Algebraic and Geometric Topology 4 (2004),
paper no. 1, pages 1-22.
The concordance genus of knots
Charles Livingston
Abstract.
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 <= g_4(K) <= g_c(K) <= 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
AMS subject classification.
Primary: 57M25, 57N70.
DOI: 10.2140/agt.2004.4.1
E-print: arXiv:math.GT/0107141
Submitted: 27 July 2003.
(Revised: 3 January 2004.)
Accepted: 7 January 2004.
Published: 9 January 2004.
Notes on file formats
Charles Livingston
Department of Mathematics, Indiana University
Bloomington, IN 47405, USA
Email: livingst@indiana.edu
AGT home page
