Dehn Surgery on Arborescent Links 3

This paper studies Dehn surgery on a large class of links, called arborescent links. It will be shown that if an arborescent link L is suuciently complicated, in the sense that it is composed of at least 4 rational tangles T (p i =q i) with all q i > 2, and none of its length 2 tangles are of the form T (1=2q 1 ; 1=2q 2), then all complete surgeries on L produce Haken manifolds. The proof needs some result on surgery on knots in tangle spaces. Let T (r=2s; p=2q) = (B; t 1 t 2 K) be a tangle with K a closed circle, and let M = B ? IntN(t 1 t 2). We will show that if s > 1 and p 6 6 1 mod 2q, then @M remains incompressible after all nontrivial surgeries on K. Two bridge links are a subclass of arborescent links. For such a link 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 has a partial fraction decomposition of the form 1=(r ? 1=s), in which case it does admit non-laminar surgeries. 0. Introduction In Dehn surgery theory, we would like to know what 3-manifolds are produced through certain surgeries on certain knots or links. More explicitly, we want to know how many surgeries yield Haken, hyperbolic, or laminar manifolds, and how many of them are \exceptional", meaning that the resulting manifolds are reducible, or have cyclic or nite fundamental group, or are small Seifert bered spaces. There have been many results on these problems for surgery on knots. See Gor] and Ga] for surveys and frontier problems. These results, however, are not ready to be generalized to surgery on links of multiple components. The major diiculty is that surgery on one component of the link may change the property of the other components. An exception is Thurston's hyperbolic surgery theorem Th], which says that if L is a hyperbolic link, then except for nitely many slopes on each component of L, all other surgeries are hy-perbolic. Another interesting result is Scharlemann's simultaneous crossing change theorem, see Sch]. There has been extensive study about surgery on a large class of knots called arborescent knots, also known as Conway's algebraic knots Co, BS], which include all Montesinos knots. The name \arborescent links" is rst …