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Graphs of tangles


Author: J. C. Gómez-Larrañaga
Journal: Trans. Amer. Math. Soc. 286 (1984), 817-830
MSC: Primary 57M25; Secondary 57N10
DOI: https://doi.org/10.1090/S0002-9947-1984-0760989-3
MathSciNet review: 760989
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Abstract: We prove that under necessary conditions a graph of tangles is a prime link. For this we generalize the result that the sum of $ 2$-string prime $ L$-tangles is a prime link. Some applications are found. We explore Property $ {\text{L}}$ for tangles in order to prove primeness of knots.


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

  • [1] S. A. Bleiler, Prime tangles and composite knots (to appear). MR 823278 (87e:57006)
  • [2] J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra, Pergamon Press, Oxford and New York, 1969, pp. 329-358. MR 0258014 (41:2661)
  • [3] R. H. Fox, A quick trip through knot theory, Topology of $ 3$-Manifolds, (M. K. Fort, ed.), Prentice-Hall, New York, 1962, pp. 120-167. MR 0140099 (25:3522)
  • [4] J. C. Gòmez-Larrañaga, Totally knotted knots are prime, Math. Proc. Cambridge Philos. Soc. 91 (1982), 467-472. MR 654092 (83k:57003)
  • [5] C. McA. Gordon, Problems in knot theory, Knot Theory Proceedings (Plans-Sur-Bex, 1977), Lecture Notes in Math., vol. 685, Springer-Verlag, Berlin and New York, 1978, pp. 309-311. MR 0645392 (58:31084)
  • [6] A. Hatcher and W. Thurston, Incompressible surfaces in $ 2$-bridge knot complements (to appear).
  • [7] J. Hempel, $ 3$-manifolds, Ann. of Math. Studies, vol. 86, Princeton Univer. Press, Princeton, N. J., 1976. MR 0415619 (54:3702)
  • [8] T. Kanenobu, A note on $ 2$-fold branched covering spaces of $ {S^3}$, Math. Ann. 256 (1981), 449-452. MR 628226 (83a:57003)
  • [9] R. C. Kirby and W. B. R. Lickorish, Prime knots and concordance, Math. Proc. Cambridge Philos. Soc. 86 (1979), 437-441. MR 542689 (80k:57011)
  • [10] W. B. R. Lickorish, Prime knots and tangles, Trans. Amer. Math. Soc. 267 (1981), 321-332. MR 621991 (83d:57004)
  • [11] M. T. Lozano, Arcbodies, preprint.
  • [12] 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)
  • [13] R. Myers, Simple knots in compact, orientable $ 3$-manifolds, Trans. Amer. Math. Soc. 273 (1982), 75-91. MR 664030 (83h:57018)
  • [14] Y. Nakanishi, Prime links, concordance and Alexander invariants, Math. Sem. Notes Kobe Univ. 8 (1980), 561-568. MR 615876 (82f:57004)
  • [15] -, Primeness of links, Math. Sem. Notes Kobe Univ. 9 (1981), 415-440. MR 650747 (83k:57004)
  • [16] D. Rolfsen, Knots and links, Publish or Perish, Berkeley, Calif., 1976. MR 0515288 (58:24236)
  • [17] C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Ergeb. Math. Grenzgeb., vol. 69, Springer-Verlag, Berlin, 1972. MR 0350744 (50:3236)
  • [18] H. Seifert, Topologie dreidimensionaler gefaserter Räume, Acta Math. 60 (1933), 147-238. MR 1555366
  • [19] F. Waldhausen, Über Involuntionen der $ 3$-sphäre, Topology 8 (1969), 81-91.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0760989-3
Keywords: Graph of tangles, primeness, link, knot, Property $ {\text{L}}$, unknotting number
Article copyright: © Copyright 1984 American Mathematical Society

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