The concordance genus of knots

Abstract

In knot concordance three genera arise naturally, $g\left(K\right),g_4\left(K\right)$, and $g_c\left(K\right)$: these are the classical genus, the 4–ball genus, and the concordance genus, defined to be the minimum genus among all knots concordant to $K$. Clearly $0\le g_4\left(K\right)\le g_c\left(K\right)\le g\left(K\right)$. Casson and Nakanishi gave examples to show that $g_4\left(K\right)$ need not equal $g_c\left(K\right)$. We begin by reviewing and extending their results.

For knots representing elements in $\mathcal{A}$, the concordance group of algebraically slice knots, the relationships between these genera are less clear. Casson and Gordon’s result that $\mathcal{A}$ is nontrivial implies that $g_4\left(K\right)$ can be nonzero for knots in $\mathcal{A}$. Gilmer proved that $g_4\left(K\right)$ can be arbitrarily large for knots in $\mathcal{A}$. We will prove that there are knots $K$ in $\mathcal{A}$ with $g_4\left(K\right)=1$ and $g_c\left(K\right)$ arbitrarily large.

Finally, we tabulate $g_c$ 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.