#### Volume 4, issue 1 (2004)

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The concordance genus of knots

### Charles Livingston

Algebraic & Geometric Topology 4 (2004) 1–22
 arXiv: math.GT/0107141
##### 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 $\mathsc{A}$, the concordance group of algebraically slice knots, the relationships between these genera are less clear. Casson and Gordon’s result that $\mathsc{A}$ is nontrivial implies that ${g}_{4}\left(K\right)$ can be nonzero for knots in $\mathsc{A}$. Gilmer proved that ${g}_{4}\left(K\right)$ can be arbitrarily large for knots in $\mathsc{A}$. We will prove that there are knots $K$ in $\mathsc{A}$ 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.

##### Keywords
concordance, knot concordance, genus, slice genus
##### Mathematical Subject Classification 2000
Primary: 57M25, 57N70
##### Publication
Received: 27 July 2003
Revised: 3 January 2004
Accepted: 7 January 2004
Published: 9 January 2004
##### Authors
 Charles Livingston Department of Mathematics Indiana University Bloomington IN 47405 USA