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

## TL;DR

This paper extends the Fox-Milnor theorem to certain ribbon 2-knots in 4-dimensional space, providing conditions under which their Alexander polynomial factorizes, despite general asymmetry.

## Contribution

It introduces a topological condition for ribbon 2-knots that ensures Alexander polynomial factorization, generalizing the classical Fox-Milnor theorem.

## Key findings

- Identifies a topological condition for ribbon 2-knots
- Shows factorization of Alexander polynomial under this condition
- Provides a new perspective on 2-knot invariants

## 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 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.