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2004 The concordance genus of knots
Charles Livingston
Algebr. Geom. Topol. 4(1): 1-22 (2004). DOI: 10.2140/agt.2004.4.1


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 0g4(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.


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Charles Livingston. "The concordance genus of knots." Algebr. Geom. Topol. 4 (1) 1 - 22, 2004.


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

zbMATH: 1055.57007
MathSciNet: MR2031909
Digital Object Identifier: 10.2140/agt.2004.4.1

Primary: 57M25, 57N70

Rights: Copyright © 2004 Mathematical Sciences Publishers


Vol.4 • No. 1 • 2004
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