In knot concordance three genera arise naturally,
, and
:
these are the classical genus, the 4–ball genus, and the concordance
genus, defined to be the minimum genus among all knots concordant to
. Clearly
.
Casson and Nakanishi gave examples to show that
need not
equal .
We begin by reviewing and extending their results.
For knots representing elements in ,
the concordance group of algebraically slice knots, the relationships
between these genera are less clear. Casson and Gordon’s result that
is nontrivial implies that
can be nonzero for knots
in . Gilmer proved that
can be arbitrarily large
for knots in . We will
prove that there are knots
in
with
and
arbitrarily large.
Finally, we tabulate
for all prime knots with 10 crossings and, with two exceptions, all prime knots with
fewer than 10 crossings. This requires the description of previously unnoticed
concordances.
