TITLE: Prime decomposition for knots in manifolds
QUESTION [6 upvotes]: A knot in S^3 is uniquely decomposed into a connected sum of prime knots. What is known for knots in other three-manifolds?

REPLY [9 votes]: 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.