Minimal Generating sets of Reidemeister moves
Noboru Ito, Yuichiro Iwamoto
TL;DR
The paper resolves the minimal generating set problem for oriented Reidemeister moves by providing a complete classification of minimal generating sets that include either a non-braid-type $II$ move or a braid-type $III$ move, across 336 cases; it proves that any five-element minimal generating set must consist of exactly one $III$, two $II$, and two $I$ moves under coherent constraints, and it lists explicit five-element minimal generating sets for all relevant families. A combination of explicit generation lemmas for moves of each type and a based-two-component-link invariant is used to establish both generating and non-generating results, including the impossibility of six-element minimal generating sets. The work also identifies 12 of 16 candidate configurations as minimal and highlights 4 open cases (Remark 5.1), thereby refining the structure of move-generation and providing a resource for future investigations and knot-theoretic applications. Overall, the results deepen the understanding of how local Reidemeister moves suffice to generate all oriented Reidemeister moves and illuminate the interplay between move types in minimal generating configurations.
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).