Comments and References for (smooth) Concordance Genus
The main reference for the concordance genus is [Livingston, C. The concordance genus of knots Algebr. Geom. Topol. 4 (2004)
1-22. /arxiv.org/abs/math.GT/0107141]. (Note that in that paper the knot 10_82 does not appear among the unknowns, a gap in that
paper.) An earlier reference, containing some of the basic concordance relations for low crossing number knots is Conway's paper,
[An enumeration of knots and links, and some of their algebraic properties,
Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), 329--358, Pergamon, Oxford, 1970] Most of the work for 11
crossing knots was done by John McAtee, preprint in preparation. Christoph Lamm provided the demonstrations that the knots
11a28,11a35, and 11a96 are slice.
Knots with Concordance genus less than the 3-ball genus or unknown.
KNOTS OF 10 CROSSINGS OR LESS
Slice Knots (Concordance genus 0)
6_1, 8_8, 8_9, 8_20, 9_27, 9_41, 9_46, 10_3,
10_22, 10_35, 10_42, 10_48, 10_75, 10_87, 10_99, 10_123, 10_129, 10_137, 10_140, 10_153, and 10_155
Concordant to the Trefoil -- 3_1 (Concordance genus 1) 8_10, 8_11, 10_40, 10_59, 10_103, 10_106, 10_143, 10_147.
Concordant to the Figure 8 -- 4_1 (Concordance genus 1)
9_24, 9_37
Concordant to the 5_1 (Concordance genus 2) 10_21, 10_62. Both have g_c(K) = 2 = g_4(K).
Concordant to the 5_2 (Concordance genus 1) 10_65, 10_67, 10_74, 10_77
Concordant to the 3_1 + 3_1 (Concordance genus 2) 10_98
Remaining cases, cr(K) < 10. The last unknown cases for prime knots of 10 or fewer crossings, 8_18, 9_40, 10_82, have been
resolved in the paper by Livingston: "The concordance genus of a knot, II." Arxiv
preprint.
KNOTS OF 11 CROSSINGS
Slice Knots (Concordance genus 0)
11a28, 11a35, 11a_36, 11a_58, 11a_87, 11a96, 11a_103, 11a_115, 11a_164,11a_165, 11a_169, 11a_201, 11n_21, 11n_37, 11n_39,
11n_42, 11n_49, 11n_50, 11n_67, 11n_73, 11n_74, 11n_83, 11n_97, 11n_116, 11n_132, 11n_139, 11n_172
Concordant to the Trefoil -- 3_1 (Concordance genus 1) 11a_196, 11a_216, 11a_283, 11a_286, 11n_106, 11n_122
Concordant to the Figure 8 -- 4_1 (Concordance genus 1) 11a_5, 11a_104, 11a_112, 11a_168, 11n_85, 11n_100,
Concordant to the 5_1 (Concordance genus 2) 11n_69, 11n_76, 11n_78,
Concordant to the 5_2 (Concordance genus 1) 11n_68 11n_71, 11n_75,
Concordant to the 6_2 (Concordance genus 2) 11a_108, 11a_139, 11a_181, 11a_199, 11a_231,
Concordant to the 6_3 (Concordance genus 2) 11a_38, 11a_44, 11a_47, 11a_187,
Concordant to the 3_1 + 3_1 (Concordance genus 2)
Concordant to the 3_1 + 4_1 (Concordance genus 2) 11a_132, 11a_157,
Unknown 11a_6, 11a_8, 11a_38, 11a_47, 11a_57, 11a_67, 11a_72,
11a_102, 11a_109, 11a_135, 11a_249, 11a_264, 11a_297, 11a_305, 11a_316, 11a_326, 11a_332, 11a_352, 11n_4, 11n_34, 11n_45, 11n_66,
11n_68, 11n_69, 11n_72, 11n_83, 11n_145, 11n_152, 11n_172,