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.
"The concordance genus of knots." Algebr. Geom. Topol. 4 (1) 1 - 22, 2004. https://doi.org/10.2140/agt.2004.4.1