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Transactions of the American Mathematical Society

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



Prime knots and tangles

Author: W. B. Raymond Lickorish
Journal: Trans. Amer. Math. Soc. 267 (1981), 321-332
MSC: Primary 57M25; Secondary 57M12
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 (Proc. Conf., Oxford, 1967), Pergamon, Oxford, 1970, pp. 329–358. MR 0258014
  • [GL] J. C. Gómez Larrańaga, Knot primeness, Doctoral Dissertation, Cambridge University, 1981.
  • [Gk] Richard E. Goodrick, Non-simplicially collapsible triangulations of 𝐼ⁿ, Proc. Cambridge Philos. Soc. 64 (1968), 31–36. MR 0220272
  • [Gn] Jean-Claude Hausmann (ed.), Knot theory, Lecture Notes in Mathematics, vol. 685, Springer-Verlag, Berlin-New York, 1978. MR 0645392
  • [HT] A. Hatcher and W. Thurston, Incompressible surfaces in $ 2$-bridge knot complements (to appear).
  • [Jo] William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450
  • [Jn] Klaus Johannson, Homotopy equivalences of 3-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR 551744
  • [KT] Paik Kee Kim and Jeffrey L. Tollefson, Splitting the PL involutions of nonprime 3-manifolds, Michigan Math. J. 27 (1980), no. 3, 259–274. MR 584691
  • [K] Rob Kirby, Problems in low dimensional manifold theory, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 273–312. MR 520548
  • [KL] Robion C. Kirby and W. B. Raymond Lickorish, Prime knots and concordance, Math. Proc. Cambridge Philos. Soc. 86 (1979), no. 3, 437–441. MR 542689, 10.1017/S0305004100056280
  • [L] Charles Livingston, Homology cobordisms of 3-manifolds, knot concordances, and prime knots, Pacific J. Math. 94 (1981), no. 1, 193–206. MR 625818
  • [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] Horst Schubert, Über eine numerische Knoteninvariante, Math. Z. 61 (1954), 245–288 (German). MR 0072483
  • [W] Friedhelm Waldhausen, Über Involutionen der 3-Sphäre, Topology 8 (1969), 81–91 (German). MR 0236916

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Keywords: Prime knot, tangle, branched cover, irreducible $ 3$-manifold
Article copyright: © Copyright 1981 American Mathematical Society