Dehn surgery on arborescent links

Abstract

This paper studies Dehn surgery on a large class of links, called arborescent links. It will be shown that if an arborescent link L L is sufficiently complicated, in the sense that it is composed of at least 4 4 rational tangles T ( p i / q i ) T(p_{i}/q_{i}) with all q i > 2 q_{i} > 2 , and none of its length 2 tangles are of the form T ( 1 / 2 q 1 , 1 / 2 q 2 ) T(1/2q_{1}, 1/2q_{2}) , then all complete surgeries on L L produce Haken manifolds. The proof needs some result on surgery on knots in tangle spaces. Let T ( r / 2 s , p / 2 q ) = ( B , t 1 ∪ t 2 ∪ K ) T(r/2s, p/2q) = (B, t_{1}\cup t_{2}\cup K) be a tangle with K K a closed circle, and let M = B − Int ⁡ N ( t 1 ∪ t 2 ) M = B - \operatorname {Int} N(t_{1}\cup t_{2}) . We will show that if s > 1 s>1 and p ≢ ± 1 p \not \equiv \pm 1 mod 2 q 2q , then ∂ M \partial M remains incompressible after all nontrivial surgeries on K K . Two bridge links are a subclass of arborescent links. For such a link L ( p / q ) L(p/q) , most Dehn surgeries on it are non-Haken. However, it will be shown that all complete surgeries yield manifolds containing essential laminations, unless p / q p/q has a partial fraction decomposition of the form 1 / ( r − 1 / s ) 1/(r-1/s) , in which case it does admit non-laminar surgeries.