Journal
ANNALS OF MATHEMATICS
Volume 195, Issue 1, Pages 207-249Publisher
Princeton Univ, Dept Mathematics
DOI: 10.4007/annals.2022.195.1.3
Keywords
Legendrian knots; Lagrangian fillings; microlocal sheaves; cluster structures; ping-pong lemma
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We prove that all maximal-tb positive Legendrian torus links (n, m) in the standard contact 3-sphere, except for (2, m), (3, 3), (3, 4) and (3, 5), admit infinitely many Lagrangian fillings in the standard symplectic 4-ball. This is proven by constructing infinite order Lagrangian concordances that induce faithful actions of the modular group PSL(2, Z) and the mapping class group M-0,M-4 into the coordinate rings of algebraic varieties associated to Legendrian links. In particular, our results imply that there exist Lagrangian concordance monoids with subgroups of exponential-growth, and yield Stein surfaces homotopic to a 2-sphere with infinitely many distinct exact Lagrangian surfaces of higher-genus. We also show that there exist infinitely many satellite and hyperbolic knots with Legendrian representatives admitting infinitely many exact Lagrangian fillings.
We prove that all maximal-tb positive Legendrian torus links (n, m) in the standard contact 3-sphere, except for (2, m), (3, 3), (3, 4) and (3, 5), admit infinitely many Lagrangian fillings in the standard symplectic 4-ball. This is proven by constructing infinite order Lagrangian concordances that induce faithful actions of the modular group PSL(2, Z) and the mapping class group M-0,M-4 into the coordinate rings of algebraic varieties associated to Legendrian links. In particular, our results imply that there exist Lagrangian concordance monoids with subgroups of exponential-growth, and yield Stein surfaces homotopic to a 2-sphere with infinitely many distinct exact Lagrangian surfaces of higher-genus. We also show that there exist infinitely many satellite and hyperbolic knots with Legendrian representatives admitting infinitely many exact Lagrangian fillings.
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