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Title: Augmentations and immersed Lagrangian fillings

For a Legendrian link with or , immersed exact Lagrangian fillings of can be lifted to conical Legendrian fillings of . When is embedded, using the version of functoriality for Legendrian contact homology (LCH) from Pan and Rutherford [J. Symplectic Geom.19(2021), no. 3, 635–722], for each augmentation of the LCH algebra of , there is an induced augmentation . With fixed, the set of homotopy classes of all such induced augmentations, , is a Legendrian isotopy invariant of . We establish methods to compute based on the correspondence between MCFs and augmentations. This includes developing a functoriality for the cellular differential graded algebra from Rutherford and Sullivan [Adv. Math.374(2020), 107348, 71 pp.] with respect to Legendrian cobordisms, and proving its equivalence to the functoriality for LCH. For arbitrary , we give examples of Legendrian torus knots with distinct conical Legendrian fillings distinguished by their induced augmentation sets. We prove that when and ,every‐graded augmentation of can be induced in this manner by an immersed Lagrangian filling. Alternatively, this is viewed as a computation of cobordism classes for an appropriate notion of ‐graded augmented Legendrian cobordism.

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Author(s) / Creator(s):
Publisher / Repository:
Oxford University Press (OUP)
Date Published:
Journal Name:
Journal of Topology
Page Range / eLocation ID:
p. 368-429
Medium: X
Sponsoring Org:
National Science Foundation
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