  This note has an experimental nature and contains no new theorems.
  We introduce certain moves for classical knot diagrams that for all the very
many examples we have tested them on give a monotonic complete simplification.
A complete simplification of a knot diagram D is a sequence of moves that
transform D into a diagram D' with the minimal possible number of crossings for
the isotopy class of the knot represented by D. The simplification is monotonic
if the number of crossings never increases along the sequence. Our moves are
certain Z1, Z2, Z3 generalizing the classical Reidemeister moves R1, R2, R3,
and another one C (together with a variant) aimed at detecting whether a knot
diagram can be viewed as a connected sum of two easier ones.
  We present an accurate description of the moves and several results of our
implementation of the simplification procedure based on them, publicly
available on the web.
