Slicing knots in definite 4-manifolds

We study the C P 2 \mathbb {CP}^2 -slicing number of knots, i.e. the smallest m ≥ 0 m\geq 0 such that a knot K ⊆ S 3 K\subseteq S^3 bounds a properly embedded, null-homologous disk in a punctured connected sum ( # m C P 2 ) × (\#^m\mathbb {CP}^2)^{\times } . We find knots for which the smooth and topological C P 2 \mathbb {CP}^2 -slicing numbers are both finite, nonzero, and distinct. To do this, we give a lower bound on the smooth C P 2 \mathbb {CP}^2 -slicing number of a knot in terms of its double branched cover and an upper bound on the topological C P 2 \mathbb {CP}^2 -slicing number in terms of the Seifert form.