Representations, sheaves, and Legendrian $(2,m)$ torus links

TL;DR

This paper explores a new category associated with Legendrian links in 3D space, conjectures its equivalence to a sheaf category, and proves this for certain torus links, advancing the understanding of Legendrian invariants.

Contribution

It introduces a generalized representation category for Legendrian links and proves its equivalence to a sheaf category for specific torus links.

Findings

Established the cohomological equivalence for Legendrian (2,m) torus links.

Generalized the positive augmentation category to an $A_ abla$ category.

Conjectured an equivalence with a sheaf category of microlocal rank n.

Abstract

We study an $A_\infty$ category associated to Legendrian links in $\mathbb{R}^3$ whose objects are $n$-dimensional representations of the Chekanov-Eliashberg differential graded algebra of the link. This representation category generalizes the positive augmentation category and we conjecture that it is equivalent to a category of sheaves of microlocal rank $n$ constructed by Shende, Treumann, and Zaslow. We establish the cohomological version of this conjecture for a family of Legendrian $(2,m)$ torus links.