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Explicit Formulas for the Alexander Polynomial of Pretzel Knots

Y. Belousov

Abstract

We provide explicit formulas for the Alexander polynomial of pretzel knots and establish several immediate corollaries, including the characterization of pretzel knots with a trivial Alexander polynomial. As an application, we construct a new family of knots that are topologically slice but not smoothly slice.

Explicit Formulas for the Alexander Polynomial of Pretzel Knots

Abstract

We provide explicit formulas for the Alexander polynomial of pretzel knots and establish several immediate corollaries, including the characterization of pretzel knots with a trivial Alexander polynomial. As an application, we construct a new family of knots that are topologically slice but not smoothly slice.

Paper Structure

This paper contains 8 sections, 5 theorems, 20 equations, 3 figures.

Table of Contents

  1. Proof of Theorem \ref{['thm: main formula']}
  2. Case \ref{['case 3 thm main']} of Theorem \ref{['thm: main formula']} and Case \ref{['case 2 lemma']} of Lemma \ref{['lem: two component pretzel']}
  3. Cases \ref{['case 1 thm main']} and \ref{['case 2 thm main']} of Theorem \ref{['thm: main formula']} and Case \ref{['case 1 lemma']} of Lemma \ref{['lem: two component pretzel']}
  4. Corollaries
  5. Determinant of pretzel knots
  6. Degree of the Alexander polynomial of pretzel knots
  7. Topologically slice pretzel knots
  8. Irreducible solutions of the system \ref{['eq: system trivial alex']} for $n=5$

Key Result

Theorem 1

Let $K=P(q_1, q_2, \dots, q_n)$ be a pretzel knot. Then the Alexander polynomial $\Delta_K(t)$ is given by the following cases:

Figures (3)

  • Figure 1: Examples of pretzel knots
  • Figure 2: Illustration of the skein relation
  • Figure 3: Example of two-component pretzel link $P(3,3,3,3)$ with different orientations

Theorems & Definitions (10)

  • Theorem 1
  • Lemma 1
  • Remark 1
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Corollary 3
  • proof
  • Conjecture 1