Minimal generating sets of Reidemeister moves

Abstract

It is well known that any two diagrams representing the same oriented link are related by a finite sequence of Reidemeister moves Ω, 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 4 oriented Reidemeister moves, which includes two Ω1 moves, one Ω2 move, and one Ω3 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 Ω3 moves is discussed.