Monotonic simplification algorithms for knots

I will survey results and ideas concerning a very naive approach to recognition problems in knot theory, the one that is based on the monotone complexity reduction of the knot presentation. This approach is successful for the cases of recognizing the trivial knot and link factorizations. Examples found by D. Kazantsev show that the direct extension of the method to the problem of recognizing satellite knots does knot work.