The homotopy cardinality of the representation category for a Legendrian knot

Abstract

Given a Legendrian knot in $(\mathbb{R}^3, \ker(dz-ydx))$ one can assign a combinatorial invariants called ruling polynomials. These invariants have been shown to recover not only a (normalized) count of augmentations but are also closely related to a categorical count of augmentations in the form of the homotopy cardinality of the augmentation category. In this article, we prove that that the homotopy cardinality of the $n$-dimensional representation category is a multiple of the $n$-colored ruling polynomial. Along the way, we establish that two $n$-dimensional representations are equivalent in the representation category if they are conjugate DGA homotopic. We also provide some applications to Lagrangian concordance.