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×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.