We know about geometrization conjecture. I am trying to understand how decomposition of prime manifold along compressible tori help classify 3-manifolds. Can we construct list of all prime manifolds this way ?
Once we decomposed given prime manifold $M$ on components having tori boundary we should been able to glue up the components back to obtain $M$. Therefore I thought that it is possible to list all possible components $C_1, C_2,...$ each of them having tori boundary (and no more incompressible torus) and have recipe of gluing such that any prime manifold can be obtained this way. This procedure should be so good, so for each prime $M$ there should be different pair (components, gluing recipe). Obviously we have already such procedure - it is surgery on link embedded in $S^3$, but is this 3-manifolds classification ? Maybe my knowledge is little here. Do we know when prime manifold is obtained as result of surgery and when two links give the same prime manifold ? If yes, then it remains to list all possible links in $S^3$.
Related question is what can be the genus of Heegaard splitting of prime 3-manifold ?