Nonorientable slice genus can be arbitrarily large

A generic cross-section of a surface smoothly embedded in four-space is a knot or a link. Every knot can be realized as the cross-section of some surface, but if that surface is required to be orientable, then work of Fox, Milnor, and Murasugi show that its genus may need to be quite large. For example, the $(2, n)$ torus knot is not a cross-section of any orientable surface with genus less than $n-1$, but is the cross-section of a Klein bottle. We give a lower bound on the first Betti number of a surface with cross-section a knot $K$ in terms of the signature of $K$ and the Heegaard-Floer $d$-invariant of the integer homology sphere given by $-1$-surgery on $K$. In particular, we show that any smooth surface in four-space with cross-section the $(2k, 2k - 1)$ torus knot has first Betti number at least $2k - 2$.

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Published 13 October 2014