Non-fillable Augmentations of Twist Knots. (11th October 2021)

• Record Type: Journal Article
• Authors: Gao, Honghao; Rutherford, Dan
• Is Part Of: International mathematics research notices. Volume 2023:Number 2(2023)
• Page Start: 1255
• Page End: 1291
• Publication Date: 2021-10-11
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnab224
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: We establish new examples of augmentations of Legendrian twist knots that cannot be induced by orientable Lagrangian fillings. To do so, we use a version of the Seidel –Ekholm–Dimitroglou Rizell isomorphism with local coefficients to show that any Lagrangian filling point in the augmentation variety of a Legendrian knot must lie in the injective image of an algebraic torus with dimension equal to the 1st Betti number of the filling. This is a Floer-theoretic version of a result from microlocal sheaf theory. For the augmentations in question, we show that no such algebraic torus can exist.