Minimal Generating sets of Reidemeister moves

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).