On the AJ conjecture for knots Thang LeAnh Tran Primary 57N10. Secondary 57M25A-polynomialcolored Jones polynomialAJ conjecturetwo-bridge knotdouble twist knotpretzel knotuniversal character ring 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 $n$. Indiana University Mathematics Journal 2015 text pdf 10.1512/iumj.2015.64.5602 10.1512/iumj.2015.64.5602 en Indiana Univ. Math. J. 64 (2015) 1103 - 1151 state-of-the-art mathematics http://iumj.org/access/