Incompressible surfaces, hierarchies and unknot recognition

We present a new algorithm to determine whether a compact orientable surface properly embedded in a compact orientable 3-manifold is incompressible. As a special case, this provides a new algorithm to detect the unknot. The central technique is the use of hierarchies. Unlike previous algorithms, no challenging invariants need to be computed and no lengthy search procedures are required. This algorithm leads to a new proof that determining whether a compact orientable 3-manifold has incompressible boundary is in NP.