Generalized normal rulings and invariants of Legendrian solid torus links

TL;DR

This paper introduces a generalized notion of normal rulings for Legendrian links in the 1-jet space of S^1, connecting them to skein modules, polynomial invariants, and augmentations of the Chekanov-Eliashberg DGA, thus unifying various invariants.

Contribution

It generalizes normal rulings for Legendrian links and links these to polynomial invariants and augmentations, providing new tools for Legendrian link classification.

Findings

Generalized normal rulings are equivalent to sharp Kauffman polynomial bounds.

Existence of generalized normal rulings corresponds to ungraded augmentations of the DGA.

Results extend to HOMFLY-PT polynomial and 2-graded rulings.

Abstract

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.