Slicing knots in definite 4-manifolds

We study the C P 2 CP² -slicing number of knots, i. e. the smallest m ≥ 0 m 0 such that a knot K ⊆ S 3 K S³ bounds a properly embedded, null-homologous disk in a punctured connected sum (# m C P 2) × (\#ᵐ CP²) ^. We find knots for which the smooth and topological C P 2 CP² -slicing numbers are both finite, nonzero, and distinct. To do this, we give a lower bound on the smooth C P 2 CP² -slicing number of a knot in terms of its double branched cover and an upper bound on the topological C P 2 CP² -slicing number in terms of the Seifert form.