Ruling polynomials and augmentations for Legendrian tangles

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