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

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.

v3: 50 pages, small changes, to appear in Journal of the London Mathematical Society