  Differential graded algebra invariants are constructed for Legendrian links
in the 1-jet space of the circle. In parallel to the theory for R^3,
Poincare-Chekanov polynomials and characteristic algebras can be associated to
such links. The theory is applied to distinguish various knots, as well as
links that are closures of Legendrian versions of rational tangles. For a large
number of two-component links, the Poincare-Chekanov polynomials agree with the
polynomials defined through the theory of generating functions. Examples are
given of knots and links which differ by an even number of horizontal flypes
that have the same polynomials but distinct characteristic algebras. Results
obtainable from a Legendrian satellite construction are compared to results
obtainable from the DGA and generating function techniques.
