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On wen knots

Celeste Damiani, Shin Satoh

Abstract

We introduce the notion of wen knots, and prove that the set of wen knots is a proper subset of the set of extended welded knots. Furthermore we prove that the complementary subset consists of welded knots up to horizontal mirror reflections. This allow us to characterise completely extended welded knots by the parity of their number of wens, that we can always reduce to 0 or 1.

On wen knots

Abstract

We introduce the notion of wen knots, and prove that the set of wen knots is a proper subset of the set of extended welded knots. Furthermore we prove that the complementary subset consists of welded knots up to horizontal mirror reflections. This allow us to characterise completely extended welded knots by the parity of their number of wens, that we can always reduce to 0 or 1.
Paper Structure (4 sections, 10 theorems, 6 equations, 8 figures)

This paper contains 4 sections, 10 theorems, 6 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Extended welded knots of odd type
  3. Extended welded knots of even type
  4. Extended welded links

Key Result

Theorem 1.1

Let $D$ and $D'$ be virtual knot diagrams with a single wen. If $D$ is related to $D'$ by a finite sequence of Reidemeister moves R1--R8 and wen moves W1--W4, then they are related by a finite sequence of R1--R8 and W1--W3, without the need of W4.

Figures (8)

  • Figure 1: Reidemeister moves R1--R8.
  • Figure 2: Wen moves W1--W4.
  • Figure 3: A virtual knot diagram $D$ and its horizontal mirror reflection $D^\dagger$.
  • Figure 4: A Gauss diagram associated to a wen knot diagram.
  • Figure 5: Wen moves W1 and W2 on Gauss diagrams
  • ...and 3 more figures

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 3.1
  • ...and 7 more