Prime knots and tangles
Author:
W. B. Raymond Lickorish
Journal:
Trans. Amer. Math. Soc. 267 (1981), 321332
MSC:
Primary 57M25; Secondary 57M12
DOI:
https://doi.org/10.1090/S00029947198106219912
MathSciNet review:
621991
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Abstract  References  Similar Articles  Additional Information
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.

F. Bonahon (to appear).
 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 J. C. Gómez Larrańaga, Knot primeness, Doctoral Dissertation, Cambridge University, 1981.
 Richard E. Goodrick, Nonsimplicially collapsible triangulations of $I^{n}$, Proc. Cambridge Philos. Soc. 64 (1968), 31–36. MR 220272, DOI https://doi.org/10.1017/s0305004100042511
 JeanClaude Hausmann (ed.), Knot theory, Lecture Notes in Mathematics, vol. 685, SpringerVerlag, BerlinNew York, 1978. MR 0645392 A. Hatcher and W. Thurston, Incompressible surfaces in $2$bridge knot complements (to appear).
 William Jaco, Lectures on threemanifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450
 Klaus Johannson, Homotopy equivalences of $3$manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR 551744
 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
 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
 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, DOI https://doi.org/10.1017/S0305004100056280
 Charles Livingston, Homology cobordisms of $3$manifolds, knot concordances, and prime knots, Pacific J. Math. 94 (1981), no. 1, 193–206. MR 625818 W. Menasco, Incompressible surfaces in the complement of alternating knots and links (to appear). K. A. Perko, A weak $2$bridged knot with at most three bridges is prime, Notices Amer. Math. Soc. 26 (1978), A648 (and a preprint). , Invariants of elevencrossing knots (to appear).
 Horst Schubert, Über eine numerische Knoteninvariante, Math. Z. 61 (1954), 245–288 (German). MR 72483, DOI https://doi.org/10.1007/BF01181346
 Friedhelm Waldhausen, Über Involutionen der $3$Sphäre, Topology 8 (1969), 81–91 (German). MR 236916, DOI https://doi.org/10.1016/00409383%2869%29900330
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Additional Information
Keywords:
Prime knot,
tangle,
branched cover,
irreducible <IMG WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img8.gif" ALT="$3$">manifold
Article copyright:
© Copyright 1981
American Mathematical Society