Generalized normal rulings and invariants of Legendrian solid torus links

For Legendrian links in the 1-jet space of $S^1$ we show that the 1-graded ruling polynomial may be recovered from the Kauffman skein module. For such links a generalization of the notion of normal ruling is introduced. We show that the existence of such a generalized normal ruling is equivalent to sharpness of the Kauffman polynomial estimate for the Thurston-Bennequin number as well as to the existence of an ungraded augmentation of the Chekanov-Eliashberg DGA. Parallel results involving the HOMFLY-PT polynomial and 2-graded generalized normal rulings are established.