Lickorish, W. B. R. The unknotting number of a classical knot, in Combinatorial methods in topology and algebraic geometry (Rocherster, N.Y., 1982), volume 44 of Cont. Math., 117-121.
Kanenobu, T. and Murakami, H. Two-bridge knots with unknotting number one, Proc. Amer. Math. Soc. 98 (1986), 499--502.
Kanenobu, T. and Murakami, H. Two-bridge knots with unknotting number one, Proc. Amer. Math. Soc. 98 (1986), 499--502.
Kanenobu, T. and Murakami, H. Two-bridge knots with unknotting number one, Proc. Amer. Math. Soc. 98 (1986), 499--502.
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Kanenobu, T. and Murakami, H. Two-bridge knots with unknotting number one, Proc. Amer. Math. Soc. 98 (1986), 499--502.
Murakami, T. and Yasuahara, A. Four Genus and four-dimensional clasp number of a knot, Proc. Amer. Math. Soc. 128 (2000) no. 12. 3693--3699.Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076, accepted by Mathematical Research Letters.
Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076.)
Kanenobu, T. and Murakami, H. Two-bridge knots with unknotting number one, Proc. Amer. Math. Soc. 98 (1986), 499--502.
Kanenobu, T. and Murakami, H. Two-bridge knots with unknotting number one, Proc. Amer. Math. Soc. 98 (1986), 499--502.
Owens, B. Unknotting information from Heegaard Floer homology, arxiv.org/math.GT/0506485.
Owens, B. Unknotting information from Heegaard Floer homology, arxiv.org/math.GT/0506485.
Kanenobu, T. and Murakami, H. Two-bridge knots with unknotting number one, Proc. Amer. Math. Soc. 98 (1986), 499--502.Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076, accepted by Mathematical Research Letters.
Kanenobu, T. and Murakami, H. Two-bridge knots with unknotting number one, Proc. Amer. Math. Soc. 98 (1986), 499--502. Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076, accepted by Mathematical Research Letters.
Kobayashi, T. Minimal genus Seifert surfaces for unknotting number $1$ knots, Kobe J. Math. 6 (1989), 53--62.
Gordon, C. McA. and Lueke, J, arxiv.org/abs/math.GT/0601265. Also Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Kanenobu, T. and Murakami, H. Two-bridge knots with unknotting number one, Proc. Amer. Math. Soc. 98 (1986), 499--502.
Owens, B. Unknotting information from Heegaard Floer homology, arxiv.org/math.GT/0506485.
Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076, accepted by Mathematical Research Letters.
Owens, B. Unknotting information from Heegaard Floer homology, arxiv.org/math.GT/0506485.
Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076, accepted by Mathematical Research Letters.
Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076, accepted by Mathematical Research Letters.
Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076, accepted by Mathematical Research Letters.
Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076, accepted by Mathematical Research Letters.
Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076, accepted by Mathematical Research Letters.
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Owens, B. Unknotting information from Heegaard Floer homology, arxiv.org/math.GT/0506485.
u=1 ruled out by: Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Miyazawa, Y. The Jones polynomial of an unknotting number one knot, Top. App. 83 (1998), 161-167.
Traczyk, P. A criterion for signed unknotting number, Contemporary Math. 233 (1999), 212-220, and Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Miyazawa, Y. The Jones polynomial of an unknotting number one knot, Top. App. 83 (1998), 161-167.
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
u=1 ruled out by: Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
u=1 ruled out by: Gordon, C. McA. and Lueke, J, arxiv.org/abs/math.GT/0601265. Also Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Gordon, C. McA. and Lueke, J, arxiv.org/abs/math.GT/0601265. Also Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Gordon, C. McA. and Lueke, J, arxiv.org/abs/math.GT/0601265. Also Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426. See also, Nakanishi, Y. A note on unknotting number. II. J. Knot Theory Ramifications 14 (2005), no. 1, 3--8.
Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076, accepted by Mathematical Research Letters. Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426 and also Gordon, C. McA. and Lueke, J, arxiv.org/abs/math.GT/0601265.
Gordon, C. McA. and Lueke, J, arxiv.org/abs/math.GT/0601265. Also Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Miyazawa, Y. The Jones polynomial of an unknotting number one knot, Top. App. 83 (1998), 161-167.
Gordon, C. McA. and Lueke, J, arxiv.org/abs/math.GT/0601265. Also Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Gordon, C. McA. and Lueke, J, arxiv.org/abs/math.GT/0601265. Also Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Gordon, C. McA. and Lueke, J, arxiv.org/abs/math.GT/0601265. Also Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Gordon, C. McA. and Lueke, J, arxiv.org/abs/math.GT/0601265. Also Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Miyazawa, Y. The Jones polynomial of an unknotting number one knot, Top. App. 83 (1998), 161-167. See also, Nakanishi, Y. A note on unknotting number. II.
Owens, B. Unknotting information from Heegaard Floer homology, arxiv.org/math.GT/0506485.
Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076, accepted by Mathematical Research Letters.
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426. Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076, accepted by Mathematical Research Letters. See also, Nakanishi, Y. A note on unknotting number. II.
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426. Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076, accepted by Mathematical Research Letters. See also, Nakanishi, Y. A note on unknotting number. II.
Miyazawa, Y. The Jones polynomial of an unknotting number one knot, Top. App. 83 (1998), 161-167.
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426. Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076, accepted by Mathematical Research Letters. See also, Nakanishi, Y. A note on unknotting number. II.
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426. Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076, accepted by Mathematical Research Letters.
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Owens, B. Unknotting information from Heegaard Floer homology, arxiv.org/math.GT/0506485.
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426. Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076, accepted by Mathematical Research Letters. See also, Nakanishi, Y. A note on unknotting number. II.
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Stoimenow, A. Polynomial values, the linking forms and unknotting numbers, arxiv.org/abs/math.GT/0405076, accepted by Mathematical Research Letters. Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426.
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
refs to be posted soon
refs to be posted soon
Gordon, C. McA. and Lueke, J, arxiv.org/abs/math.GT/0601265. Also Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Gordon, C. McA. and Lueke, J, arxiv.org/abs/math.GT/0601265. Also Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
refs to be posted soon
Gordon, C. McA. and Lueke, J, arxiv.org/abs/math.GT/0601265.
Tanaka, T, unknotting numbes of quasipositive knots, Top. Appl 88 (1998) 239-246. A. Stoimenow: Positive knots, closed braids and the Jones polynomial Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2(2) (2003), 237-285.
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Tanaka, T, unknotting numbes of quasipositive knots, Top. Appl 88 (1998) 239-246.
Ozsvath, P. and Szabo, Z. Knots with unknotting number one and Heegaard Floer homology, arxiv.org/math.GT/0401426
Miyazawa, Y. The Jones polynomial of an unknotting number one knot, Top. App. 83 (1998), 161-167. (Warning, this was listed as 10_164 in the paper, not taking into account the duplication in early tables.)
Miyazawa, Y. The Jones polynomial of an unknotting number one knot, Top. App. 83 (1998), 161-167. (Warning, this was listed as 10_164 in the paper, not taking into account the duplication in early tables.)
11n_9, 11n_16, 11n_31, 11n_77, 11n_183,
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.
11a_362 was shown by Kobayashi to have unknotting number greater than 1 in [T. Kobayashi, Minimal genus Seifert surfaces for unknotting number 1 knots, Kobe J Math 6 (1989) 53-62]. (Thanks to Josh Green for pointing out that 11a_362 is the pretzel knot P(5,3,3), and thus falls to the work of Kobayashi.)
The work of Gordon and Luecke arxiv.org/abs/math.GT/0601265 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: 11a k 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. 11n k 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 11a k for k= 44, 47, 57, 231, and 11n k for k= 71, 73, 74, 75, are Montesinos knots of length 4 so they were actually ruled out earlier by Motegi (Rev. Mat. Univ. Complut. Madrid 9 (1996))."
(Note added 6 Feb, 2009) During the fall, 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 intresection forms of smooth manifolds, are sufficient to rule out unknotting number 1 for the remaining 100 cases for 11 alternating crossing knots. While that work was in progress, Slaven Jabuka announced the resolution of several cases of unknotting number 1. Since then Eric Staron has, independently, also ruled out unknotting number 1 for most of the remaining cases of 11 alternating crossing knots. Details of that work are available directly from Eric, at the University of Texas.