Minimal generating sets of Reidemeister moves

TL;DR

The paper identifies a minimal set of four oriented Reidemeister moves that can generate all other moves, analyzes various move sets, and discusses the non-equivalence of certain O3 moves.

Contribution

It introduces a minimal generating set of four oriented Reidemeister moves and examines which other move sets are sufficient to generate all moves.

Findings

A minimal generating set of four moves is established.

Many commonly used move sets are shown not to be sufficient.

Different O3 moves are found to be non-equivalent.

Abstract

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.