The
–solvablefiltration
of the smooth knot concordance group (denoted by
) due
to Cochran, Orr and 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 attributes of a knot imply membership in various levels of the
filtration. We show the counterpart of this fact for two new filtrations of
due
to Cochran, Harvey and Horn; the
positive and
negative filtrations, denoted by
and
respectively. In particular, we show that if a knot
bounds a
Cassontower of height
in
with only positive (resp. negative) kinks in the base-level kinky disk, then
(resp. ).
En route to this result we show that if a knot
bounds a Casson
tower of height
in
,
it bounds an embedded (symmetric)
grope of height
and is therefore
–solvable.
We also define a variant of Casson towers and show that if
bounds a
tower of type
in
, it is
–solvable.
If
bounds
such a tower with only positive (resp. negative) kinks in the base-level kinky disk
then
(resp. ).
Our results show that either every knot which bounds a Casson
tower of height three is topologically slice or there exists a knot in
which is not topologically slice. We also give a
–dimensional
characterization, up to concordance, of knots which bound kinky disks in
with only positive (resp. negative) kinks; such knots form a subset of
(resp. ).
