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Totally knotted knots are prime

Published online by Cambridge University Press:  24 October 2008

J. C. Gomez-Larran¯aga
J. C. Gomez-Larran¯aga
Affiliation:
Instituto de Matemáticas – U.N.A.M., MéxicoD.F.
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Extract

Throughout, the word knot means a subspace of the 3-sphere S3 homeomorphic with the 1-sphere S1. Any knot can be expressed as a connected sum of a finite number of prime knots in a unique way (13), we consider the trivial knot a non-prime knot. (For higher dimensional knots, factorization and uniqueness have been studied in (1).) However given a knot it is difficult to determine if it is prime or not. We prove that totally knotted knots, see definition in §2, are prime in theorem 1, give a class of examples in theorem 2 and investigate how the last result can be applied to the conjecture that the family Y of unknotting number one knots are prime. (See problem 2 in (5).) At the end, prime tangles as defined by W. B. R. Lickerish are used to prove that in a certain family of knots, related somewhat to Y, there is just one non-prime knot: the square knot. The paper should be interpreted as being in the piecewise linear category. Standard definitions of 3-manifolds and knot theory may be found in (6) and (11) respectively.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 91 , Issue 3 , May 1982 , pp. 467 - 472
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

REFERENCES

(1)Bayer, E., Hillman, J. A. and Kearton, C. The factorisation of simple knots. (To appear.)Google Scholar
(2)Conway, J. H. An enumeration of knots and links, and some of their algebraic properties. In Computational problems in abstract algebra (Pergamon Press, Oxford and New York, 1969), pp. 329–358.Google Scholar
(3)Fox, R. H. A quick trip through knot theory. Topology of 3-manifolds, ed. Fort, M. K. (Prentice-Hall, New York, 1962), pp. 120–167.Google Scholar
(4)Gomez-Larran¯aga, J. C. Knot primeness. Thesis, Cambridge University (1981).Google Scholar
(5)Gordon, C. McA. Problems in knot theory. Knot Theory Proceedings, Plans-sur-Bex, Switzerland, 1977 (Springer Lectures Notes in Mathematics, 685, 1978), pp. 309–311.Google Scholar
(6)Hempel, J.3-manifolds. Ann. of Math. Studies 86 (1976).Google Scholar
(7)Kirby, R. C. and Lickorish, W. B. R.Prime knots and concordance. Math. Proc. Camb. Phil. Soc. 86 (1979), 437–441.CrossRefGoogle Scholar
(8)Montesinos, J. M.Variedades de Seifert que son recubridores ciclicos ramificados de dos hojas. Bol.Soc. Mat. Mex. 18 (1973), 1–32.Google Scholar
(9)Nakagawa, Y.Elementary disks and their equivalences. Quart. J. Math. Oxford (2), 27 (1976), 355–369.CrossRefGoogle Scholar
(10)Neumann, W. D. and Raymond, F. Seifert manifolds, plumbing, μ-invariant and orientation reversing maps. Algebraic and Geometric Topology Proceedings, Santa Barbara, 1977 (Springer Lectures Notes in Mathematics, no. 664, 1978), pp. 163–196.Google Scholar
(11)Rolfsen, D.Knots and Links (Publish or Perish, Berkeley, Cal., 1976).Google Scholar
(12)Seifert, H.Topologie dreidimensionaler gefaserter Raume. Acta Math. 60 (1933), 147–238.CrossRefGoogle Scholar
(13)Schubert, H.Die eindeutige Zerlegbarkeit eines knotens in Primknotten. S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 3 (1949), 57–104.Google Scholar
(14)Schubert, H.Knoten und Vollringe. Acta Math. 90 (1953), 131–286.Google Scholar
(15)Waldhausen, F.Über Involutionen der 3-Sphare. Topology 8 (1969), 81–91.CrossRefGoogle Scholar