Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-99-02131-5
Published electronically: February 5, 1999
MathSciNet review: 1458339
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [BW] M. Brittenham and Y-Q. Wu, The classification of Dehn surgeries on 2-bridge knots, preprint.
  • [BS] F. Bonahon and L. Siebenmann, Geometric splittings of knots, and Conway's algebraic knots, preprint.
  • [Co] J. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra, Pergamon, New York and Oxford, 1970, pp. 329-358. MR 41:2661
  • [CGLS] M. Culler, C. Gordon, J. Luecke and P. Shalen, Dehn surgery on knots, Annals of Math. 125 (1987), 237-300. MR 88a:57026
  • [De1] C. Delman, Essential laminations and Dehn surgery on 2-bridge knots, Topology and its Appl. 63 (1995), 201-221. MR 96c:57029
  • [De2] -, Constructing essential laminations which survive all Dehn surgeries, preprint.
  • [Fl] W. Floyd, Incompressible surfaces in 3-manifolds: The space of boundary curves, Low-dimensional topology and Kleinian groups, London Math. Soc. LN, 112, 1986, pp. 131-143. MR 89f:57019
  • [Ga] D. Gabai, Eight problems in the geometric theory of foliations and laminations on 3-manifolds, Proceedings of Georgia Topology Conference, Part II, 1997, pp. 1-33.
  • [Ga2] -, Genera of the arborescent links, Mem. Amer. Math. Soc. 339 (1986), 1-98. MR 87h:57010
  • [GO] D. Gabai and U. Oertel, Essential laminations in 3-manifolds, Ann. Math. 130 (1989), 41-73. MR 90h:57012
  • [Gor] C. Gordon, Dehn surgery on knots, Proceedings of the International Congress of Mathematicians, Kyoto (1990), 631-642. MR 93e:57006
  • [GL] C. Gordon and J. Luecke, Knots are determined by their complements, Jour. Amer. Math. Soc. 2 (1989), 371-415. MR 90a:57006a
  • [HT] A. Hatcher and W. Thurston, Incompressible surfaces in 2-bridge knot complements, Invent. Math. 79 (1985), 225-246. MR 86g:57003
  • [He] J. Hempel, 3-manifolds, Ann. Math. Studies 86, Princeton Univ. Press, 1976. MR 54:3702
  • [Oe] U. Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984), 209-230. MR 85j:57008
  • [Sch] M. Scharlemann, Unlinking via simultaneous crossing changes, Trans. Amer. Math. Soc. 336 (1993), 855-868. MR 93k:57021
  • [St] E. Starr, Curves in handlebodies, Thesis, UC Berkeley (1992).
  • [Th] W. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982), 357-381. MR 83h:57019
  • [Wu1] Y-Q. Wu, The classification of nonsimple algebraic tangles, Math. Ann. 304 (1996), 457-480. MR 97b:57010
  • [Wu2] -, Dehn surgery on arborescent knots, J. Diff. Geo. 43 (1996), 171-197. MR 97j:57013
  • [Wu3] -, Imcompressibility of surfaces in surgered 3-manifolds, Topology 31 (1992), 271-279. MR 94e:57027

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 57N10, 57M25, 57M50

Retrieve articles in all journals with MSC (1991): 57N10, 57M25, 57M50


Additional Information

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

DOI: https://doi.org/10.1090/S0002-9947-99-02131-5
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

American Mathematical Society