Alexander and writhe polynomials for virtual knots

We give a new interpretation of the Alexander polynomial $Delta_0$ for virtual knots due to Sawollek and Silver and Williams, and use it to show that, for any virtual knot, $Delta_0$ determines the writhe polynomial of Cheng and Gao (equivalently, Kauffmans affine index polynomial). We also use it to define a second-order writhe polynomial, and give some applications.