# Topological Four-Genus

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.

## Specific Knots

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.

12n694
Lukas Lewark points out that a single crossing change converts this knot into 10n_38 , which can be shown to have topological 4-genus 2 using the menthods of [9].

11n45, 11n145
These have four-genus 1 (smoothly and topologically), as a proved by Julia Collins [10].

## References

[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 and Lukas Lewark and Filip Misev and Paula TruĂ¶l, "3-braid knots with maximal 4-genus," Arxiv preprint

[10] Collins, J., "On the concordance orders of knots," Arxiv preprint.

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