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

Generalized normal rulings and invariants of Legendrian solid torus links