A Fox–Milnor theorem for the Alexander polynomial of knotted 2-spheres in 𝑆⁴

Abstract

For knots in 𝑆³, 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 𝑆⁴ 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.