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

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 $\mathcal {K}$ .