Hyperbolicity of Augmented Links in the Thickened Torus

TL;DR

This paper demonstrates that augmenting a hyperbolic link in the thickened torus allows for a decomposition into angled tetrahedra, proving the hyperbolicity of the augmented link through geometric structures.

Contribution

It introduces a new decomposition method of augmented links in the thickened torus into torihedra and tetrahedra, establishing hyperbolicity via angled structures.

Findings

Decomposition of link complements into torihedra and tetrahedra.

Construction of an angled structure on the triangulation.

Proof that augmented links are hyperbolic.

Abstract

For a hyperbolic link K in the thickened torus with no bigons, we show that there is a decomposition of the complement of a link L, obtained from augmenting K, into torihedra. We further decompose the torihedra into angled pyramids and finally angled tetrahedra. These fit into an angled structure on a triangulation of the link complement, and thus by [5], this shows that L is hyperbolic.