Twisted Alexander polynomials, character varieties and Reidemeister torsions of double branched covers

We give an extension of Fox's formula of the Alexander polynomial for a double branched cover over the three-sphere. Our formula provides the Reidemeister torsion of the double branched cover along a knot for a non-trivial 1-dimensional representation. In our formula, the Reidemeister torsion is given by the product of two factors derived from the knot group. One of the factors is determined by the twisted Alexander polynomial and the other is determined by a rational function on the character variety of the knot group. As an application, we show that these products distinguish the isotopy classes of two-bridge knots up to their mirror images.