Title:  Normal rulings, augmentations, and the colored HOMFLY-PT polynomial

Abstract:  Normal rulings are certain decompositions of front diagrams of Legendrian links in $R^3$ that were discovered independently by Chekanov & Pushkar and Fuchs in the context of generating families and augmentations of the Legendrian DG-algebra respectively. They can be used to define combinatorial invariants of Legendrian links called ruling polynomials.  In this talk, I will survey some results connecting normal rulings with augmentations  and with 2-variable topological knot polynomials (HOMFLY-PT and Kauffman).  In particular, I will discuss recent joint work with C. Leverson and J. Murray relating counts of higher rank augmentations to the n- colored HOMFLY-PT and Kauffman polynomials.