  The n-solvable filtration $\{\mathcal{F}_n\}_{n=0}^\infty$ of the smooth knot
concordance group (denoted by $\mathcal{C}$), due to Cochran-Orr-Teichner, has
been instrumental in the study of knot concordance in recent years. Part of its
significance is due to the fact that certain geometric characterizations of a
knot imply membership in various levels of the filtration. We show the
counterpart of this fact for two new filtrations of $\mathcal{C}$ due to
Cochran-Harvey-Horn, the positive and negative filtrations, denoted by
$\{\mathcal{P}_n\}_{n=0}^\infty$ and $\{\mathcal{N}_n\}_{n=0}^\infty$
respectively. In particular, we show that if a knot K bounds a Casson tower of
height n+2 in the 4-ball with only positive (resp. negative) kinks in the
base-level kinky disk, then K is in $\mathcal{P}_n$ (resp. $\mathcal{N}_n$). En
route to this result we show that if a knot K bounds a Casson tower of height
n+2 in the 4-ball, it bounds an embedded (symmetric) grope of height n+2, and
is therefore, n-solvable (this also implies that topologically slice knots
bound arbitrarily tall gropes in the 4-ball). We also define a variant of
Casson towers and show that if K bounds a tower of type (2,n) in the 4-ball, it
is n-solvable. If K bounds such a tower with only positive (resp. negative)
kinks in the base-level kinky disk then K is in $\mathcal{P}_n$ (resp.
$\mathcal{N}_n$). Our results show that either every knot which bounds a Casson
tower of height three is topologically slice or there exists a knot which is
not topologically slice but lies in each $\mathcal{F}_n$. We also give a
3-dimensional characterization, up to concordance, of knots which bound kinky
disks in the 4-ball with only positive (resp. negative) kinks; such knots form
a subset of $\mathcal{P}_0$ (resp. $\mathcal{N}_0$).
