  Theory is developed for linear-quadratic at infinity generating families for
Legendrian knots in R^3. It is shown that the unknot with maximal
Thurston--Bennequin invariant of -1 has a unique linear-quadratic at infinity
generating family, up to fiber-preserving diffeomorphism and stabilization.
  From this, invariant generating family polynomials are constructed for
2-component Legendrian links where each component is a maximal unknot.
Techniques are developed to compute these polynomials, and computations are
done for two families of Legendrian links: rational links and twist links. The
polynomials allow one to show that some topologically equivalent links with the
same classical invariants are not Legendrian equivalent. It is also shown that
for these families of links the generating family polynomials agree with the
polynomials arising from a linearization of the differential graded algebra
associated to the links.
