A Fox–Milnor theorem for the Alexander polynomial of knotted 2-spheres in $S^4$

Abstract

For knots in $S^3$, it is well-known that the Alexander polynomial of a ribbon knot factorizes as $f(t)f(t^{-1})$ for some polynomial $f(t)$. By contrast, the Alexander polynomial of a ribbon 2-knot in $S^4$ is not even symmetric in general. Via an alternative notion of ribbon 2-knots, we give a topological condition on a 2-knot that implies the factorization of the Alexander polynomial.