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Associated to Legendrian links in the standard contact three-space, Rulingpolynomials are Legendrian isotopy invariants, which also compute augmentationnumbers, that is, the points-counting of augmentation varieties for Legendrianlinks (up to a normalized factor) \cite{HR15}. In this article, we generalizethis picture to Legendrian tangles, which are morally the pieces obtained bycutting Legendrian link fronts along 2 vertical lines. Moreover, we show thatthe Ruling polynomials for Legendrian tangles satisfy the composition axiom. Inthe special case of Legendrian knots, our arguments provide new proofs to themain results in \cite{HR15}. In the end, we also introduce generalized Rulingpolynomials for Legendrian tangles, to account for non-acyclic augmentations inthe "Ruling polynomials compute augmentation numbers" picture.

Ruling polynomials and augmentations for Legendrian tangles