KCH representations, augmentations, and $A$-polynomials

We describe a correspondence between augmentations of knot contact homology 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.

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Received 28 May 2015

Accepted 30 September 2016

Published 28 November 2017