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]

[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 10_{139} and 10_{152} 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.