On the AJ conjecture for knots

TL;DR

This paper verifies the AJ conjecture, linking the A-polynomial and colored Jones polynomial, for certain classes of hyperbolic knots, including some two-bridge and pretzel knots, extending previous results.

Contribution

It confirms the AJ conjecture for new classes of knots and explicitly computes the universal character ring for specific pretzel knots.

Findings

AJ conjecture holds for some two-bridge and pretzel knots

Universal character ring of (-2,3,2n+1)-pretzel knots is reduced for all n

Extended previous results on the AJ conjecture for twist knots

Abstract

We confirm the AJ conjecture [Ga04] that relates the A-polynomial and the colored Jones polynomial for those hyperbolic knots satisfying certain conditions. In particular, we show that the conjecture holds true for some classes of two-bridge knots and pretzel knots. This extends the result of the first author in [Le06] where he established the AJ conjecture for a large class of two-bridge knots, including all twist knots. Along the way, we explicitly calculate the universal character ring of the knot group of the (-2,3,2n+1)-pretzel knot and show that it is reduced for all integers n.