Conjugation and the prime decomposition of knots in closed, oriented -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 -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 -manifolds and prove the following results. (1) The prime decomposition is, roughly speaking, uniquely determined up to conjugation, (2) there is a prime knot in such that if is a conjugation of , and (3) if a knot has a prime decomposition which does not contain , then it is the unique prime decomposition of .

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1989-0997679-2

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