Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Primeness and sums of tangles


Author: Mario Eudave Muñoz
Journal: Trans. Amer. Math. Soc. 306 (1988), 773-790
MSC: Primary 57M25
DOI: https://doi.org/10.1090/S0002-9947-1988-0933317-1
MathSciNet review: 933317
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider knots and links obtained by summing a rational tangle and a prime tangle. For a given prime tangle, we show that there are at most three rational tangles that will induce a composite or splittable link. In fact, we show that there is at most one rational tangle that will give a splittable link. These results extend Scharlemann's work.


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

  • [B] 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 823278 (87e:57006)
  • [BS$ _{1}$] S. A. Bleiler and M. Scharlemann, Tangles, property $ P$, and a problem of J. Martin, Math. Ann. 273 (1986), 215 225. MR 817877 (87h:57007)
  • [BS$ _{2}$] -, A projective plane in $ {{\mathbf{R}}^4}$ with three critical points is standard, MSRI preprint.
  • [C] J. H. Conway, An enumeration of knots and links, and some of their algebraic propertis, Computational Problems in Abstract Algebra, Pergamon Press, Oxford and New York, 1969, pp. 329-358. MR 0258014 (41:2661)
  • [E] M. Eudave-Muñoz, Cirugía en nudos fuertemente invertibles, An. Inst. Mat. Univ. Nac. Autónoma México 26 (1986), 41-57. MR 906326 (88j:57008)
  • [GL] C. McA. Gordon and J. Luecke, Only integral Dehn surgeries can yield reducible manifolds, preprint. MR 886439 (89a:57003)
  • [KT] P. K. Kim and J. L. Tollefson, Splitting the P. L. involutions of nonprime $ 3$-manifolds, Michigan Math. J. 27 (1980), 259-274. MR 584691 (81m:57007)
  • [L] W. B. R. Lickorish, Prime knots and tangles, Trans. Amer. Math. Soc. 267 (1981), 321-332. MR 621991 (83d:57004)
  • [M$ _{1}$] J. M. Montesinos, Variedades de Seifert que son recubridores cíclicos ramificados de dos hojas, Bol. Soc. Mat. Mexicana (2) 18 (1973), 1-32. MR 0341467 (49:6218)
  • [M$ _{2}$] -, Surgery on links and double branched covers of $ {S^3}$, Ann. of Math. Studies, no. 84, Princeton Univ. Press, Princeton, N. J., 1975, pp. 227-260.
  • [S$ _{1}$] M. Scharlemann, Smooth spheres in $ {{\mathbf{R}}^4}$ with four critical points are standard, Invent. Math. 79 (1985), 125-141. MR 774532 (86e:57010)
  • [S$ _{2}$] -, Unknotting number one knots are prime, Invent. Math. 82 (1985), 37-55. MR 808108 (86m:57010)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57M25

Retrieve articles in all journals with MSC: 57M25


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0933317-1
Keywords: Prime tangle, rational tangle, prime knot and link, composite knot and link, splittable link
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society