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

We study an A ∞ category associated to Legendrian links in 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.