Character varieties, $A$–polynomials and the AJ conjecture

Abstract

We establish some facts about the behavior of the rational-geometric subvariety of the ${SL}_2\left(ℂ\right)$ or ${PSL}_2\left(ℂ\right)$ character variety of a hyperbolic knot manifold under the restriction map to the ${SL}_2\left(ℂ\right)$ or ${PSL}_2\left(ℂ\right)$ 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 a 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.