Legendrian skein algebras and Hall algebras

We compare two associative algebras which encode the “quantum topology” of Legendrian curves in contact threefolds of product type $S\times \mathbb{R}$. The first is the skein algebra of graded Legendrian links and the second is the Hall algebra of the Fukaya category of $S$. We construct a natural homomorphism from the former to the latter, which we show is an isomorphism if $S$ is a disk with marked points and injective if $S$ is the annulus.