Legendrian skein algebras and Hall algebras

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.

arXiv commentv2: minor improvements

References in corpus (9)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Flat surfaces and stability structures
• Higher Segal spaces I
• Legendrian knots and constructible sheaves
• Triangulated surfaces in triangulated categories
• The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra
• Homological mirror symmetry for open Riemann surfaces from pair-of-pants decompositions
• Satellites of Legendrian knots and representations of the Chekanov-Eliashberg algebra
• The cardinality of the augmentation category of a Legendrian link

Cited by in corpus (2)

• Exact structures and degeneration of Hall algebras
• Flags and tangles