2023/04/07 by Murray, Justin
#53D42 #FOS: Mathematics #Geometric Topology (math.GT) #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2304.03761
Given a Legendrian knot in (ℝ3, 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.