Hyperbolicity of arborescent tangles and arborescent links

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 ⩾ 1 , or T contains Q n for some n ⩾ 2 . We use these results to prove a theorem of Bonahon and Siebenmann which says that a large arborescent link L is non-hyperbolic if and only if it contains Q 2 .