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The Conway polynomials and Self Delta-equivalence of pretzel links

Yasutaka Nakanishi, Tetsuo Shibuya, Tatsuya Tsukamoto

Abstract

In this paper, we study the self delta-equivalence of pretzel links. If the number of components is 2, then we know the complete invariants in terms of the Conway polynomial for classification. We calculate the values. For pretzel links with more than or equal to 3 components, we give a necessary and sufficient condition to be self delta-equivalent.

The Conway polynomials and Self Delta-equivalence of pretzel links

Abstract

In this paper, we study the self delta-equivalence of pretzel links. If the number of components is 2, then we know the complete invariants in terms of the Conway polynomial for classification. We calculate the values. For pretzel links with more than or equal to 3 components, we give a necessary and sufficient condition to be self delta-equivalent.

Paper Structure

This paper contains 8 sections, 21 theorems, 7 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. the Conway polynomial of links
  4. Self Delta-equivalence of pretzel links
  5. $2$-component pretzel links
  6. $2$-component odd pretzel links
  7. $2$-component even pretzel links
  8. Self Delta-equivalence of slice pretzel links

Key Result

Proposition 1.1

A pair of knots $($or links$)$ are $\Delta$-equivalent if and only if they have the same number of components and the same linking numbers between the corresponding components. $\blacktriangleleft$$\blacktriangleleft$

Figures (8)

  • Figure 1: $\Delta$-move
  • Figure 2: a pretzel link diagram $P(k_1, k_2, \ldots, k_u)$
  • Figure 3: anti-parallel and parallel strands
  • Figure 4: $L(-3\mathrm{r}, 2\mathrm{s}, 1\mathrm{r})$
  • Figure 5: transformation of vertical twists into horizontal twists
  • ...and 3 more figures

Theorems & Definitions (40)

  • Proposition 1.1
  • Claim 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Claim 1.5
  • Proposition 1.6
  • Lemma 1.7
  • proof
  • Theorem 1.8
  • Theorem 1.9
  • ...and 30 more