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arxiv: 1901.00474 · v1 · pith:2PYZV7OHnew · submitted 2019-01-02 · 🧮 math.GT

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

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keywords polynomialalexanderknotribbonknotsalternativeconditioncontrast
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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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