Geometry of Generalized Augmented Braids

TL;DR

This paper investigates the geometric properties of generalized augmented links derived from positive braids with full twists, providing criteria for when these links exhibit hyperbolic geometry.

Contribution

It characterizes the conditions under which generalized augmented links of positive braids with full twists are hyperbolic, extending understanding of their geometric structures.

Findings

Identifies conditions for hyperbolicity of augmented links

Provides a characterization based on braid properties

Enhances understanding of link geometries

Abstract

A generalized augmented link of a knot $K$ is a link obtained by adding trivial components to $K$ that bound $n$-punctured disks. In this paper we consider that $K$ is given by a positive braid with at least one full twist. We characterize when the generalized augmented links of $K$ have hyperbolic geometry.