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.

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

8_{16}, 8_{18}, 9_{17}, 9_{31}, 9_{32}, 9_{40}, 9_{47}

Ref. [8]

9_{48}, 10_{117}, 10_{144}

Ref. [3]

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}, 10_{70}, 10_{97}, 10_{148}, 10_{151}

Ref. [10]

10_{139}

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

10_{145}, 10_{154}, 10_{161}

Ref. [13], [4], [12]

101_{52}

Ref. [6], [11], [1], [12]

11a_{28}, 11a_{35}, 11a_{36}, 11a_{96}, 11a_{164}

Ribbon Knot: Personal communication with Christoph Lamm.

11a_{316}, 11a_{326}, 11n_{4}

Ribbon Knot: Personal communication with Alexander Stoimenow.

11n_{34}

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 12a_{990}.

12a_{631}

Axel Seeliger proved that this knot is slice. Diagram

12n_{113}, 12n_{190}, 12n_{233}, 12n_{345}, 12n_{707}, 12n_{822}, and
12n_{829}

Ref. [2]

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.

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