References for Topological 4-genus

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_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_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_161

Tanaka, T. Unknotting numbes of quasipositive knots, Top. Appl 88 (1998) 239-246. 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.

11a61

Alexander Stoimenow has found a genus 1 concordance to an Alexander polynomial 1 knot, and thus the topological 4-genus is at most 1.

11a96

Ribbon Knot: Personal communication with Christoph Lamm.

11a164

Ribbon Knot: Personal communication with Christoph Lamm.

11a304

Alexander Stoimenow has found a genus 1 concordance to an Alexander polynomial 1 knot, and thus the topological 4-genus is at most 1.

11a316

Ribbon Knot: Personal communication with Alexander Stoimenow.

11a326

Ribbon Knot: Personal communication with Alexander Stoimenow.

11n4

Ribbon Knot: Personal communication with Alexander Stoimenow.

11n31

Alexander Stoimenow has found a genus 1 concordance to an Alexander polynomial 1 knot, and thus the topological 4-genus is at most 1.

Alexander Stoinenow has done a computer search to identify slice knots of 12 crossings. That search has found the following (smooth) 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}

Alexander Stoinenow has done a computer search to identify knots of 12 crossings that have genus 1 or genus 2 concordances to knots of Alexander polynomial 1, and thus are of topological genus 1 or 2. That search has found the following examples.

genus 1: 12a_{441}, 12n{433,520,577,719}

genus 2: 12n{624}

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 slice disk for 12a990. Thus, the only remaining unknown case is 12a631.