Decompositions of low-dimensional objects

We develop a new version of the famous Diamond Lemma of M. H. A. Newman (1942) and use it to describe several results on decompositions of different geometric objects. 1. A spherical splitting theorem for knotted graphs in 3-manifolds. 2. Counterexamples to the folklore prime decomposition theorem for 3-orbifolds. 3. A theorem on annular splittings of 3-manifolds, which is independent of the JSJ-decomposition theorem. 4. Prime decomposition theorem for homologically trivial knots in thick surfaces.