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$A_\infty$ Sabloff Duality via the LSFT Algebra

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arxiv 2410.20523 v2 pith:SL3Q3JXC submitted 2024-10-27 math.SG math.GT

classification math.SGmath.GT
keywords mathcalinftyalgebralsftsabloffbimodulescategoryduality
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abstract

We use Ng's LSFT algebra to upgrade Sabloff duality of Legendrian knots to a quasi-isomorphism of $A_\infty$ bimodules over the positive augmentation category $\mathcal{A}ug_+$. We also extend the Ekholm-Etnyre-Sabloff exact sequence to an exact sequence of $\mathcal{A}ug_+$-bimodules, using a quotient category $\mathcal{C}$ of short Reeb chords. In addition, we define curved augmentations of the LSFT algebra and show that they can be used to construct a homotopy inverse of the $A_\infty$ Sabloff map, together with all higher homotopies. The above results suggest a conjectural recipe for an explicit weak relative Calabi-Yau structure on the quotient $A_\infty$ functor $\pi:\mathcal{A}ug_+\to \mathcal{C}$.

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  1. Weak Relative Calabi-Yau Structures for Legendrian Contact Homology

    math.SG 2025-09 conditional novelty 5.0 of 10

    For Legendrian knots in standard contact R3, the projection from the simply perturbed positive augmentation category to the circle category carries a weak right relative Calabi-Yau structure of dimension 2.

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