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The H(n)-move is an unknotting operation for virtual and welded links

Danish Ali, Zhiqing Yang, Abid Hussain, Mohd Ibrahim Sheikh

TL;DR

This work extends the $H(n)$-move unknotting operation from classical knots to virtual and welded links, establishing that any virtual knot diagram can be unknotted by a finite sequence of $H(n)$-moves together with generalized Reidemeister moves. It provides constructive realizations of crossing virtualization using $H(2)$ and $H(3)$-moves and shows that forbidden moves ($F_U$, $F_O$) and CF-moves can be realized via $H(n)$-moves plus appropriate Reidemeister moves, linking various unknotting notions. The results also demonstrate that the $H(n)$-move unknotting framework extends to welded knots, offering a unified approach to unknotting in virtual and welded knot theories. Overall, the paper bridges classical and generalized unknotting operations, giving explicit move sequences to reduce diagrams to the trivial knot.

Abstract

An unknotting operation is a local move such that any knot diagram can be transformed into a diagram of the trivial knot by a finite sequence of these operations plus some Reidemeister moves. It is known that for all $n \geq 2$ the $H(n)$-move is an unknotting operation for classical knots and links. In this paper, we extend the classical unknotting operation $H(n)$-move to virtual knots and links. Virtualization and forbidden move are well-known unknotting operations for virtual knots and links. We also show that virtualization and forbidden move can be realized by a finite sequence of generalized Reidemeister moves and $H(n)$-moves.

The H(n)-move is an unknotting operation for virtual and welded links

TL;DR

This work extends the -move unknotting operation from classical knots to virtual and welded links, establishing that any virtual knot diagram can be unknotted by a finite sequence of -moves together with generalized Reidemeister moves. It provides constructive realizations of crossing virtualization using and -moves and shows that forbidden moves (, ) and CF-moves can be realized via -moves plus appropriate Reidemeister moves, linking various unknotting notions. The results also demonstrate that the -move unknotting framework extends to welded knots, offering a unified approach to unknotting in virtual and welded knot theories. Overall, the paper bridges classical and generalized unknotting operations, giving explicit move sequences to reduce diagrams to the trivial knot.

Abstract

An unknotting operation is a local move such that any knot diagram can be transformed into a diagram of the trivial knot by a finite sequence of these operations plus some Reidemeister moves. It is known that for all the -move is an unknotting operation for classical knots and links. In this paper, we extend the classical unknotting operation -move to virtual knots and links. Virtualization and forbidden move are well-known unknotting operations for virtual knots and links. We also show that virtualization and forbidden move can be realized by a finite sequence of generalized Reidemeister moves and -moves.
Paper Structure (2 sections, 6 theorems, 12 figures)

This paper contains 2 sections, 6 theorems, 12 figures.

Table of Contents

  1. Introduction
  2. $H(n)$-move is unknotting operation for virtual links

Key Result

Theorem 2.1

Given any integer $n (\geq 2)$, any virtual knot diagram can be transformed into a trivial knot diagram by a finite sequence of $H(n)$-moves plus some generalized Reidemeister moves.

Figures (12)

  • Figure 1: Classical crossings and a virtual crossing
  • Figure 2: Classical Reidemeister moves
  • Figure 3: Virtual Reidemeister moves and a mixed Reidemeister move
  • Figure 4: Crossing virtualization
  • Figure 5: An $H(n)-move$
  • ...and 7 more figures

Theorems & Definitions (11)

  • Theorem 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • ...and 1 more