Description:
The famous Kneser-Milnor Theorem states that every compact orientable 3-manifold can be presented as a connected sum of prime factors, which are unique up to order. Is a similar prime decomposition theorem true for 3- orbifolds? In 1994 Carlo Petronio posed this question and answered it posi- tively for 3-orbifolds containing neither bad nor nonseparating 2-suborbifolds. The question also makes sense for knotted graphs in 3-manifolds. We show that in general the answer is negative in both cases. Moreover, the set of all possible counterexamples admits an acceptable description. We conjecture that, in a certain sense, it is generated by a n̄ite number of basic counterexamples.