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A generating set of Reidemeister moves of oriented virtual knots

Danish Ali

TL;DR

The paper addresses verifying invariants for oriented virtual knots by establishing a generating set for oriented virtual Reidemeister moves. It enumerates the 17 oriented virtual moves and proves that the set $G= {V1a, V2a, V3a, V4g}$ suffices to generate all oriented virtual moves, using a structured proof that partitions moves into $S_1$–$S_4$ and derives each from $G$ through a sequence of lemmas. This result reduces invariant-verification work and provides a foundation for future study of virtual knot invariants and their computational properties. The work extends classical generating-set results (e.g., Polyak, Caprau–Scott) to the virtual setting, clarifying minimality and informing potential virtual quandle axioms.

Abstract

In oriented knot theory, verifying a quantity is an invariant involves checking its invariance under all oriented Reidemeister moves, a process that can be intricate and time-consuming. A generating set of oriented moves simplifies this by requiring verification for only a minimal subset from which all other moves can be derived. While generating sets for classical oriented Reidemeister moves are well-established, their virtual counterparts are less explored. In this study, we enumerate the oriented virtual Reidemeister moves, identifying seventeen distinct moves after accounting for redundancies due to rotational and combinatorial symmetries. We prove that a four-element subset serves as a generating set for these moves. This result offers a streamlined approach to verifying invariants of oriented virtual knots and lays the groundwork for future advancements in virtual knot theory, particularly in the study of invariants and their computational properties.

A generating set of Reidemeister moves of oriented virtual knots

TL;DR

The paper addresses verifying invariants for oriented virtual knots by establishing a generating set for oriented virtual Reidemeister moves. It enumerates the 17 oriented virtual moves and proves that the set suffices to generate all oriented virtual moves, using a structured proof that partitions moves into – and derives each from through a sequence of lemmas. This result reduces invariant-verification work and provides a foundation for future study of virtual knot invariants and their computational properties. The work extends classical generating-set results (e.g., Polyak, Caprau–Scott) to the virtual setting, clarifying minimality and informing potential virtual quandle axioms.

Abstract

In oriented knot theory, verifying a quantity is an invariant involves checking its invariance under all oriented Reidemeister moves, a process that can be intricate and time-consuming. A generating set of oriented moves simplifies this by requiring verification for only a minimal subset from which all other moves can be derived. While generating sets for classical oriented Reidemeister moves are well-established, their virtual counterparts are less explored. In this study, we enumerate the oriented virtual Reidemeister moves, identifying seventeen distinct moves after accounting for redundancies due to rotational and combinatorial symmetries. We prove that a four-element subset serves as a generating set for these moves. This result offers a streamlined approach to verifying invariants of oriented virtual knots and lays the groundwork for future advancements in virtual knot theory, particularly in the study of invariants and their computational properties.

Paper Structure

This paper contains 2 sections, 14 theorems, 11 equations, 6 figures.

Table of Contents

  1. Introduction
  2. A generating set of oriented virtual knot diagrams

Key Result

Theorem 1.2

Let $D$ and $D'$ be two diagrams in $\mathbb{R}^2$, representing the same oriented link. Then one may pass from $D$ to $D'$ by isotopy and a finite sequence of four oriented Reidemeister moves $C1a, C1b, C2a$ and $C3a$, shown in Fig.gscrm.

Figures (6)

  • Figure 1: Classical Reidemeister moves
  • Figure 2: Virtual Reidemeister moves
  • Figure 3: Oriented classical Reidemeister moves
  • Figure 4: A generating set of classical Reidemeister moves
  • Figure 5: Oriented virtual Reidemeister moves
  • ...and 1 more figures

Theorems & Definitions (26)

  • Definition 1.1: Generating set
  • Theorem 1.2: polyak, Theorem 1.1
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • ...and 16 more