The topological 4-genus of a knot is the minimum genus of a topological, locally flat surface embedded in the 4-ball with boundary the knot. Bounds arise from the p-signatures and, to obstruct being of 4-genus 0 (slice), the Alexander polynomial. The additional bounds that arise in the smooth case don't apply here. For instance, there are knots of Alexander polynomial 1 that are not smoothly slice, but are topologically slice by Freedman's work.
816, 818, 917, 931, 932, 940, 947
Ref. [5]
948, 10117, 10144
Ref. [3]
1051
Selahi Durusoy has found a single crossing change (in the given diagram) that converts 1051 into 88,
which is slice.
1054, 1070, 1097, 10148, 10151
Ref. [7]
10{139, 145, 161}, 11a211, 11n{9, 77, 183}, 12a_{153 255, 1414, 534, 542, 624, 636, 719,
1118}, 12n_{59, 91, 105, 110, 120, 136, 148, 175, 187, 199, 207, 217, 220, 228, 239, 242, 328, 329, 366, 374, 402, 404, 417,
426, 472, 512, 518, 528, 574, 575, 591, 594, 640, 647, 660, 679, 680, 688, 689, 691, 692, 693, 694, 696,
725, 801, 850, 851, 888}.
Mark Brittenham has computed new information regarding these knots. Some details are contained in
excerpts from his letter concerning the unknotting number: Mark Brittenham's notes.
10161
Ref. [8], [6]
11a28, 11a35, 11a36, 11a96, 11a164
Ribbon Knot: Personal communication with Christoph Lamm.
11a61, 11a304, 11n31
Alexander Stoimenow found a genus 1 concordance to an Alexander polynomial 1 knot, and thus the topological 4-genus
is at most 1.
11a316, 11a326, 11n4
Ribbon Knot: Personal communication with Alexander Stoimenow.
11n80, 12a_{187, 230, 0317, 450, 570, 908, 1185, 1189, 1208}, 12a787, 12n_{52, 63, 225, 542, 558,
665, 886} 12n{269, 505, 598, 602, 756}, 12n276
Ref. [4]
12a_{3, 54, 77, 100, 173, 183, 189, 211, 221, 245, 258, 279, 348, 348, 377, 425, 427, 435, 447, 456, 458, 464, 473,
477, 484, 484, 606, 646, 667, 715, 786, 819, 879, 887, 975, 979, 1011, 1019, 1019, 1029, 1034, 1083, 1087, 1105, 1105,
1119, 1202, 1202, 1225, 1225, 1269, 1277, 1283}
12n_{4, 19, 23, 24, 43, 48, 49, 51, 56, 57, 62, 66, 87, 106, 145, 170, 214, 256, 257, 268, 279, 288, 309, 312,
313, 318, 360, 380, 393, 394, 397, 399, 414, 420, 430, 440, 462, 501, 504, 553, 556, 582, 605, 636, 657, 670, 676, 702,
706, 708, 721, 768, 782, 802, 817, 838, 870, 876}
Alexander Stoimenow did a computer search to identify slice knots of 12 crossings and found these
(smooth) ribbon knots.
genus 1: 12a{441}, 12n{433,520,577,719}
genus 2: 12n{624}
Alexander Stoimenow did a computer search to identify knots of 12 crossings that have genus 1 or genus 2
concordances to knots of Alexander polynomial 1, and thus are of topological genus 1 or 2. That search found these examples.
12n113, 12n190, 12n233, 12n345, 12n707, 12n822, and
12n829
Ref. [1]
12n321 (1), 12n411 (1), 12n519 (1), 12n750 (2), 12n830 (2), and
12n293 (1)
Ref. [2]
Duncan McCoy used a computer search to find low genus surfaces in the four-ball bounded by knots and used the results
to determine the smooth four-genus of over 600 11 and 12 crossing knots. This also determined the topological four-genus
for many of these knots. See smooth four-genus references and [4] for more
details.
Herald-Kirk-Livingston have obstructed topological slicing of 16 of the remaining 18 possible slice knots of 12 or fewer
crossings and have found a slice disk for 12a990. Thus, the only remaining unknown case is 12a631.
12a631
12a1142
12n694
Axel Seeliger proved that this knot is slice. Diagram
has topological slice genus 2. This follows from [4], as pointed out be Stepan Orevkov. Diagram
Lukas Lewark points out that a single crossing change converts this knot into 10n38, which can be shown to have topological 4-genus 2 using the menthods of [9].
[1] Boileau, M., Boyer, S., and Gordon, C., "Branched covers of quasipositive links and L-spaces," Arxiv preprint.
[2] Feller, P., "The degree of the Alexander polynomial is an upper bound for the topological slice genus," Arxiv preprint.
[3] Fujino, Y., Miyazawa, Y., and Nakajima, K., "H(n)-unknotting nubmer of a knot," Reports of knots and low-dimensional manifolds (1997), 72-85.
[4] Lewark, L. and McCoy, D., "On calculating the slice genera of 11- and 12-crossing knots," Arxiv preprint.
[5] Murakami, H. and Nakanishi, Y., "Triple points and knot cobordism," Kobe J. Math, v 1. (1984), 1-16.
[6] Rasmussen, J., "Khovanov homology and the slice genus," Arxiv preprint.
[7] Shibuya, T., Memoirs of the Osaka Institute of Technology, Vol 45 (2000), 1-10.
[8] Tanaka, T., "Unknotting numbers of quasipositive knots," Top. Appl 88 (1998), 239-246.
[9] Sebastian Baader, Lukas Lewark, Filip Misev and Paula Truol, "3-braid knots with maximal 4-genus," Arxiv preprint.
