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

TL;DR

This paper proves that the $C_n$ exact Lagrangian fillings of Legendrian $(2,n)$ torus links are all distinct up to isotopy by computing and comparing their induced augmentations.

Contribution

It demonstrates that the previously constructed $C_n$ fillings are pairwise non-isotopic by analyzing their augmentations, providing new insights into the topology of these fillings.

Findings

All $C_n$ fillings are pairwise non-isotopic.

Augmentations distinguish different Lagrangian fillings.

Explicit computation of augmentations for each filling.

Abstract

For a Legendrian $(2,n)$ torus knot or link with maximal Thurston-Bennequin number, Ekholm, Honda, and K\'alm\'an 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.