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Register a new userAugmentations, annuli, and Alexander polynomials
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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.
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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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