2022/02/10 by Carmen Caprau, Caprau, Carmen, Scott Bradley +1 · 3 citations
Computer Science · Mathematics · #57K10 #57K12 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Logic, programming, and type systems
paper · pdf · doi:10.48550/arxiv.2202.04839
openalex publication_date 2022/02/10 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28
Polyak proved that all oriented versions of Reidemeister moves for knot and link diagrams can be generated by a set of just four oriented Reidemeister moves, and that no fewer than four oriented Reidemeister moves generate them all. We refer to a set containing four oriented Reidemeister moves that collectively generate all of the other oriented Reidemeister moves as a minimal generating set. Polyak also proved that a certain set containing two Reidemeister moves of type 1, one move of type 2, and one move of type 3 form a minimal generating set for all oriented Reidemeister moves. We expand upon Polyak's work by providing an additional eleven minimal, 4-element, generating sets of oriented Reidemeister moves, and we prove that these twelve sets represent all possible minimal generating sets of oriented Reidemeister moves. We also consider the Reidemeister-type moves that relate oriented spatial trivalent graph diagrams with trivalent vertices that are sources and sinks and prove that a minimal generating set of oriented Reidemeister-type moves for spatial trivalent graph diagrams contains ten moves.