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Minimal generating sets of moves for diagrams of isotopic knots and spatial trivalent graphs

Carmen Caprau, Bradley Scott

TL;DR

The paper addresses the problem of identifying minimal generating sets for oriented Reidemeister moves in knot diagrams and extends the framework to Reidemeister-type moves for spatial trivalent graphs. Building on Polyak's result that four moves suffice, it proves there are exactly twelve distinct minimal 4-move generating sets for knot diagrams, organized into two collections A and H, each with two Ω1 moves, one Ω2 move, and one Ω3 move (a in A or h in H). It further shows that Ω2c and Ω2d cannot occur in any minimal generating set, and provides a complete classification of all minimal sets. Extending to spatial trivalent graphs, the paper shows a minimal generating set requires 10 moves (two Ω1, one Ω2, one Ω3, four Ω4, two Ω5), and computes a total of 768 minimal generating sets by counting combinations across move-types. These results yield a complete catalog of minimal generating sets, enabling streamlined verification of knot and spatial-graph invariants and facilitating algorithmic approaches to diagram isotopy.

Abstract

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.

Minimal generating sets of moves for diagrams of isotopic knots and spatial trivalent graphs

TL;DR

The paper addresses the problem of identifying minimal generating sets for oriented Reidemeister moves in knot diagrams and extends the framework to Reidemeister-type moves for spatial trivalent graphs. Building on Polyak's result that four moves suffice, it proves there are exactly twelve distinct minimal 4-move generating sets for knot diagrams, organized into two collections A and H, each with two Ω1 moves, one Ω2 move, and one Ω3 move (a in A or h in H). It further shows that Ω2c and Ω2d cannot occur in any minimal generating set, and provides a complete classification of all minimal sets. Extending to spatial trivalent graphs, the paper shows a minimal generating set requires 10 moves (two Ω1, one Ω2, one Ω3, four Ω4, two Ω5), and computes a total of 768 minimal generating sets by counting combinations across move-types. These results yield a complete catalog of minimal generating sets, enabling streamlined verification of knot and spatial-graph invariants and facilitating algorithmic approaches to diagram isotopy.

Abstract

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.
Paper Structure (6 sections, 22 theorems, 4 equations, 8 figures)

This paper contains 6 sections, 22 theorems, 4 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Minimal generating sets of Reidemeister moves
  3. The Oriented Reidemeister Moves
  4. Generating the Oriented Reidemeister Moves
  5. Non-Generating Reidemeister Moves
  6. Generating sets of moves for isotopic spatial trivalent graph diagrams

Key Result

Theorem 1

Pol Let $D$ and $D'$ be two diagrams in $\mathbb{R}^2$ representing the same oriented knot. Then one may pass from $D$ to $D'$ by isotopy and a finite sequence of four oriented Reidemeister moves $\Omega1a, \Omega 1b, \Omega2a,$ and $\Omega3a$.

Figures (8)

  • Figure 1: Reidemeister-type moves for spatial trivalent graph diagrams
  • Figure 2: Oriented Reidemeister moves of type 1
  • Figure 3: Oriented Reidemeister moves of type 2
  • Figure 4: Oriented Reidemeister moves of type 3
  • Figure 5: Smoothing a diagram according to orientation
  • ...and 3 more figures

Theorems & Definitions (46)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Remark 3
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 36 more