KCH representations, augmentations, and $A$-polynomials

Abstract

We describe a correspondence between augmentations and certain representations of the knot group. The correspondence makes the 2-variable augmentation polynomial into a generalization of the classical $A$-polynomial. It also associates to an augmentation a rank, which is bounded by the bridge number and shares its behavior under connect sums. We also study augmentations with rank equal to the braid index.