The topological 4-genus of a knot is the minimum genus of a topological, locally flat, surface embedded in the 4-ball with boundary the knot. Bounds arise from the p-signatures and, to obstruct being of 4-genus 0 (slice) the Alexander polynomial. The additional bounds that arise in the smooth case don't apply here. For instance, there are knots of Alexander polynomial 1 that are not smoothly slice, but are topologically slice by Freedman's work.
Further information on particular knots.