Character varieties, A-polynomials, and the AJ Conjecture

TL;DR

This paper investigates the behavior of character varieties and A-polynomials of hyperbolic knot manifolds, establishing properties and proving the AJ conjecture for specific classes of knots, including certain 2-bridge knots.

Contribution

It provides new insights into the rational-geometric subvariety behavior under restriction maps and proves the AJ conjecture for a class of knots including some 2-bridge knots.

Findings

Properties of the rational-geometric subvariety under restriction maps.

Verification of the AJ conjecture for certain hyperbolic knots.

Results on the behavior of A-polynomials for these knots.

Abstract

We establish some facts about the behavior of the rational-geometric subvariety of the $SL_2(\c)$ or $PSL_2(\c)$ character variety of a hyperbolic knot manifold under the restriction map to the $SL_2(\c)$ or $PSL_2(\c)$ 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.