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

TL;DR

This paper explores the relationship between satellites of Legendrian knots and the Chekanov-Eliashberg algebra, establishing a link between finite-dimensional representations of the algebra and rulings of satellite knots, with implications for knot invariants.

Contribution

It generalizes the correspondence between rulings and augmentations to finite-dimensional representations of the DGA for Legendrian knots.

Findings

Finite-dimensional representations of the DGA correspond to rulings of satellite knots.

Existence of ungraded finite-dimensional representations depends only on topological type and Thurston-Bennequin number.

Results connect algebraic properties of the DGA with topological features of Legendrian knots.

Abstract

We study satellites of Legendrian knots in R^3 and their relation to 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.