We produce a family \(\{K_p\}\) of topologically slice knots which generates an infinite rank subgroup of the first quotient \(T_0/T_1\) of the bipolar filtration of the concordance group.  In addition to being topologically slice, each of these knots is smoothly slice in \(D^4 \# \mathbb{CP}(2)\#\cdots\#\mathbb{CP}(2)\) and in \(D^4 \# \overline{\mathbb{CP}(2)}\#\cdots\#\overline{\mathbb{CP}(2)}\) (equipped with a possibly exotic smooth structure).  In addition, we show that no nontrivial linear combination of the \(K_p\) is smoothly concordant to a knot with Alexander polynomial one.