# Filtering smooth concordance classes of topologically slice knots

Recall that the n-solvable filtration of the smooth knot concordance group, suggested by work of Cochran-Orr-Teichner, has the property that any topologically slice knot lies in every term of the filtration. With the aim of studing topologically slice knots, we investigate a new filtration, $$\{B_n\}$$, that is simultaneously a refinement of the n-solvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. As is the case for the n-solvable filtration, each $$B_n/B_{n+1}$$ has infinite rank. But our primary interest is in the induced filtration, $$\{T_n\}$$, on the subgroup, $$T$$, of knots that are topologically slice. We prove that $$T/T_0$$ is large, detected by gauge-theoretic invariants and the $$\tau$$, $$s$$, and $$\epsilon$$-invariants; while the non-triviality of $$T_0/T_1$$ can be detected by certain $$d$$-invariants. All of these concordance obstructions vanish for knots in $$T_1$$. Nonetheless, going beyond this, our main result is that $$T_1/T_2$$ has positive rank. Moreover under a "weak homotopy-ribbon" condition, we show that each $$T_n/T_{n+1}$$ has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity.