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

Delphine Moussard; Emmanuel Wagner

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arXiv · 1901.00474

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

Delphine Moussard·Emmanuel Wagner

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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.

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Delphine Moussard, Emmanuel Wagner. 2019-01-02. A Fox-Milnor theorem for the Alexander polynomial of knotted $2$-spheres in $S^4$. https://arxiv.org/abs/1901.00474

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