Positive links

Abstract

Given a link $L\subset S^3$, we ask whether the components of $L$ bound disjoint, nullhomologous disks properly embedded in a simply connected positive-definite smooth $4$–manifold; the knot case has been studied extensively by Cochran, Harvey and Horn. Such a $4$–manifold is necessarily homeomorphic to a (punctured) ${\#}_kℂP\left(2\right)$. We characterize all links that are slice in a (punctured) ${\#}_kℂP\left(2\right)$ in terms of ribbon moves and an operation which we call adding a generalized positive crossing. We find obstructions in the form of the Levine–Tristram signature function, the signs of the first author’s generalized Sato–Levine invariants, and certain Milnor’s invariants. We show that the signs of coefficients of the Conway polynomial obstruct a $2$–component link from being slice in a single punctured $ℂP\left(2\right)$ and conjecture these are obstructions in general. These results have applications to the question of when a $3$–manifold bounds a $4$–manifold whose intersection form is that of some ${\#}_kℂP\left(2\right)$. For example, we show that any homology $3$–sphere is cobordant, via a smooth positive-definite manifold, to a connected sum of surgeries on knots in $S^3$.