Isomorphism in the augmentation category

2026/02/13 by Honghao Gao, Hanming Liu · 1 voice
Mathematics · #math.SG

paper · pdf

Abstract

Given a Legendrian submanifold in any dimension, we prove that two augmentations are isomorphic within the positive augmentation category exactly when they differ by a combination of a dga homotopy and a dilation. This extends the corresponding statement for Legendrian knots and links, but instead of relying on the dga for consistent copies, we make use of quantum flow tree techniques. Consequently, we can strengthen and clarify a result of the first author as follows: for knot contact homology, the augmentation category is not in general equivalent to the microlocal rank 1 sheaf category.

Citations

• Augmentation categories in higher dimensions
• Obstructions to reversing Lagrangian surgery in Lagrangian fillings
• Augmentations and sheaves for links
• Augmentations are Sheaves
• Bilinearised Legendrian contact homology and the augmentation category
• A Duality Exact Sequence for Legendrian Contact Homology
• Legendrian Contact Homology in P X R
• Framed knot contact homology
• Knot and braid invariants from contact homology I
• Knot and braid invariants from contact homology II, with an appendix written jointly with Siddhartha Gadgil

Discussions

• Honghao Gao, Hanming Liu: Isomorphism in the augmentation category https://arxiv.org/abs/2602.12555 https://arxiv.org/pdf/2602.12555 https://arxiv.org/html/2602.12555 [bsky, 0 points, 0 comments]

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• Augmentations and Rulings of Legendrian Knots
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• Decompositions of augmentation varieties via weaves and rulings