Infinitely many Lagrangian fillings

Lagrangian Skeleta and Plane Curve Singularities

“…By Proposition 2.2, A(f ) is a well-defined invariant of the complex topological singularity. For these Legendrian links Λ = Λ f , the Couture-Perron algorithm [29] implies that there exist a Legendrian front π(Λ f ) ⊆ R 2 given by the (−1)-closure of a positive braid β∆ 2 , where ∆ is the full twist; equivalently the front is the rainbow closure of the positive braid β [20]. Hence, there is a set of non-negatively graded Reeb chords generating the DGA A (Λ f ) and ob(Aug + (Λ f )) coincides with the set of k-valued augmentations of A (Λ f ) where exactly one base point per component has been chosen, k a field.…”

“…Thanks to the computational techniques available for augmentation varieties, the moduli of objects ob(Aug + (Λ(f ))) is readily computable for (−1)-framed closures of positive braids as in Section 3 above, confer Computation 3.2. Similarly Θ(f ) could be computed directly, or by means of the isomorphism to the wrapped Fukaya category 20 of W (Λ f ).…”

“…These maps are not necessarily invertible in µsh(L(f n ) 0 ). 20 Should the reader be willing to use the surgery formula, this wrapped Fukaya category may be presented as modules over the Legendrian DGA of Λ f . (This is only informative and not needed for the present purposes.)…”

“…In this case, the standard Lagrangian flat disk is the unique filling: there is precisely one exact Lagrangian filling, up to Hamiltonian isotopy. The recent developments [20,22,23,49] show that such finiteness is actually rare: e.g. the max-tb torus links (n, m) admit infinitely many exact Lagrangian filling, up to Hamiltonian isotopy, if n, m ≥ 4.…”

“…The recent developments [20,49,93,94] and the existence of the Lagrangian skeleta in Theorem 1.1 shyly hint towards the fact that, possibly, Lagrangian fillings are classified by the cluster algebra A(Q(f )). That is, every cluster chart in A(Q(f )) is induced by precisely one exact Lagrangian filling.…”