For a knot K, the concordance genus is the minimum genus among all knots concordant to K. Casson gave the first example of a knot of concordance genus greater than the 4-ball genus by demonstrating that the knot 62 has 4-ball genus 1, since it has unknotting number 1, but it cannot be concordant to a knot of genus 1 for the following reason: Its Alexander polynomial is irreducible, of degree 4. If it were concordant to a knot of genus 1, its polynomial times a polynomial of degree at most 2, would factor as f(t)*f(t-1), clearly an impossibility.
For other results concerning the concordance genus, see the reference below. Many of the initial results for 11 crossing knots were found by John McAtee. These were checked and expanded on by Kate Kearney in [2].
For knots that are concordant to lower genus knots, the simplest such knot is listed in a document linked to the number in the table.
Clearly the concordance genus is dependent on the category, smooth or topological, locally flat. The first known example of this occurs for knots that are topologically slice, and thus topological concordance genus = 0, but are not smoothly slice, so smooth concordance genus > 0. Since, as of yet, this distinction is extremely rare, we concentrate on the smooth case. (The only known distinction on the chart occurs with 11n34, which is topologically slice, but we do not know its smooth concordance genus.)
Initial work for 11 crossing knots was done by John McAtee. Kate Kearney verified and extended McAtee's work on 11 crossing knots, now available in [2].
11a28, 11a35, and 11a96
Christoph Lamm provided the demonstrations that these knots are slice.
10 or fewer crossings
Slice Knots (Concordance genus 0)
61, 88, 89, 820, 927, 941, 946,
103, 1022, 1035, 1042, 1048, 1075, 1087,
1099, 10123, 10129, 10137, 10140, 10153, and 10155
Concordant to the Trefoil: 31 (Concordance genus 1)
810, 811, 1040, 1059, 10103, 10106,
10143, 10147.
Concordant to the Figure 8: 41 (Concordance genus 1)
924, 937
Concordant to the 51 (Concordance genus 2)
1021, 1062. Both have gc(K) = 2 = g4(K).
Concordant to the 52 (Concordance genus 1)
1065, 1067, 1074, 1077
Concordant to the 31 + 31 (Concordance genus 2)
1098
Remaining cases, cr(K) < 10.
The last unknown cases for prime knots of 10 or fewer crossings, 818, 940,
1082, were resolved in [4].
11 crossings
Slice Knots (Concordance genus 0)
11a28, 11a35, 11a36, 11a58, 11a87, 11a96,
11a103, 11a115, 11a164, 11a165, 11a169, 11a201,
11n21, 11n37, 11n39, 11n42, 11n49, 11n50, 11n67,
11n73, 11n74, 11n83, 11n97, 11n116, 11n132,
11n139, 11n172
Concordant to the Trefoil: 31 (Concordance genus 1)
11a196, 11a216, 11a283, 11a286, 11n106, 11n122
Concordant to the Figure 8: 41 (Concordance genus 1)
Concordant to the 51 (Concordance genus 2)
Concordant to the 52 (Concordance genus 1)
Concordant to the 62 (Concordance genus 2)
Concordant to the 63 (Concordance genus 2)
Concordant to the 31 + 31 (Concordance genus 2)
Concordant to the 31 + 41 (Concordance genus 2)
Unknown
11a6, 11a8, 11a57, 11a67, 11a72, 11a102,
11a109, 11a135, 11a249, 11a264, 11a297, 11a305,
11a332, 11a352, 11n34, 11n45, 11n66, 11n69,
11n145, 11n152
11a5, 11a104, 11a112, 11a168, 11n85, 11n100
11n69, 11n76, 11n78
11n68 11n71, 11n75
11a57, 11a102, 11a139, 11a199, 11a231
11a38, 11a44, 11a47, 11a187
11a132, 11a157
[1] Conway, J., "An enumeration of knots and links, and some of their algebraic properties," Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), Pergamon, Oxford (1970), 329-358.
[2] Kearney, K., "The Concordance2 Genus of 11-Crossing Knots," Arxiv preprint.
[3] Livingston, C., "The concordance genus of knots," Algebr. Geom. Topol. 4 (2004), 1-22.
Arxiv preprint
(Note: the knot 1082 does not appear among the unknowns, a gap in that paper.)
[4] Livingston, C., "The concordance genus of a knot, II," Arxiv preprint.