  It is well known that any two diagrams representing the same oriented link
are related by a finite sequence of Reidemeister moves O1, O2 and O3. Depending
on orientations of fragments involved in the moves, one may distinguish 4
different versions of each of the O1 and O2 moves, and 8 versions of the O3
move. We introduce a minimal generating set of four oriented Reidemeister
moves, which includes two O1 moves, one O2 move, and one O3 move. We then study
which other sets of up to 5 oriented moves generate all moves, and show that
only few of them do. Some commonly considered sets are shown not to be
generating. An unexpected non-equivalence of different O3 moves is discussed.
