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

Hyperbolicity of arborescent tangles and arborescent links