Differential graded algebra invariants are constructed for
Legendrian links in the $1$-jet space of the circle. In parallel to
the theory for $\R^3$, Poincar\'e--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 Poincar\'e--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.