## Algebraic & Geometric Topology

### The concordance genus of knots

Charles Livingston

#### Abstract

In knot concordance three genera arise naturally, $g(K),g4(K)$, and $gc(K)$: 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 $0≤g4(K)≤gc(K)≤g(K)$. Casson and Nakanishi gave examples to show that $g4(K)$ need not equal $gc(K)$. 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 $g4(K)$ can be nonzero for knots in $A$. Gilmer proved that $g4(K)$ can be arbitrarily large for knots in $A$. We will prove that there are knots $K$ in $A$ with $g4(K)=1$ and $gc(K)$ arbitrarily large.

Finally, we tabulate $gc$ 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 4, Number 1 (2004), 1-22.

Dates
Received: 27 July 2003
Revised: 3 January 2004
Accepted: 7 January 2004
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882463

Digital Object Identifier
doi:10.2140/agt.2004.4.1

Mathematical Reviews number (MathSciNet)
MR2031909

Zentralblatt MATH identifier
1055.57007

#### Citation

Livingston, Charles. The concordance genus of knots. Algebr. Geom. Topol. 4 (2004), no. 1, 1--22. doi:10.2140/agt.2004.4.1. https://projecteuclid.org/euclid.agt/1513882463

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