The concordance genus of knots

In knot concordance three genera arise naturally, $g (K), g_{4} (K)$ , and $g_{c} (K)$ : 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 $0 \leq g_{4} (K) \leq g_{c} (K) \leq g (K)$ . Casson and Nakanishi gave examples to show that $g_{4} (K)$ need not equal $g_{c} (K)$ . We begin by reviewing and extending their results.

For knots representing elements in $A$ , the concordance group of algebraically slice knots, the relationships between these genera are less clear. Casson and Gordon’s result that $A$ is nontrivial implies that $g_{4} (K)$ can be nonzero for knots in $A$ . Gilmer proved that $g_{4} (K)$ can be arbitrarily large for knots in $A$ . We will prove that there are knots $K$ in $A$ with $g_{4} (K) = 1$ and $g_{c} (K)$ 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

References

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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

Volume 4, issue 1 (2004)

Charles Livingston

Abstract