomorphic curve version of the Legendrian contact algebra of certain Legendrians submanifolds in standard contact $J^1(M).$ This includes all 1- and 2-dimensional Legendrians, and some higher dimensional ones. We present various applications including a Mayer-Vietoris sequence for linearized contact homology similar to Siveks and a connect sum formula for the augmentation variety introduced by Ng. The main tool is the theory of gradient flow trees developed by Ekholm.
The Thurston-Bennequin invariant provides one notion of self-linking for any homologically-trivial Legendrian curve in a contact three-manifold. Here we discuss related analytic notions of self-linking for Legendrian knots in Euclidean space. Our definition is based upon a reformulation of the elementary Gauss linking integral and is motivated by ideas from supersymmetric gauge theory. We recover the Thurston-Bennequin invariant as a special case.
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