This talk is a more in-depth look at knot contact homology (following the colloquium from the previous day, though we'll start from scratch). We'll examine how this knot invariant can be used to produce a three-variable knot polynomial, the augmentation polynomial, which specializes to a "stable" version of the A-polynomial (in work of Chris Cornwell) and is conjecturally related to colored HOMFLY polynomials. We'll then discuss a conjectural interpretation of this polynomial that uses string theory and mirror symmetry (in joint work with Mina Aganagic, Tobias Ekholm, and Cumrun Vafa).