Highly twisted plat diagrams

TL;DR

This paper proves that a broad class of highly twisted plat knots and links are hyperbolic by demonstrating their complements are unannular and atoroidal, using a novel Euler characteristic approach.

Contribution

It introduces a new method to establish hyperbolicity of certain knots and links, extending previous results with a different proof technique.

Findings

All 3-highly twisted 2m-plat knots and links with m ≥ 2 are hyperbolic.

The complements are shown to be unannular and atoroidal.

The proof employs an Euler characteristic argument, offering a new approach.

Abstract

We prove that the knots and links in the infinite set of $3$-highly twisted $2m$-plats, with $m \geq 2$, are all 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.