Minimal Generating sets of Reidemeister moves

TL;DR

This paper classifies minimal generating sets of Reidemeister moves, identifying which are minimal and providing a comprehensive overview of their properties, including specific cases involving move types II and III.

Contribution

It offers a complete classification of minimal generating sets of Reidemeister moves, especially those involving move types II and III, and identifies candidates whose minimality remains unresolved.

Findings

12 sets proven minimal

4 sets' minimality remains unsolved

Complete classification of sets with move types II and III

Abstract

We determine whether each known generating set of arbitrary oriented Reidemeister moves is minimal. We then provide a complete classification of minimal generating sets that include a coherent Reidemeister move of type II. We also classify all minimal generating sets that include a braid-type Reidemeister move of type III. Beyond these two cases, we identify 16 possible candidates for minimal generating sets. Among them, we prove that 12 are indeed minimal, whereas the minimality and even the generating property of the remaining 4 sets remains unsolved (Remark 5.1).