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A knot in S^3 is uniquely decomposed into a connected sum of prime knots. What is known for knots in other three-manifolds?

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What definition of connect-sum do you want to use? I suppose the most natural one would be to take connect-sum of the ambient manifolds along common 3-balls that intersect the knots in unknotted arcs. Is that what you're interested in? – Ryan Budney Apr 7 2010 at 6:20
Perhaps that will work. But even better to say: this raises another question, namely what all possible definitions of connected sum(s) are! – John Vrem Apr 7 2010 at 6:46
@Ryan- I would ask also whether there is a little discs operad action in such a context. What would happen with your work with Fred Cohen for long knots in other 3-manifolds? – Daniel Moskovich Apr 7 2010 at 7:02
There is a 2-cubes/discs action for "knots" in a variety of 3-manifolds but they're maybe not what most people would like to call knots. Things like self-embeddings of $\mathbb{R}\times \Sigma$ in itself with support contained in $[-1,1]\times \Sigma$ where $\Sigma$ is a compact 2-manifold. Presumably there's more global algebraic structure lurking in the background for general knots in 3-manifolds but I haven't found a strong formalism for it. – Ryan Budney Apr 7 2010 at 19:38

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This was addressed in Miyazaki, Conjugation and the prime decomposition of knots in closed, oriented 3-manifolds, using the definition of connected sum suggested by Ryan Budney in the comments.

The main results are that a prime decomposition of K exists iff a meridian of K is not null-homotopic in the complement of K, and if a prime decomposition of K does not contain a particular knot R in S1×S2 then it is the unique decomposition of K, whereas knots with this summand R can admit several prime decompositions.

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