Minimal Sets of Reidemeister Moves

It is well known that any two diagrams representing the same oriented link are related by a finite sequence of Reidemeister moves Ω1, Ω2 and Ω3. Depending on orientations of fragments involved in the moves, one may distinguish 4 different versions of each of the Ω1 and Ω2 moves, and 8 versions of the Ω3 move. We introduce a minimal generating set of four oriented Reidemeister moves, which includes two moves of type Ω1, one move of type Ω2 and one move of type Ω3. We then study other sets of moves, considering various sets with one move of type Ω3, and show that only few sets generate all Reidemeister moves. An unexpected non-equivalence of different Ω3 moves is discussed.