Legendrian skein algebras and Hall algebras

TL;DR

This paper establishes a deep connection between Legendrian skein algebras and Hall algebras of Fukaya categories, providing isomorphisms in specific cases and advancing the understanding of quantum topology in contact threefolds.

Contribution

It constructs a natural homomorphism between Legendrian skein algebras and Hall algebras, proving it is an isomorphism for disks with marked points and injective for annuli.

Findings

Homomorphism from skein algebra to Hall algebra constructed

Isomorphism established for disks with marked points

Injectivity shown for annulus case

Abstract

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.