Smooth Four-Genus

The smooth 4-genus of a knot is the minimum genus of a smooth surface embedded in the 4-ball with boundary the knot. Bounds are determined by the p-signatures and, to avoid being of 4-genus 0 (slice), the Alexander polynomial. In addition, there are bounds determined by gauge theoretic invariants, which apply only in the smooth category. The knots for which these smooth techniques are required are marked in the table with reference links.

Specific Knots

Some values of the 4-genus can be deduced from the concordance genus references.

816, 818, 917, 931, 932, 940, 947
   Ref. [8]

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. [10]

10139
   Ref. [5], [6], [4], [12], [3]

10145, 10154, 10161
   Ref. [13], [4], [12]

10152
   Ref. [6], [11], [1], [12]

11a28, 11a35, 11a36, 11a96, 11a164
   Ribbon Knot: Personal communication with Christoph Lamm.

11a316, 11a326, 11n4
   Ribbon Knot: Personal communication with Alexander Stoimenow.

11n34
   Lisa Piccirillo proved that this Conway knot is not smoothly slice and thus has four-genus 1 [9].

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} and
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 Stoinenow did a computer search to identify slice knots of 12 crossings and found these ribbon knots.

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 smooth slice disk for 12a990.

12a631
   Axel Seeliger proved that this knot is slice. Diagram

11a_{1, 3, 4, 6, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 29, 30, 32, 33, 37, 38, 39, 42, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 59, 60, 61, 63, 64, 65, 66, 67, 68, 72, 75, 76, 79, 81, 83, 84, 85, 89, 90, 92, 93, 97, 99, 102, 105, 107, 108, 109, 110, 111, 118, 119, 125, 126, 128, 130, 131, 132, 133, 134, 135, 137, 141, 144, 145, 147, 148, 151, 152, 153, 154, 155, 156, 157, 158, 159, 161, 162, 163, 166, 170, 171, 172, 173, 174, 175, 176, 178, 181, 183, 185, 188, 193, 197, 199, 202, 205, 211, 214, 217, 218, 219, 221, 228, 229, 231, 232, 239, 248, 249, 251, 252, 253, 254, 258, 262, 265, 268, 269, 270, 271, 273, 274, 277, 278, 279, 281, 284, 285, 288, 293, 294, 296, 297, 301, 303, 304, 305, 312, 313, 314, 315, 317, 322, 323, 324, 325, 327, 331, 332, 333, 346, 347, 349, 350, 352},
11n_{3, 5, 6, 7, 11, 15, 17, 23, 24, 29, 30, 32, 33, 36, 40, 44, 46, 51, 54, 58, 60, 65, 66, 79, 91, 92, 94, 98, 99, 102, 112, 113, 115, 117, 119, 120, 127, 128, 129, 133, 137, 138, 140, 142, 146, 148, 150, 155, 157, 160, 161, 162, 163, 165, 166, 167, 168, 170, 173, 177, 178, 179, 182},
12a_{4, 10, 39, 45, 49, 50, 65, 66, 75, 76, 86, 89, 103, 104, 108, 120, 125, 127, 128, 129, 135, 147, 148, 150, 160, 161, 163, 164, 166, 167, 168, 175, 177, 178, 181, 193, 194, 195, 196, 200, 204, 212, 231, 244, 247, 255, 259, 260, 265, 289, 291, 292, 296, 298, 302, 311, 312, 327, 338, 339, 342, 353, 354, 357, 364, 370, 372, 375, 376, 379, 380, 381, 395, 396, 399, 400, 412, 413, 414, 423, 424, 434, 436, 438, 448, 449, 454, 459, 462, 463, 465, 468, 481, 482, 489, 493, 494, 496, 503, 505, 534, 542, 544, 545, 549, 554, 564, 580, 581, 582, 597, 598, 601, 609, 621, 634, 639, 642, 643, 644, 649, 665, 668, 669, 677, 680, 684, 687, 689, 690, 691, 692, 693, 704, 706, 719, 725, 730, 735, 741, 749, 750, 752, 757, 767, 769, 771, 783, 784, 789, 791, 810, 812, 815, 816, 818, 824, 825, 826, 827, 833, 835, 841, 842, 845, 852, 853, 862, 870, 871, 873, 878, 886, 895, 896, 898, 899, 901, 908, 911, 912, 914, 916, 921, 939, 940, 941, 942, 957, 967, 971, 981, 983, 988, 989, 999, 1000, 1012, 1014, 1016, 1025, 1028, 1039, 1040, 1050, 1061, 1066, 1085, 1095, 1103, 1109, 1110, 1115, 1116, 1118, 1124, 1127, 1138, 1142, 1145, 1147, 1148, 1149, 1150, 1151, 1160, 1163, 1165, 1171, 1174, 1175, 1179, 1185, 1194, 1200, 1201, 1205, 1226, 1254, 1256, 1259, 1275, 1278, 1279, 1281, 1282, 1284, 1285, 1286, 1288}, and
12n_{47, 60, 61, 75, 80, 84, 92, 101, 109, 113, 115, 116, 118, 137, 140, 147, 157, 159, 167, 171, 176, 190, 192, 193, 197, 200, 202, 204, 206, 208, 211, 212, 216, 219, 227, 233, 236, 247, 248, 253, 258, 260, 267, 270, 291, 304, 307, 324, 334, 345, 351, 359, 376, 379, 383, 388, 391, 396, 409, 410, 411, 439, 441, 442, 443, 451, 454, 456, 460, 469, 475, 480, 489, 495, 496, 500, 514, 519, 520, 522, 524, 525, 531, 532, 537, 543, 554, 564, 569, 577, 583, 595, 596, 601, 606, 608, 621, 626, 630, 631, 672, 673, 675, 678, 681, 685, 698, 699, 700, 701, 707, 717, 726, 730, 734, 735, 737, 742, 759, 769, 777, 783, 794, 796, 797, 804, 805, 808, 809, 811, 813, 814, 815, 818, 822, 824, 829, 833, 844, 846, 854, 855, 856, 859, 861, 862, 863, 867, 869, 873, 875}.
   Duncan McCoy has done extensive computer searches to find low genus surfaces bounded by knots and has resolved the four-genus of these 11 and 12 crossing knots. See [7] for details.

12n113, 12n190, 12n233, 12n345, 12n707, 12n822, and 12n829
   Ref. [2]

12a0787, 12n{269, 505, 598, 602, 756}
   Lukas Lewark and Duncan McCoy have provided the smooth four-genus of 2 for these knots using Larry Taylor's lower bound.

11n34
   Lisa Piccirillo proved that this Conway knot is not smoothly slice and thus has four-genus 1 [9].

References

[1] A'Campo, N., "Generic immersions of curves, knots, monodromy and gordian numbers," Inst. Hautes Etudes Sci. Bubl. Math. 88 (1998), 151-169.

[2] Boileau, M., Boyer, S., and Gordon, C., "Branched covers of quasipositive links and L-spaces," Arxiv preprint.

[3] Fujino, Y., Miyazawa, Y., and Nakajima, K., "H(n)-unknotting number of a knot," Reports of knots and low-dimensional manifolds (1997), 72-85.

[4] Gibson, W. and Ishikawa, M., "Links and gordian numbers associated with generic immersions of intervals," Topology and its Applications, 123 (2002), 609-636.

[5] Kawamura, T., On unknotting numbers and four-dimensional clasp numbers of links, Ph.D. Thesis, University of Tokyo (2000).

[6] Kawamura, T., "The unknotting numbers of 10139 and 10152 are 4," Osaka J. Math. 35 (1998), 539-546.

[7] Lewark, L. and McCoy, D., "On calculating the slice genera of 11- and 12-crossing knots," Arxiv preprint.

[8] Murakami, H. and Nakanishi, Y., "Triple points and knot cobordism," Kobe J. Math, v 1. (1984), 1-16.

[9] Piccirillo, L., "The Conway knot is not slice," Arxiv preprint.

[10] Shibuya, T., Memoirs of the Osaka Institute of Technology, Vol 45 (2000) 1-10.

[11] Stoimenow, A., "Positive knots, closed braids and the Jones polynomial," Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2(2) (2003), 237-285.

[12] Rasmussen, J., "Khovanov homology and the slice genus," Arxiv preprint.

[13] Tanaka, T., "Unknotting numbers of quasipositive knots," Top. Appl 88 (1998), 239-246.