Minimal number of type III Reidemeister moves

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*].

Figure: Minimal number of type III moves

In Fig.  (from [CESS05*]), a series of Reidemeister moves are sketched for well-known diagrams of trefoil, firgure-eight, and the $(2,4)$-torus link, from top to bottom. From these figures, it is seen that these different diagrams are related by $2$, $3$ and $3$ type III moves, respectively. It was proved that at least these numbers of type III moves are actually needed.