Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



Conjugation and the prime decomposition of knots in closed, oriented $ 3$-manifolds

Author: Katura Miyazaki
Journal: Trans. Amer. Math. Soc. 313 (1989), 785-804
MSC: Primary 57M99; Secondary 57M25
MathSciNet review: 997679
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider the prime decomposition of knots in closed, oriented $ 3$-manifolds. (For classical knots one can easily prove the uniqueness of prime decomposition by using a standard innermost disk argument.) We define a new relation, conjugation, between oriented knots in closed, oriented $ 3$-manifolds and prove the following results. (1) The prime decomposition is, roughly speaking, uniquely determined up to conjugation, (2) there is a prime knot $ \mathcal{R}$ in $ {S^1} \times {S^2}$ such that $ \mathcal{R}\char93 {\mathcal{K}_1} = \mathcal{R}\char93 {\mathcal{K}_2}$ if $ {\mathcal{K}_1}$ is a conjugation of $ {\mathcal{K}_2}$, and (3) if a knot $ \mathcal{K}$ has a prime decomposition which does not contain $ \mathcal{R}$, then it is the unique prime decomposition of $ \mathcal{K}$ .

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

  • [1] John Hempel, 3-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
  • [2] William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450
  • [3] W. Magnuss, A. Karrass, and D. Solitar, Combinatorial group theory, Wiley, New York, 1966.
  • [4] Horst Schubert, Die eindeutige Zerlegbarkeit eines Knotens in Primknoten, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (1949), no. 3, 57–104 (German). MR 0031733
  • [5] Itiro Tamura, Fundamental theorems in global knot theory, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 40 (1993), no. 3, 561–606. MR 1269028
  • [6] Koichi Yano, The support of global graph links, J. Math. Soc. Japan 37 (1985), no. 4, 683–702. MR 806308,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57M99, 57M25

Retrieve articles in all journals with MSC: 57M99, 57M25

Additional Information

Keywords: Prime knot, prime decomposition of knot, Haken's finiteness theorem, conjugation of knot, inducing-pair, annulus theorem
Article copyright: © Copyright 1989 American Mathematical Society