Alexander polynomials of ribbon knots and virtual knots

We find that Alexander polynomial of a ribbon knot in $ \mathbb{Z}HS^3 $ is determined by the intrinsic singularity information of its ribbon, and give a formula to calculate Alexander polynomial of a ribbon knot by that. We define half Alexander polynomial $ A_R (t) $, an invariant of oriented ribbons, and in fact the Alexander polynomial of the ribbon knot is $ A_R (t) A_R (t^{-1}) $. We give two useful simplified formulas for half Alexander polynomial. We characterize completely the polynomials arising as half Alexander polynomials of ribbons. The above study unexpectedly leads us to discover new formulas for Alexander polynomial of general knots and virtual knots in terms of Gauss diagrams.