Pretzel knots up to nine crossings

TL;DR

This paper investigates the Jones polynomial of pretzel links, establishing finiteness results for links with a given span, and provides an algorithm to identify pretzel knots, including classifying all pretzel knots up to nine crossings.

Contribution

It introduces a finiteness theorem for pretzel links with fixed Jones polynomial span and presents an algorithm to determine if a knot is pretzel, classifying pretzel knots up to nine crossings.

Findings

Finiteness of pretzel links with fixed Jones polynomial span

Identification of all pretzel knots up to nine crossings

Discovery that $8_{12}$ is the first non-pretzel knot

Abstract

There are infinitely many pretzel links with the same Alexander polynomial (actually with trivial Alexander polynomial). By contrast, in this note we revisit the Jones polynomial of pretzel links and prove that, given a natural number S, there is only a finite number of pretzel links whose Jones polynomials have span S. More concretely, we provide an algorithm useful for deciding whether or not a given knot is pretzel. As an application we identify all the pretzel knots up to nine crossings, proving in particular that $8_{12}$ is the first non-pretzel knot.