Highly twisted diagrams

TL;DR

This paper proves that knots and links with certain highly twisted diagrams are hyperbolic by showing their complements are unannular and atoroidal, using a novel Euler characteristic approach.

Contribution

It introduces a new proof technique for hyperbolicity of knots with 3-highly twisted diagrams, extending previous results for 6-highly twisted diagrams.

Findings

Knots with 3-highly twisted diagrams are hyperbolic.

The proof uses unannular and atoroidal complement properties.

A new Euler characteristic method is developed.

Abstract

We prove that the knots and links that admit a 3-highly twisted irreducible diagram with more than two twist regions are hyperbolic. This should be compared with a result of Futer-Purcell for 6-highly twisted diagrams. While their proof uses geometric methods our proof is achieved by showing that the complements of such knots or links are unannular and atoroidal. This is done by using a new approach involving an Euler characteristic argument.