Knot contact homology and representations of knot groups

TL;DR

This paper explores the relationship between knot contact homology and knot group representations, revealing connections between augmentation and A-polynomials, and providing insights into knot ranks and classifications.

Contribution

It introduces a new perspective linking linear representations of knot groups to augmentations of knot contact homology, clarifying polynomial relationships and knot ranks.

Findings

For 2-bridge knots, the augmentation polynomial and A-polynomial agree.

The polynomials differ for non-2-bridge torus knots and certain pretzel knots.

Provides a lower bound on the meridional rank of knots.

Abstract

We study certain linear representations of the knot group that induce augmentations of knot contact homology. This perspective on augmentations enhances our understanding of the relationship between the augmentation polynomial and the A-polynomial of the knot. For example, we show that for 2-bridge knots the polynomials agree and that this is never the case for (non-2-bridge) torus knots, nor for a family of 3-bridge pretzel knots. In addition, we obtain a lower bound on the meridional rank of the knot. As a consequence, our results give another proof that torus knots and a family of pretzel knots have meridional rank equal to their bridge number.