Some invariants of pretzel links

TL;DR

This paper investigates the properties of pretzel links, establishing their hyperbolic nature with specific exceptions, and introduces new methods to compute their invariants, including Conway polynomials and genus, revealing their geometric and topological characteristics.

Contribution

It provides a new computation tree for Conway polynomials of pretzel links and demonstrates that genus and canonical genus coincide for these links, advancing understanding of their invariants.

Findings

Most pretzel knots are hyperbolic, except for eight torus knots.

A new computation tree for Conway polynomials of pretzel links is introduced.

Genus and canonical genus of pretzel links are shown to be equal.

Abstract

We show that nontrivial classical pretzel knots L(p,q,r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n-pretzel links using a new computation tree. As applications, we compute the genera of n-pretzel links using these polynomials and find the basket number of pretzel links by showing that the genus and the canonical genus of a pretzel link are the same.