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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Dehn surgery on arborescent links


Author: Ying-Qing Wu
Journal: Trans. Amer. Math. Soc. 351 (1999), 2275-2294
MSC (1991): Primary 57N10; Secondary 57M25, 57M50
Published electronically: February 5, 1999
MathSciNet review: 1458339
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Abstract: This paper studies Dehn surgery on a large class of links, called arborescent links. It will be shown that if an arborescent link $L$ is sufficiently 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 $T(r/2s, p/2q) = (B, t_{1}\cup t_{2}\cup K)$ be a tangle with $K$ a closed circle, and let $M = B - \operatorname{Int} N(t_{1}\cup t_{2})$. We will show that if $s>1$ and $p \not \equiv \pm 1$ mod $2q$, then $\partial M$ 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.


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Additional Information

Ying-Qing Wu
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
Email: wu@math.uiowa.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02131-5
PII: S 0002-9947(99)02131-5
Received by editor(s): March 15, 1996
Received by editor(s) in revised form: April 17, 1997
Published electronically: February 5, 1999
Article copyright: © Copyright 1999 American Mathematical Society