Satellites of Legendrian knots and representations of the Chekanov–Eliashberg algebra

Abstract

We develop a close relation between satellites of Legendrian knots in ${ℝ}^3$ and the Chekanov–Eliashberg differential graded algebra of the knot. In particular, we generalize the well-known correspondence between rulings of a Legendrian knot in ${ℝ}^3$ and augmentations of its DGA by showing that the DGA has finite-dimensional representations if and only if there exist certain rulings of satellites of the knot. We derive several consequences of this result, notably that the question of existence of ungraded finite-dimensional representations for the DGA of a Legendrian knot depends only on the topological type and Thurston–Bennequin number of the knot.