Exact Lagrangian Fillings of Legendrian $(2,n)$ torus links

Abstract

For a Legendrian $(2,n)$ torus knot or link with maximal Thurston-Bennequin number, Ekholm, Honda, and Kálmán constructed $C_n$ exact Lagrangian fillings, where $C_n$ is the $n$-th Catalan number. We show that these exact Lagrangian fillings are pairwise non-isotopic through exact Lagrangian isotopy. To do that, we compute the augmentations induced by the exact Lagrangian fillings $L$ to ${\mathbb{Z}}_2[H_1(L)]$ and distinguish the resulting augmentations.

19 pages, 15 figures