Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Union and tangle


Author: Yasutaka Nakanishi
Journal: Proc. Amer. Math. Soc. 124 (1996), 1625-1631
MSC (1991): Primary 57M25
DOI: https://doi.org/10.1090/S0002-9939-96-03453-3
MathSciNet review: 1342035
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] S. A. Bleiler, Prime tangles and composite knots, Knot Theory and Manifolds, Lecture Notes in Math., vol. 1144, Springer-Verlag, Berlin and New York, 1985, pp. 1--13. MR 87e:57006
  • [2] S. A. Bleiler and M. Scharlemann, Tangles, property P, and a problem of J. Martin, Math. Ann. 273 (1986), 215--225. MR 87h:57007
  • [3] S. A. Bleiler and M. Scharlemann, A projective plane in $\hbox {\bf R}^{4}$ with three critical points are standard, Topology 27 (1988), 519--540. MR 90e:57006
  • [4] M. Boileau and L. Siebenmann, A planar classification of pretzel knots and Montesinos knots, Prépublications, Orsay, 1980.
  • [5] F. Bonahon, Involutions et fibrés de Seifert dans les variétés de dimension 3, Thèse de 3e cycle, Orsay, 1979.
  • [6] M. Eudave-Muñoz, Primeness and sums of tangles, Trans. Amer. Math. Soc. 306 (1988), 773--790. MR 89g:57005
  • [7] T. Kanenobu, Hyperbolic links with Brunnian properties, J. Math. Soc. Japan 38 (1986), 295--308. MR 88a:57024
  • [8] S. Kinoshita and H. Terasaka, On union of knots, Osaka Math. J. 9 (1957), 131--153. MR 20:4846
  • [9] W. B. R. Lickorish, Prime knots and tangles, Trans. Amer. Math. Soc. 267 (1981), 321--332. MR 83d:57004
  • [10] W. Magnus, A. Karrass, and D. Solitar, Combinatorial group theory: presentations of groups in terms of generators and relators, Interscience Publ., Wiley & Sons, New York, 1966. MR 34:7617
  • [11] J. M. Montesinos, Una familia infinita de nudos representados no separables, Revista Math. Hisp.-Amer. (IV) 33 (1973), 32--35. MR 52:6698
  • [12] J. M. Montesinos, Revêtements ramifiés de n{\oe}uds, espaces fibrés de Seifert et scindements de Heegaard, Prépublications, Orsay, 1979.
  • [13] Y. Nakanishi, Unknotting numbers and knot diagrams with the minimum crossings, Math. Seminar Notes Kobe Univ. 11 (1983), 257--258. MR 85h:57008
  • [14] M. Scharlemann, Smooth spheres in $\hbox {\bf R}^{4}$ with four critical points are standard, Invent. Math. 79 (1985), 125--141. MR 86e:57010
  • [15] T. Shibuya, Primeness of union of knots, preprint, 1993.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 57M25

Retrieve articles in all journals with MSC (1991): 57M25


Additional Information

Yasutaka Nakanishi
Affiliation: Department of Mathematics, Kobe University, 1-1, Rokkodai-cho, Nada-ku, Kobe 657, Japan
Email: nakanisi@math.s.kobe-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-96-03453-3
Keywords: Union, tangle, primeness, Montesinos knot
Received by editor(s): November 3, 1994
Communicated by: Ronald J. Stern
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society