Seifert fibered spaces are an important class of three-manifolds which may be viewed as circle bundles over surface orbifolds. We focus on the case when these manifolds admit contact forms whose Reeb orbits realize the Seifert fibers. In this setting, and we construct a combinatorial invariant modeled on Legendrian contact homology for Legendrian knots in these contact manifolds. This construction may be used to prove the existence of an infinite family of non-simple knot types representing torsion homology classes. This is joint work with J. Sabloff.