Slicing knots in definite 4-manifolds

Abstract

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

33 pages, 3 footnotes