The idea of quandle cocycle invariants was used to determine the minimal number of Reidemeister moves needed to move one diagram to another diagram of the same knot for some examples in [CESS05*].
In Fig. (from [CESS05*]),
a series of Reidemeister moves are sketched for
well-known diagrams of trefoil, firgure-eight, and the
-torus link,
from top to bottom. From these figures, it is seen that
these different diagrams are related by
,
and
type III moves,
respectively. It was proved that at least these numbers of type III moves
are actually needed.