Minimal Generating sets of Reidemeister moves

2025/11/25 by Ito, Noboru, Iwamoto, Yuichiro · 1 citation
Mathematics · #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology

paper · doi:10.48550/arxiv.2511.19918

openalex publication_date 2025/11/25 · openalex created_date 2025/11/28 · openalex updated_date 2026/07/28

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).

Citations

• Minimal generating sets of moves for diagrams of isotopic knots and spatial trivalent graphs
• Every Reidemeister move is needed for each knot type
• Invariants of knot diagrams and relations among Reidemeister moves

Cited by

• Parallelization of Welded Links

Related

• Minimal generating sets of moves for diagrams of isotopic knots and spatial trivalent graphs
• A short proof of the equivalence of any Reidemeister oriented move 3
• Every Reidemeister move is needed for each knot type
• Natural Moves for Knots and Links
• On link diagrams that are minimal with respect to Reidemeister moves I and II