Concordance Genus

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.)

Specific Knots

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.

Knots with concordance genus less than the 3-ball genus or unknown.

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)
11a5, 11a104, 11a112, 11a168, 11n85, 11n100

Concordant to the 51 (Concordance genus 2)
11n69, 11n76, 11n78

Concordant to the 52 (Concordance genus 1)
11n68 11n71, 11n75

Concordant to the 62 (Concordance genus 2)
11a57, 11a102, 11a139, 11a199, 11a231

Concordant to the 63 (Concordance genus 2)
11a38, 11a44, 11a47, 11a187

Concordant to the 31 + 31 (Concordance genus 2)

Concordant to the 31 + 41 (Concordance genus 2)
11a132, 11a157

Unknown

11a6, 11a8, 11a57, 11a67, 11a72, 11a102, 11a109, 11a135, 11a249, 11a264, 11a297, 11a305, 11a332, 11a352, 11n34, 11n45, 11n66, 11n69, 11n145, 11n152

References

[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 Concordance 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.