Dehn surgery on arborescent links

This paper studies Dehn surgery on a large class of links, called arborescent links. It will be shown that if an arborescent link is sufficiently complicated, in the sense that it is composed of at least rational tangles with all , and none of its length 2 tangles are of the form , then all complete surgeries on produce Haken manifolds. The proof needs some result on surgery on knots in tangle spaces. Let be a tangle with a closed circle, and let . We will show that if and mod , then remains incompressible after all nontrivial surgeries on . Two bridge links are a subclass of arborescent links. For such a link , most Dehn surgeries on it are non-Haken. However, it will be shown that all complete surgeries yield manifolds containing essential laminations, unless has a partial fraction decomposition of the form , in which case it does admit non-laminar surgeries.