Hyperbolicity of arborescent tangles and arborescent links

TL;DR

This paper characterizes when arborescent links and tangles are hyperbolic by explicitly identifying essential surfaces and conditions under which their complements are non-hyperbolic, extending understanding of their geometric structures.

Contribution

It provides a complete classification of non-hyperbolic arborescent links and tangles based on their structural composition, including explicit criteria involving rational tangles.

Findings

Arborescent tangle complements with non-negative Euler characteristic are explicitly characterized.

A large arborescent link is non-hyperbolic if and only if it contains Q2.

The paper proves a theorem linking the presence of Q2 in arborescent links to their non-hyperbolicity.

Abstract

In this paper, we study the hyperbolicity of arborescent tangles and arborescent links. We will explicitly determine all essential surfaces in arborescent tangle complements with non-negative Euler characteristic, and show that given an arborescent tangle T, the complement X(T) is non-hyperbolic if and only if T is a rational tangle, T=Q_m * T' for some m greater than or equal to 1, or T contains Qn for some n greater than or equal to 2. We use these results to prove a theorem of Bonahon and Seibenmann which says that a large arborescent link L is non-hyperbolic if and only if it contains Q2.