Knot Table: Smooth Four-Genus

Specific Knots

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

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].

12a_{153}, 12n_{{239, 512}}
The lower bounds of 2 were established by Brittenham and Hermiller See [15].

11n₈₀, 12a_{{187, 230, 317, 450, 570, 624, 636, 905, 1189, 1208}}, 12n_{{52, 63, 225, 555, 558, 665, 886}}
These have four-genus 1, as a consequences of the work of Brittenham and Hermiller [15]. (Thank you to Lukas Lewark for noting these examples.) See also the work of Karageorghis and Swenton in [14].

11n₄₅, 11n₁₄₅
These have four-genus 1 (smoothly and topologically), as a proved by Julia Collins [16].

13 CONTENT

Initial estimates of the smooth four-genus were found be Chuck Livingston, as follows. Ciprian Manolescu provided a list of 13 crossing ribbon knots that was based on his work in [18] and work of Dunfield-Gong; that work included a proof that all but a few of the remaining knots are not slice. (The initial list of possibly slice knots was 13n_{65, 866, 3727, 3871, 3872, 3897, 3936, 4582} but using techniques of Piccirillo, 13n_866 was shown not to be slice.) Invariants such as the signature function and Heegaard Floer invariants provided lower bounds on the four-genus. Crossing changes were used to build singular concordances to knots of known four-genus (via the work of Brittenham-Hermiller [15]) and these could be used to build bounding surfaces.

Alexander Stoimenow extended that work. To improve some of the upper bounds, he constructed concordances to knots of known four-genus, including those in the Brittenham-Hermiller work [15]. By applying obstructions derived from gauge theoretic methods to find lower bounds, including results of Lewark-McCoy [7]. References to his earlier work include [17].

The following have smooth four-genus 1.

13a_{12, 19, 51, 97, 113, 119, 124, 175, 233, 238, 268, 285, 334, 338, 339, 365, 382, 424, 444, 489, 522, 562, 579, 582, 598, 619, 630, 652, 656, 681, 701, 800, 862, 863, 906, 916, 980, 998, 1005, 1017, 1041, 1069, 1084, 1248, 1262, 1294, 1302, 1346, 1361, 1405, 1406, 1433, 1442, 1451, 1492, 1538, 1542, 1548, 1549, 1553, 1644, 1666, 1679, 1747, 1788, 1887, 1888, 2102, 2160, 2219, 2250, 2251, 2300, 2319, 2327, 2328, 2362, 2386, 2396, 2410, 2413, 2422, 2450, 2451, 2465, 2511, 2550, 2553, 2591, 2703, 2769, 2795, 2861, 2882, 2886, 2917, 2928, 2938, 2959, 2978, 2994, 3002, 3004, 3038, 3044, 3159, 3161, 3166, 3175, 3179, 3181, 3195, 3198, 3218, 3257, 3274, 3287, 3370, 3389, 3421, 3451, 3452, 3456, 3537, 3573, 3607, 3679, 3694, 3695, 3701, 3706, 3776, 3795, 3808, 3813, 3821, 3832, 3919, 3977, 4014, 4030, 4072, 4102, 4115, 4116, 4139, 4141, 4154, 4169, 4208, 4236, 4255, 4276, 4278, 4294, 4337, 4338, 4373, 4401, 4433, 4481, 4513, 4515, 4533, 4541, 4542, 4573, 4581, 4634, 4700, 4738, 4755, 4785, 4793, 4811, 4816, 4843, 4856, 4873}

13n_{31, 41, 57, 81, 102, 112, 168, 180, 197, 199, 264, 311, 320, 367, 494, 545, 632, 717, 828, 869, 879, 938, 1032, 1089, 1098, 1111, 1116, 1122, 1126, 1152, 1228, 1233, 1246, 1285, 1312, 1317, 1350, 1351, 1388, 1425, 1429, 1431, 1439, 1458, 1475, 1482, 1567, 1575, 1596, 1667, 1670, 1672, 1691, 1695, 1710, 1771, 1791, 1837, 1857, 1911, 1917, 1962, 2008, 2039, 2042, 2064, 2066, 2080, 2164, 2165, 2256, 2268, 2301, 2372, 2378, 2383, 2388, 2453, 2456, 2463, 2486, 2519, 2530, 2542, 2558, 2570, 2575, 2611, 2613, 2619, 2658, 2667, 2682, 2712, 2820, 2825, 2832, 2844, 2847, 2853, 2869, 2915, 2924, 2929, 2949, 2991, 3083, 3089, 3101, 3125, 3153, 3197, 3216, 3226, 3228, 3295, 3330, 3334, 3376, 3388, 3447, 3455, 3470, 3579, 3595, 3622, 3678, 3721, 3724, 3728, 3758, 3760, 3797, 3801, 3802, 3806, 3819, 3824, 3825, 3828, 3834, 3836, 3846, 3868, 3889, 3918, 3929, 3949, 3967, 3968, 3977, 4040, 4043, 4054, 4073, 4090, 4092, 4106, 4118, 4126, 4132, 4152, 4175, 4204, 4209, 4221, 4226, 4269, 4270, 4297, 4300, 4343, 4366, 4371, 4406, 4410, 4419, 4425, 4463, 4471, 4483, 4527, 4567, 4594, 4628, 4649, 4650, 4671, 4698, 4702, 4715, 4717, 4724, 4727, 4775, 4796, 4809, 4835, 4852, 4854, 4861, 4887, 4891, 4914, 4926, 4928, 4939, 4948, 4957, 4995, 5022, 5026, 5030, 5036, 5060, 5062, 5097, 5110}

The following have smooth four-genus 1.

13a_{2304}, 13a_{4427} (Alexander Stiomenow, May 10, 2024, October 22, 2024).

The following have smooth four-genus 2.

13a_{5, 17, 109, 146, 159, 473, 720, 793, 820, 1020, 1099, 1150, 1259, 1332, 1419, 1476, 1544, 1575, 1581, 1609, 1784, 1868, 1901, 1986, 2079, 2143, 2151, 2181, 2228, 2233, 2372, 2377, 2436, 2507, 2604, 2607, 2667, 2711, 2745, 2822, 2875, 2897, 3105, 3118, 3122, 3157, 3174, 3484, 3504, 3513, 3595, 3597, 3691, 3692, 3693, 3930, 4005, 4034, 4088, 4106, 4122, 4191, 4196, 4216, 4225, 4233, 4295, 4304, 4422, 4454, 4458, 4493, 4531, 4554, 4705, 4727, 4778, 4781, 4807, 4857, 4877}

13n_{152, 365, 492, 1420, 1478, 1524, 2335, 2612, 2681, 2689, 2729, 3263, 3639, 3720, 3729, 4381, 4497, 4623, 4648, 4672, 4879, 4964}

13n_{46, 128, 391, 839, 993, 1345, 1846, 3251, 3811 3853, 3881, 4029, 4360, 4456} (Alexander Stiomenow, May 10, 2024).

13n_3727 has been shown not to be slice by Dunfield-Gong.

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.

[14] Karageorghis, L. and Swenton, F., "Determining the doubly slice genera of primes knots with up to 12 crossings, " Arxiv preprint.

[15] Brittetnham, M., Hermiller, S., "The smooth 4-genus of (the rest of) the prime knots through 12 crossings , " Arxiv preprint.

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

[17] Stoimenow, A., "Some inequalities between knot invariants," Internat.~J.~Math. {\bf 13(4)} (2002), 373--393.

[18] Gukov, S., Halverson, J., Manolescu, C., Ruehle, F., "Searching for ribbons with machine learning," Arxiv preprint. https://arxiv.org/abs/2304.09304