article

Title: On the AJ conjecture for knots

Authors: Anh T. Tran and Thang T.Q. Le

Issue: Volume 64 (2015), Issue 4, 1103-1151

Abstract:

$We confirm the AJ conjecture [S. Garoufalidis, \textit{On the characteristic and deformation varieties of a knot}, Proceedings of the Casson Fest, Geom. Topol. Monogr., vol. 7, Geom. Topol. Publ., Coventry, 2004, pp. 291-309 (electronic)] that relates the $A$-poly\-nomial and the colored Jones polynomial for 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 [Th. T. Q. Le, \textit{The colored Jones polynomial and the A-polynomial of knots}, Adv. Math. \textbf{207} (2006), no. 2, 782-804], who established the AJ conjecture for a large class of two-bridge knots, including all twist knots. Along the way, we explicitly calculate the universal $\mathrm{SL}_2(\mathbb{C})$-character ring of the knot group of the $(-2,3,2n+1)$-pretzel knot, and show it is reduced for all integers $n$.$