On Ruling polynomials of Legendrian links

The ruling polynomial is a Legendrian invariant that is closely related to the augmentation variety of the Legendrian. We characterize all graded and ungraded ruling polynomials, and construct Legendrian links realizing each possible polynomial. The graded augmentation varieties of the Legendrians we construct all have trivial cluster algebra structures. Finally, we construct Legendrian knots admitting $k$ exact Lagrangian fillings with $χ(L)=n$ that are pairwise smoothly non-isotopic for $n\leq 1$, and $k\geq 0.$