I’ve posted a new preprint to the ArXiv titled “Legendrian skein algebras and Hall algebras”. From the abstract:
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.
The heart of the paper is a statement about the relation between Maurer-Cartan elements and smoothing of self-intersection points of Legendrian curves, which is actually something I have been thinking about on and off since my days as a grad student and which found a satisfying application here.