Union and tangle

Nakanishi, Yasutaka

doi:10.1090/S0002-9939-96-03453-3

Union and tangle
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by Yasutaka Nakanishi PDF

Proc. Amer. Math. Soc. 124 (1996), 1625-1631 Request permission

Abstract:

Shibuya proved that any union of two nontrivial knots without local knots is a prime knot. In this note, we prove it in a general setting. As an application, for any nontrivial knot, we give a knot diagram such that a single unknotting operation on the diagram cannot yield a diagram of a trivial knot.

References

Steven A. Bleiler, Prime tangles and composite knots, Knot theory and manifolds (Vancouver, B.C., 1983) Lecture Notes in Math., vol. 1144, Springer, Berlin, 1985, pp. 1–13. MR 823278, DOI 10.1007/BFb0075008
Steven Bleiler and Martin Scharlemann, Tangles, property $P$, and a problem of J. Martin, Math. Ann. 273 (1986), no. 2, 215–225. MR 817877, DOI 10.1007/BF01451402
Steven Bleiler and Martin Scharlemann, A projective plane in $\textbf {R}^4$ with three critical points is standard. Strongly invertible knots have property $P$, Topology 27 (1988), no. 4, 519–540. MR 976593, DOI 10.1016/0040-9383(88)90030-4
M. Boileau and L. Siebenmann, A planar classification of pretzel knots and Montesinos knots, Prépublications, Orsay, 1980.
F. Bonahon, Involutions et fibrés de Seifert dans les variétés de dimension 3, Thèse de 3e cycle, Orsay, 1979.
Mario Eudave Muñoz, Primeness and sums of tangles, Trans. Amer. Math. Soc. 306 (1988), no. 2, 773–790. MR 933317, DOI 10.1090/S0002-9947-1988-0933317-1
Taizo Kanenobu, Hyperbolic links with Brunnian properties, J. Math. Soc. Japan 38 (1986), no. 2, 295–308. MR 833204, DOI 10.2969/jmsj/03820295
Shin’ichi Kinoshita and Hidetaka Terasaka, On unions of knots, Osaka Math. J. 9 (1957), 131–153. MR 98386
W. B. Raymond Lickorish, Prime knots and tangles, Trans. Amer. Math. Soc. 267 (1981), no. 1, 321–332. MR 621991, DOI 10.1090/S0002-9947-1981-0621991-2
Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
José M. Montesinos, An infinite family of non-separable represented knots, Rev. Mat. Hisp.-Amer. (4) 33 (1973), 32–35 (Spanish). MR 385839
J. M. Montesinos, Revêtements ramifiés de nœuds, espaces fibrés de Seifert et scindements de Heegaard, Prépublications, Orsay, 1979.
Yasutaka Nakanishi, Unknotting numbers and knot diagrams with the minimum crossings, Math. Sem. Notes Kobe Univ. 11 (1983), no. 2, 257–258. MR 749196
Martin Scharlemann, Smooth spheres in $\textbf {R}^4$ with four critical points are standard, Invent. Math. 79 (1985), no. 1, 125–141. MR 774532, DOI 10.1007/BF01388659
T. Shibuya, Primeness of union of knots, preprint, 1993.