Local Knots, $\hat{I}{½}^{+}$-Sharp Knots, and Rational Slice Genus

Abstract

Hom and Wu introduced the knot concordance invariant $\hat{I}{½}^{+}$ for knots in $S^3$ and proved that it gives a lower bound for the slice genus. Wu and Yang extended $\hat{I}{½}^{+}$ to knots in rational homology $3$-spheres, where it gives a lower bound for the rational slice genus, an analogue of the slice genus for knots in rational homology $3$-spheres. We call a knot $\hat{I}{½}^{+}$-sharp if this bound is realized as an equality. An open question asks whether a local knot in a $3$-manifold $Y$, that is, a knot contained in a $3$-ball, can bound a surface of smaller genus in $Y\times I$ than in $S^3\times I$. Using the Heegaard Floer invariant $\hat{I}{½}^{+}$, we show that this does not occur for local knots arising from $\hat{I}{½}^{+}$-sharp knots: if $K\subset S^3$ is $\hat{I}{½}^{+}$-sharp and $Y$ is a rational homology $3$-sphere, then the induced local knot in $Y$ has rational slice genus equal to the slice genus of $K$. The proof proceeds by establishing an additivity result for the rational slice genus.

16 pages, 1 figure