# Unknotting Number

The unknotting number of a knot is the minimal number of crossing changes required to convert a knot into the unknot. Literature on the unknotting number is extensive.

Many people contributed to the list of values of unknotting numbers presented here. References are given when published for individual knots.

Special mention should go to Slavik Jablan and Radmila Sazdanovic for the values of the unknotting numbers of 11 crossing knots. They did the initial calculations, developing lower bounds based on the nontriviality of the knots and the signature. Upper bounds were found using explicit calculations.

The lower bounds obtained by Jablan and Sazdanovic were found using the signature of the knot and the fact that these knots are nontrivial. According to the Bernhard-Jablan Conjecture (e.g. see [15]), a minimal sequence of unknotting crossing changes can be found from a sequence determined by particular diagrams, thus offering a conjectural algorithm for determining the unknotting number. The algorithm produces an unknotting sequence and thus gives an upper bound on the unknotting number, regardless of the truth of the conjecture.

## Specific Knots

74
Ref. [8]

84, 86, 88, 812, 95, 98, 931
Ref. [6]

810, 1048, 1052, 1054 (u=1 ruled out), 1057, 1058, 1064, 1068, 1070, 1077 (u=1 ruled out), 10110, 10112, 10116, 10117, 10125, 10126, 10130, 10135, 10138, 10158, 10162
Ref. [14]

816
Ref. [11], [16]

818, 937, 940, 946, 948, 949, 10103
Ref. [16]

910, 913, 935, 938, 1053, 10101, 10120
Ref. [13]

915, 917
Ref. [6], [16]

925
Ref. [7]

929
Ref. [3], [14]

108
Ref. [1]

1065, 1069, 1089, 10108, 10163, 10165 (Warning: these last two were listed as 10164 in the paper, not taking into account the duplication in early tables.)
Ref. [9]

1067
Ref. [19], [14]

1079
Ref. [14], u=1 ruled out by [3]

1081, 1087, 1090, 1093, 1094, 1096
Ref. [3], [14]

1083
Ref. [3], [14], [12]

1086
Ref. [16], [14], [3]

1097
Ref. [9], [12]

10105, 10106, 10109, 10121
Ref. [14], [16], [12]

10131
Ref. [16], [14]

10139, 10145, 10152
Ref. [2]

10148, 10151
Ref. [3], [14]

10153
Ref. [3]

10154
Ref. [18], [17], [2]

10161
Ref. [18], [2]

11n9, 11n16, 11n31, 11n77, 11n183
The lower bound comes from the Khovanov-Bar-Natan-Rasmussen invariant. The realization was done by direct calculation, carried out by Slavik Jablan and Radmila Sazdanivic. 11n77 also follows from [2].

11a362
This was shown by Kobayashi to have unknotting number greater than 1 in [7]. (Thanks to Josh Green for pointing out that 11a362 is the pretzel knot P(5,3,3), and thus falls to the work of Kobayashi.)

The work of Gordon and Luecke in [3] has now been applied to 11 crossing knots. Gordon summarizes as follows:
"... This rules out the following 75 11-crossing knots listed in Knotinfo as possibly having unknotting number 1:
11ak for k = 3, 14, 15, 17, 18, 19, 24, 25, 26, 27, 28, 29, 30, 38, 44, 47, 52, 54, 57, 66, 67, 68, 72, 76, 79, 81, 102, 115, 126, 130, 132, 141, 147, 151, 152, 156, 157, 173, 231, 232, 251, 252, 253, 254, 262, 265, 294, 316, 323, 347.
11nk for k = 4, 5, 6, 7, 11, 24, 32, 33, 36, 37, 40, 44, 46, 65, 66, 67, 68, 71, 73, 74, 75, 80, 97, 98, 99.
The knots 11ak for k= 44, 47, 57, 231, and 11nk for k= 71, 73, 74, 75, are Montesinos knots of length 4 so they were actually ruled out earlier by Motegi [10]."

11a292
This knot can be shown to have unknotting number three using results from Brendan Owens in [13].

(Note added 6 Feb, 2009) During the fall of 2008, Josh Greene reported that new results of his, based on a combination of tools coming from Heegaard-Floer theory and Donaldson's original restrictions on the intersection forms of smooth manifolds, are sufficient to rule out unknotting number 1 for the remaining 100 cases for 11 crossing alternating knots. While that work was in progress, Slaven Jabuka announced the resolution of several cases of unknotting number 1 [5]. Since then Eric Staron has, independently, also ruled out unknotting number 1 for most of the remaining cases of 11 crossing alternating knots. Details of that work are available directly from Eric, at the University of Texas.

11a123, 11n133, 12a311, 12a327, 12a386, 12a433, 12a561, 12a563, 12a569, 12a664, 12a683, 12a725, 12a780, 12a907, 12a921, 12a1194, 12n147, 12n494, 12n496, 12n626, 12n654.
Stefan Friedl and Maciej Borodzik have identified these knots as having unknotting number 3.

Mark Brittenham has computed new bounds and values for a large number of knots. A complete list, with further details of the techniques used in references, can be found in Mark Brittenham's notes.

11a192, 11a341, 11a360 11a365
Ref. [20]

11n162, 12n805, 12n814, 12n844, 12n856.
Ana Wright has shown these knots have unknotting number greater than 1 (which determines the unknotting number of 11n162 and constrains the others to be 2 or 3).

11n139 and 11n141 cannot have unknotting number 1. As identified by Alexander Stoimenow, these follows from [7]

12a554, 12a750.
Lukas Lewark points out that the unknotting number of these is bounded below by the Nakanishi index, 3.

## References

"The rational Witt class and the unknotting number of a knot,"

[1] Adams, C., The Knot Book, page 62.

[2] Gibson, W. and Ishikawa, M., "Links and gordian numbers assoicated with generic immersions of intervals," Topology and its Applications, 123 (2002), 609-636.

[4] Jablan, S. and Sazdanovic, R.

[5] Jabuka, S., "The rational Witt class and the unknotting number of a knot," Arxiv preprint.

[6] Kanenobu, T. and Murakami, H., "Two-bridge knots with unknotting number one," Proc. Amer. Math. Soc. 98 (1986), 499-502.

[7] Kobayashi, T., "Minimal genus Seifert surfaces for unknotting number 1 knots," Kobe J. Math. 6 (1989), 53-62.

[8] Lickorish, W. B. R., "The unknotting number of a classical knot," Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982), volume 44 of Cont. Math., 117-121.

[9] Miyazawa, Y., "The Jones polynomial of an unknotting number one knot," Top. App. 83 (1998), 161-167.

[10] Motegi, K., "A Note on Unlinking Numbers of Montesinos Links," Rev. Mat. Univ. Complut. Madrid 9 (1996), 151-164.

[11] Murakami, T. and Yasuhara, A., "Four Genus and four-dimensional clasp number of a knot," Proc. Amer. Math. Soc. 128 no. 12 (2000), 3693-3699.

[12] Nakanishi, Y., "A note on unknotting number. II." J. Knot Theory Ramifications 14 no. 1 (2005), 1, 3-8.

[13] Owens, B., "Unknotting information from Heegaard Floer homology," Adv. Math. 217 (2008), 2353-2376. Arxiv preprint.

[14] Ozsvath, P. and Szabo, Z., "Knots with unknotting number one and Heegaard Floer homology," Arxiv preprint.

[15] Stoimenow, A., "On unknotting numbers and knot trivadjacency," Mathematica Scandinavica 94(2) (2004), 227-248 (available online here).

[16] Stoimenow, A., "Polynomial values, the linking forms and unknotting numbers," Arxiv preprint.

[17] Stoimenow, A., "Positive knots, closed braids and the Jones polynomial," Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2(2) (2003), 237-285.

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

[19] Traczyk, P., "A criterion for signed unknotting number," Contemporary Math. 233 (1999), 212-220.

[20] Daemi, A. and Scaduto,C., "Chern-Simons functional, singular instantons, and the four-dimensional clasp number," Arxiv preprint.