Every knot in R^3 is the cross-section of an embedded surface F in R^4 where F has the property that it bounds an unknotted solid handlebody in R^4. For example, if G is a genus g Seifert surface for K, then G x [-1 , 1] is an unknotted solid handlebody in R^4 with boundary as desired. Its boundary is of genus 2g.
The double slice genus of K, gds(K) is the minumum genus among all such surfaces F. In the case that gds(K) = 0, K is called doubly slice. Clearly 0 ≤ gds(K) ≤ 2g3(K). References [2,3] provide background and initial work studying the double slice genus. The most extensive work and the data here was provided by Karageorghis and Swenton, published in [1].
Knots with double slice genus 0 are called doubly slice. There is extensive literature on this topic which we do not cite here.
The values for the double slice genus of 13 crossing knots was provided by Malcolm Gabbard.
12n{553}, 12n{556} have double slice genus 1. Ref. [5].
The following are all 2 by Ref. [6]. 9_37, 10_74, 11n_148, 12a_554, 12a_896, 12a_921, 12a_1050, 12n_554 to be 2.
The following are all 2 by Ref. [6]. 13a{30, 94, 140, 189, 319, 343, 361, 395, 398, 405, 447, 557, 689, 797, 815, 909, 919, 923, 1080, 1092, 1100, 1200, 1238, 1326, 1380, 1446, 1565, 1633, 1672, 1738, 1781, 1806, 1816, 1909, 1936, 1959, 1971, 1993, 2015, 2024, 2196, 2423, 2446, 2484, 2715, 2720, 2727, 2800, 2801, 2898, 2913, 3043, 3242, 3249, 3258, 3333, 3423, 3476, 3641, 3763, 3911, 3986, 4074, 4130, 4173, 4178, 4474, 4497, 4585, 4677, 4730, 4745, 4749}
The following are all 2 by Ref. [6]. 13n{138, 323, 353, 404, 416, 435, 493, 527, 560, 563, 571, 590, 598, 619, 710, 715, 728, 794, 819, 837, 840, 844, 921, 942, 960, 963, 968, 984, 985, 990, 997, 1010, 1039, 1045, 1055, 1062, 1073, 1133, 1134, 1193, 1213, 1215, 1272, 1286, 1306, 1346, 1368, 1370, 1466, 1467, 1525, 1542, 1551, 1552, 1568, 1570, 1585, 1587, 1686, 1689, 1693, 1700, 1848, 1901, 1946, 1957, 1970, 1994, 2004, 2010, 2068, 2072, 2093, 2134, 2145, 2181, 2182, 2212, 2249, 2296, 2331, 2343, 2380, 2381, 2411, 2459, 2472, 2488, 2498, 2499, 2504, 2533, 2585, 2597, 2676, 2683, 2757, 2758, 2764, 2850, 2863, 2892, 2932, 2953, 2956, 2968, 2976, 3044, 3116, 3130, 3155, 3170, 3265, 3266, 3282, 3287, 3304, 3331, 3399, 3400, 3412, 3449, 3458, 3463, 3524, 3526, 3542, 3564, 3573, 3574, 3658, 3660, 3661, 3663, 3665, 3669, 3719, 3752, 3769, 3782, 3790, 3791, 3833, 3856, 3857, 3863, 3864, 3896, 3902, 3923, 3976, 3985, 4025, 4068, 4339, 4370, 4416, 4434, 4510, 4549, 4564, 4571, 4602, 4620, 4627, 4645, 4699, 4719, 4737, 4764, 4798, 4802, 4935, 5109}
[1] Karageorghis, C. and Swenton, F., "Determining the doubly slice genera of prime knots with up to 12 crossings." Arxiv preprint.
[2] Livingston, C. and Meier, J., "Doubly slice knots with low crossing number," New York Journal of Math., 21 (2015), 1007-1026.
[3] McDonald, C., "Band number and the double slice genus," New York Journal of Math., 25 (2019), 964-974.
[4] Brittetnham, M., Hermiller, S., "The smooth 4-genus of (the rest of) the prime knots through 12 crossings ," Arxiv preprint.
[5] Brejevs, V. and Feller, P., "The twisting number of a ribbon knot is bounded below by its doubly slice genus," Arxiv preprint.
[6] Hans U. Boden, Ceyhun Elmacioglu, Anshul Guha, Homayun Karimi, William Rushworth, Yun-chi Tang, and Bryan Wang Peng Jun, "On knots that divide ribbon knotted surfaces," Arxiv preprint.