We study and show the equivalence between two Legendrian isotopy invariants associated to a (bordered) Legendrian graph: the augmentation number via point counting over a finite field for the augmentation variety of the associated Chekanov–Eliashberg differential graded algebra, and the ruling polynomial via combinatorics of the decompositions of the associated front projection.
"Augmentations and ruling polynomials for Legendrian graphs." Algebr. Geom. Topol. 22 (5) 2079 - 2185, 2022. https://doi.org/10.2140/agt.2022.22.2079