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.

References

[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.

[3] Gordon, C. McA. and Lueke, J., "Knots with unknotting number 1 and essential Conway spheres," Arxiv preprint.

[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.