Two-bridge knots with unknotting number one

Kanenobu, Taizo; Murakami, Hitoshi

doi:10.1090/S0002-9939-1986-0857949-8

Two-bridge knots with unknotting number one
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by Taizo Kanenobu and Hitoshi Murakami PDF

Proc. Amer. Math. Soc. 98 (1986), 499-502 Request permission

Abstract:

We determine all two-bridge knots with unknotting number one. In fact we prove that a two-bridge knot has unknotting number one iff there exist positive integers $p$, $m$, and $n$ such that $(m,n) = 1$ and $2mn = p \pm 1$, and it is equivalent to $S(p,2{n^2})$ in Schubert’s notation. It is also shown that it can be expressed as $C(a,{a_1},{a_2}, \ldots ,{a_k}, \pm 2, - {a_k}, \ldots , - {a_2}, - {a_1})$ using Conway’s notation.

References

J. W. Alexander and G. B. Briggs, On types of knotted curves, Ann. of Math. (2) 28 (1926/27), no. 1-4, 562–586. MR 1502807, DOI 10.2307/1968399
Steven A. Bleiler, A note on unknotting number, Math. Proc. Cambridge Philos. Soc. 96 (1984), no. 3, 469–471. MR 757839, DOI 10.1017/S0305004100062381
J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) Pergamon, Oxford, 1970, pp. 329–358. MR 0258014

Incompressible surfaces and surgery on knots

Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen, Dehn surgery on knots, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 1, 43–45. MR 788388, DOI 10.1090/S0273-0979-1985-15357-1

Involutions and isotopies of lens spaces

W. B. Raymond Lickorish, The unknotting number of a classical knot, Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982) Contemp. Math., vol. 44, Amer. Math. Soc., Providence, RI, 1985, pp. 117–121. MR 813107, DOI 10.1090/conm/044/813107
José M. Montesinos, Surgery on links and double branched covers of $S^{3}$, Knots, groups, and $3$-manifolds (Papers dedicated to the memory of R. H. Fox), Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N.J., 1975, pp. 227–259. MR 0380802
Louise Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), 737–745. MR 383406
Yasutaka Nakanishi, A note on unknotting number, Math. Sem. Notes Kobe Univ. 9 (1981), no. 1, 99–108. MR 634000
Horst Schubert, Knoten mit zwei Brücken, Math. Z. 65 (1956), 133–170 (German). MR 82104, DOI 10.1007/BF01473875

Exercices sur les nœuds rationnels

Wilbur Whitten, Knot complements and groups, Topology 26 (1987), no. 1, 41–44. MR 880506, DOI 10.1016/0040-9383(87)90019-X