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The factorization of simple knots

Published online by Cambridge University Press:  24 October 2008

E. Bayer ,
J. A. Hillman  and
C. Kearton
E. Bayer
Affiliation:
University of Geneva
J. A. Hillman
Affiliation:
University of Geneva
C. Kearton
Affiliation:
University of Durham
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Abstract

For high-dimensional simple knots we give two theorems concerning unique factorization into irreducible knots, and provide examples to show that the hypotheses are necessary in each case.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 90 , Issue 3 , November 1981 , pp. 495 - 506
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

REFERENCES

(1)Bayer, E.Factorisation is not unique for higher dimensional knots. Comm. Math. Helv. 55 (1980), 583–592.CrossRefGoogle Scholar
(2)Browder, W. and Levine, J.Fibering manifolds over a circle. Comm. Math. Helv. 40 (1965), 153–160.CrossRefGoogle Scholar
(3)Cassels, J. W. S.Rational quadratic forms. Academic Press (1978). LMS monograph no. 13.Google Scholar
(4)Crowell, R.The group G′/G″ of a knot group G. Duke Math. Jour. 30 (1963), 349–354.CrossRefGoogle Scholar
(5)Hillman, J. A.Blanchfield pairings with squarefree Alexander polynomials. Math. Zeit. 176 (1981), 551–563.CrossRefGoogle Scholar
(6)Hillman, J. A. Finite knot modules and the factorisation of certain simple knots. Math. Ann. (to appear).Google Scholar
(7)Kearton, C.Classification of simple knots by Blanchfield duality. Bull. Amer. Math. Soc. 79 (1973), 952–955.CrossRefGoogle Scholar
(8)Kearton, C.Blanchfield duality and simple knots. Trans. Amer. Math. Soc. 202 (1975), 141–160.CrossRefGoogle Scholar
(9)Kervaire, M. A.Les nœuds de dimensions supérieures. Bull. Soc. Math. France. 93 (1965), 225–271.Google Scholar
(10)Kojima, S.Classification of simple knots by Levine pairings. Comm. Math. Helv. 54 (1979), 356–367.CrossRefGoogle Scholar
(11)Levine, J.Unknotting spheres in codimension two. Topology 4 (1965), 9–16.CrossRefGoogle Scholar
(12)Levine, J.An algebraic classification of some knots of codimension two. Comm. Math. Helv. 45 (1970), 185–198.CrossRefGoogle Scholar
(13)Levine, J.Algebraic structure of knot modules (Springer, Lecture Notes in Mathematics, no. 772 (1980)).CrossRefGoogle Scholar
(14)Maeda, T.On a composition of knot groups. II. Algebraic bridge index. Mathematics Seminar Notes Kobe University 5 (1977), 457–464.Google Scholar
(15)O'meara, O. T.Introduction to quadratic forms. (Springer, 1971).Google Scholar
(16)Schubert, H.Die eindeutige Zerlegbarkeit eines knotens in Primknoten. S.-B. Heidelberger Akad. Wiss. Math. Nat. Kl. 3 (1949), 57–104.Google Scholar
(17)Serre, J. P.Algèbre locale-multiplicités (Springer, Lecture Notes in Mathematics, no. 11, 1965).Google Scholar
(18)Sosinskii, A. B.Decomposition of knots. Math. USSR-Sb. 10 (1970), 139–150.CrossRefGoogle Scholar
(19)Trotter, H. F.Homology of group systems with applications to knot theory. Annals of Math. 76 (1962), 464–498.CrossRefGoogle Scholar
(20)Trotter, H. F.On S-equivalence of Seifert matrices. Invent. Math. 20 (1973), 173–207.CrossRefGoogle Scholar
(21)Trotter, H. F.Knot modules and Seifert matrices (Springer, Lecture Notes in Mathematics, no. 685, Knot Theory (1978)), pp. 291–299.Google Scholar