References for Smooth 4-genus


Some values of the 4-genus can be deduced from the concordance genus references.
8_16

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


8_18

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


9_17

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


9_31

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


9_32

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


9_40

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


9_47

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


9_48

Fujino, Y. Miyazawa, Y. and Nakajima: K. H(n)-unknotting nubmer of a knot, Reports of knots and low-dimensional manifolds, (1997) 72-85.


10_51

Selahi Durusoy has found a single crossing change (in the given diagram) that converts 10_51 into 8_8, which is slice.


10_54

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


10_70

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


10_97

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


10_117

Fujino, Y. Miyazawa, Y. and Nakajima: K. H(n)-unknotting nubmer of a knot, Reports of knots and low-dimensional manifolds, (1997) 72-85.


10_139

Kawamura, T. On unknotting nubmers and four-dimensional clasp numbers of links, Ph.D. Thesis, Univ of Tokyo (2000). Kawamura, T. The unknotting nubmers of 10_139 and 10_152 are 4, Osaka J. Math. 35 (1998) 539-546. See also, Gibson, W. and Ishikawa, M. Links and gordian numbers assoicated with generic immersions of intervals, Topology and its Applications, 123 (2002) 609-636. See also recent proof by Rasmussen, J. Khovanov homology and the slice genus, alrxiv.org/math.GT/0402131Reportsreprot of knots and low-dimensional manifolds, (1997) 72-85.


10_144

Fujino, Y. Miyazawa, Y. and Nakajima: K. H(n)-unknotting nubmer of a knot, Reports of knots and low-dimensional manifolds, (1997) 72-85.


10_145

Tanaka, T. Unknotting numbers of quasipositive knots, Top. Appl 88 (1998) 239-246. See also, Gibson, W. and Ishikawa, M. Links and gordian numbers assoicated with generic immersions of intervals, Topology and its Applications, 123 (2002) 609-636. Recent proof by Rasmussen, J. Khovanov homology and the slice genus, alrxiv.org/math.GT/0402131


10_148

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


10_151

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


10_152

Kawamura, T. The unknotting nubmers of 10_139 and 10_152 are 4, Osaka J. Math. 35 (1998) 539-546. A. Stoimenow: Positive knots, closed braids and the Jones polynomial Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2(2) (2003), 237-285. A'Campo, N. Generic immersions of curves, knots, monodromy and gordian numbers, Inst. Hautes Etudes Sci. Bubl. Math. 88 (1998), 151-169. Rasmussen, J. Khovanov homology and the slice genus, alrxiv.org/math.GT/0402131.


10_154

Tanaka, T. Unknotting numbes of quasipositive knots, Top. Appl 88 (1998) 239-246. See also, Gibson, W. and Ishikawa, M. Links and gordian numbers assoicated with generic immersions of intervals, Topology and its Applications, 123 (2002) 609-636. Recent proof by Rasmussen, J. Khovanov homology and the slice genus, alrxiv.org/math.GT/0402131


10_161

Tanaka, T. Unknotting numbes of quasipositive knots, Top. Appl 88 (1998) 239-246. See also, Gibson, W. and Ishikawa, M. Links and gordian numbers assoicated with generic immersions of intervals, Topology and its Applications, 123 (2002) 609-636. Recent proof by Rasmussen, J. Khovanov homology and the slice genus, alrxiv.org/math.GT/0402131


11a28

Ribbon Knot: Personal communication with Christoph Lamm.


11a35

Ribbon Knot: Personal communication with Christoph Lamm.


11a36

Ribbon Knot: Personal communication with Christoph Lamm.


11a96

Ribbon Knot: Personal communication with Christoph Lamm.


11a_164

Ribbon Knot: Personal communication with Christoph Lamm.


11a316

Ribbon Knot: Personal communication with Alexander Stoimenow.


11a326

Ribbon Knot: Personal communication with Alexander Stoimenow.


11n4

Ribbon Knot: Personal communication with Alexander Stoimenow.


Lisa Piccirillo proved that the Conway knot 11n34 is not smoothly slice, and thus has four-genus 1: Arxiv preprint
Alexander Stoinenow has done a computer search to identify slice knots of 12 crossings. That search has found the following ribbon knots:

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}


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. Thus, the only remaining unknown case of a possibly smooth slice knot is 12a631.


Axel Seeliger proved that 12a631 is slice. Diagram



Duncan McCoy has done extensive computer searches to find low genus surfaces bounded by knots and has resolved the four-genus of the following 11 and 12 crossing knots. See Arxiv preprint for details.

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}

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}
Steve Boyer, Cameron Gordon, and Michel Boileau, have informed us that in an upcoming paper they determine the smooth and topological four-genus of the following knots: 12n113 12n190,12n233, 12n345, 12n707, 12n822, 12n829.
Lukas Lewark and Duncan Mccoy have provided the smooth four-genus of 2 for the knots 12a0787 and 12n{269, 505, 598, 602, 756}, using Larry Taylor's lower bound.
Lisa Piccirillo proved that the Conway knot 11n34 is not smoothly slice, and thus has four-genus 1: Arxiv preprint