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


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.

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