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Transactions of the American Mathematical Society
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
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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}$ .


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DOI: http://dx.doi.org/10.1090/S0002-9947-1989-0997679-2
PII: S 0002-9947(1989)0997679-2
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