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Virtualized Delta, sharp, and pass moves for oriented virtual knots and links

Takuji Nakamura, Yasutaka Nakanishi, Shin Satoh, Kodai Wada

Abstract

We study virtualized Delta, sharp, and pass moves for oriented virtual links, and give necessary and sufficient conditions for two oriented virtual links to be related by the local moves. In particular, they are unknotting operations for oriented virtual knots. We provide lower bounds for the unknotting numbers and prove that they are best possible.

Virtualized Delta, sharp, and pass moves for oriented virtual knots and links

Abstract

We study virtualized Delta, sharp, and pass moves for oriented virtual links, and give necessary and sufficient conditions for two oriented virtual links to be related by the local moves. In particular, they are unknotting operations for oriented virtual knots. We provide lower bounds for the unknotting numbers and prove that they are best possible.
Paper Structure (5 sections, 27 theorems, 33 equations, 33 figures)

This paper contains 5 sections, 27 theorems, 33 equations, 33 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm11']}
  3. Proof of the equivalence of (i) and (ii) in Theorem \ref{['thm12']}
  4. Proof of the equivalence of (i) and (iii) in Theorem \ref{['thm12']}
  5. $v\Delta^\wedge$-, $v\Delta^\circ$-, $v\sharp$-, and $vp$-unknotting numbers

Key Result

Theorem 1.1

Let $L$ and $L'$ be oriented $n$-component virtual links with $n\geq 2$. Then the following are equivalent.

Figures (33)

  • Figure 1.1: A virtualized $\Delta$-move for an oriented virtual knot or link
  • Figure 2.1: Virtualized $\Delta$-moves
  • Figure 2.2: Proof of Lemma \ref{['lem-cc']}(i)
  • Figure 2.3: Proof of Lemma \ref{['lem-cc']}(ii)
  • Figure 2.4: Proof of Lemma \ref{['lem-delta']}(i)
  • ...and 28 more figures

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 44 more