Character varieties, A-polynomials, and the AJ Conjecture

We establish some facts about the behavior of the rational-geometric subvariety of the $S{L}_2(\stackrel{¸}{)}$ or $PS{L}_2(\stackrel{¸}{)}$ character variety of a hyperbolic knot manifold under the restriction map to the $S{L}_2(\stackrel{¸}{)}$ or $PS{L}_2(\stackrel{¸}{)}$ character variety of the boundary torus, and use the results to get some properties about the A-polynomials and to prove the AJ conjecture for certain class of knots in $S^3$ including in particular any $2$-bridge knot over which the double branched cover of $S^3$ is a lens space of prime order.