Conjugation and the prime decomposition of knots in closed, oriented $3$-manifolds
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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 $\mathcal {R}\# {\mathcal {K}_1} = \mathcal {R}\# {\mathcal {K}_2}$ 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
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Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 313 (1989), 785-804
- MSC: Primary 57M99; Secondary 57M25
- DOI: https://doi.org/10.1090/S0002-9947-1989-0997679-2
- MathSciNet review: 997679