Hyperbolicity of arborescent tangles and arborescent links

Details Hyperbolicity of arborescent tangles and arborescent links Hyperbolicity of arborescent tangles and arborescent links

2008-01-30

arxiv.org/abs/0801.4704v2

Mathematics

Geometric Topology

26 pages, 18 figures

Scientific paper

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.

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