We investigate the properties of knots in
which
bound punctured Klein bottles, such that a pushoff of the knot has zero
linking number with the knot, ie has zero framing. This is motivated by
the many results in the literature regarding slice knots of genus one, for
example, the existence of homologically essential zero self-linking simple closed
curves on genus one Seifert surfaces for algebraically slice knots. Given a knot
bounding a punctured
Klein bottle with zero
framing, we show that ,
the core of the orientation preserving band in any disk–band form of
, has zero self-linking.
We prove that such a
is slice in a –homology
if and
only if
is as well, a stronger result than what is currently known for genus
one slice knots. As an application, we prove that given knots
and
and any odd
integer ,
the –cables
of and
are
–concordant
if and only if
and are
–concordant. In
particular, if the –cable
of a knot is
slice, is slice in
a –homology
ball.
