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Transactions of the American Mathematical Society

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Prime knots and tangles


Author: W. B. Raymond Lickorish
Journal: Trans. Amer. Math. Soc. 267 (1981), 321-332
MSC: Primary 57M25; Secondary 57M12
DOI: https://doi.org/10.1090/S0002-9947-1981-0621991-2
MathSciNet review: 621991
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Abstract: A study is made of a method of proving that a classical knot or link is prime. The method consists of identifying together the boundaries of two prime tangles. Examples and ways of constructing prime tangles are explored.


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

  • [B] F. Bonahon (to appear).
  • [C] 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)
  • [GL] J. C. Gómez Larrańaga, Knot primeness, Doctoral Dissertation, Cambridge University, 1981.
  • [Gk] R. E. Goodrick, Non-simplicially collapsible triangluations on $ {I^n}$, Proc. Cambridge Philos. Soc. 64 (1968), 31-36. MR 0220272 (36:3338)
  • [Gn] 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)
  • [HT] A. Hatcher and W. Thurston, Incompressible surfaces in $ 2$-bridge knot complements (to appear).
  • [Jo] W. Jaco, Lectures on three-manifold topology, CBMS Regional Conf. Ser. in Math., Number 43, Amer. Math. Soc., Providence, R. I., 1980. MR 565450 (81k:57009)
  • [Jn] K. Johannson, Homotopy equivalences of $ 3$-manifolds with boundaries, Lecture Notes in Math., vol. 761, Springer-Verlag, Berlin and New York, 1979. MR 551744 (82c:57005)
  • [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)
  • [K] R. Kirby, Problems in low dimensional manifold theory, Proc. Sympos. Pure Math., vol. 32, Amer. Math. Soc., Providence, R. I., 1978, pp. 273-312. MR 520548 (80g:57002)
  • [KL] 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)
  • [L] C. Livingston, Homology cobordisms of $ 3$-manifolds, knot concordances, and prime knots, Pacific J. Math. (to appear). MR 625818 (83e:57008)
  • [M] W. Menasco, Incompressible surfaces in the complement of alternating knots and links (to appear).
  • [P1] K. A. Perko, A weak $ 2$-bridged knot with at most three bridges is prime, Notices Amer. Math. Soc. 26 (1978), A-648 (and a preprint).
  • [P2] -, Invariants of eleven-crossing knots (to appear).
  • [S] H. Schubert, Über eine numerische Knoteninvariante, Math. Z. 61 (1954), 245-288. MR 0072483 (17:292a)
  • [W] F. Waldhausen, Über Involutionen der $ 3$-Sphäre, Topology 8 (1969), 81-91. MR 0236916 (38:5209)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0621991-2
Keywords: Prime knot, tangle, branched cover, irreducible $ 3$-manifold
Article copyright: © Copyright 1981 American Mathematical Society

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