A knot in S^3 is uniquely decomposed into a connected sum of prime knots. What is known for knots in other threemanifolds?
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This was addressed in Miyazaki, Conjugation and the prime decomposition of knots in closed, oriented 3manifolds, 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 nullhomotopic in the complement of K, and if a prime decomposition of K does not contain a particular knot R in S^{1}×S^{2} then it is the unique decomposition of K, whereas knots with this summand R can admit several prime decompositions. 

