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A Generalization of Pretzel Links via Spatial Graphs

Kotaro Shoji

Abstract

In this paper, we introduce \textit{graph-pretzel links}, a generalization of classical pretzel links based on spatial graph projections. As our main result, we investigate a subfamily associated with the complete graph on four vertices to construct an infinite family of distinct ribbon knots. Furthermore, although they all share a trivial Alexander polynomial, they can be distinguished from one another by their Jones polynomials.

A Generalization of Pretzel Links via Spatial Graphs

Abstract

In this paper, we introduce \textit{graph-pretzel links}, a generalization of classical pretzel links based on spatial graph projections. As our main result, we investigate a subfamily associated with the complete graph on four vertices to construct an infinite family of distinct ribbon knots. Furthermore, although they all share a trivial Alexander polynomial, they can be distinguished from one another by their Jones polynomials.
Paper Structure (7 sections, 6 theorems, 11 equations, 6 figures)

This paper contains 7 sections, 6 theorems, 11 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Definition and Properties of the link class
  3. The case for the complete graph on four vertices
  4. Proof of Main Theorems
  5. Proof of Theorem \ref{['maintheorem']}
  6. Proof of Theorem \ref{['maintheoremtwo']}
  7. Concluding Remarks

Key Result

Theorem 1.1

Let $K_{n}:=P(\begin{tikzpicture}[scale=1.5,rotate=180] \coordinate (A) at (0,0); \coordinate (B) at (0,1); \coordinate (C) at (1,-0.5); \coordinate (D) at (-1,-0.5); \draw (A)--(B)--(C)--(D)--(A)--(C)--(D)--(B); \fill (A) circle (1.2pt); \fil Furthermore, $J_{K_n}(q)\neq J_{K_m}(q)$ if $n\neq m$. In particular, $K_n$ is not isotopic to $K_m

Figures (6)

  • Figure 1: 1/8 twist
  • Figure 2:
  • Figure 3: Broken lines represent twist regions. $n_i$ shows the number of twists.
  • Figure 4:
  • Figure 5: $n$ in the diagram is a negative integer
  • ...and 1 more figures

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 3.1
  • ...and 7 more