On unknotting operations of two-bridge knots. (English) Zbl 0726.57009
In this paper we completely classify, up to homeomorphism, the forms of unknotting operations of a two-bridge knot with unknotting number one. We show that some class of two-bridge knots has two different unknotting operations.
Reviewer: K.Taniyama
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
References:
| [1] | Bleiler, S., Scharlemann, M.: Tangles, propertyP, and a problem of J. Martin. Math. Ann.273, 215-225 (1986) · Zbl 0563.57002 · doi:10.1007/BF01451402 |
| [2] | Brody, E.J.: The topological classification of the lens spaces. Ann. Math.71, 163-184 (1960) · Zbl 0119.18901 · doi:10.2307/1969884 |
| [3] | Burde, G., Zieschang, H.: Knots. De Gruyter studies in Math. 5. Berlin New York: De Gruyter 1985 · Zbl 0568.57001 |
| [4] | Conway, J.H.: An enumeration of knots and links, and some of their algebraic properties. computational problems in abstract algebra, pp. 329-358. Oxford New York: Pergamon Press 1969 |
| [5] | Culler, M., Gordon, C.McA., Luecke, J., Shalen, P.B.: Dehn surgery on knots. Ann. Math.125, 237-300 (1987) · Zbl 0633.57006 · doi:10.2307/1971311 |
| [6] | Kanenobu, T., Murakami, H.: Two-bridge knots with unknotting number one. Proc. Am. Math. Soc.98, 499-502 (1986) · Zbl 0613.57002 · doi:10.1090/S0002-9939-1986-0857949-8 |
| [7] | Kobayashi, T.: Minimal genus Seifert surfaces for unknotting number 1 knots. Kobe J. Math.6, 53-62 (1989) · Zbl 0688.57007 |
| [8] | Montesinos, J.M.: Surgery on links and double branched covers ofS 3, Knots, groups, and 3-manifolds. (Ann. Math. Studies vol. 84, pp. 227-259) Princeton: Princeton Univ. Press 1975 |
| [9] | Moser, L.: Elementary surgery along a torus knot. Pac. J. Math.38, 737-745 (1971) · Zbl 0202.54701 |
| [10] | Scharlemann, M., Thompson, A.: Link genus and the Conway moves. Comment. Math. Helv.64, 527-535 (1989) · Zbl 0693.57004 · doi:10.1007/BF02564693 |
| [11] | Schubert, H.: Knoten mit zwei Br?cken. Math. Z.65, 133-170 (1956) · Zbl 0071.39002 · doi:10.1007/BF01473875 |
| [12] | Seifert, H.: Schlingknoten. Math. Z.52, 62-80 (1949) · Zbl 0033.13701 · doi:10.1007/BF02230685 |
| [13] | Siebenmann, L.: Exercices sur les n?uds rationnels. Xeroxed notes. Orsay, 1975 |
| [14] | Tollefson, J.L.: Involutions of Seifert fiber spaces. Pac. J. Math.74, 519-529 (1978) · Zbl 0395.57022 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
