Dehn surgery on arborescent links
HTML articles powered by AMS MathViewer
- by Ying-Qing Wu PDF
- Trans. Amer. Math. Soc. 351 (1999), 2275-2294 Request permission
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
- M. Brittenham and Y-Q. Wu, The classification of Dehn surgeries on 2-bridge knots, preprint.
- F. Bonahon and L. Siebenmann, Geometric splittings of knots, and Conway’s algebraic knots, preprint.
- J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) Pergamon, Oxford, 1970, pp. 329–358. MR 0258014
- Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987), no. 2, 237–300. MR 881270, DOI 10.2307/1971311
- Charles Delman, Essential laminations and Dehn surgery on $2$-bridge knots, Topology Appl. 63 (1995), no. 3, 201–221. MR 1334307, DOI 10.1016/0166-8641(95)00085-U
- —, Constructing essential laminations which survive all Dehn surgeries, preprint.
- William J. Floyd, Incompressible surfaces in $3$-manifolds: the space of boundary curves, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 131–143. MR 903862
- 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.
- David Gabai, Genera of the arborescent links, Mem. Amer. Math. Soc. 59 (1986), no. 339, i–viii and 1–98. MR 823442, DOI 10.1090/memo/0339
- David Gabai and Ulrich Oertel, Essential laminations in $3$-manifolds, Ann. of Math. (2) 130 (1989), no. 1, 41–73. MR 1005607, DOI 10.2307/1971476
- Cameron McA. Gordon, Dehn surgery on knots, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 631–642. MR 1159250
- C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371–415. MR 965210, DOI 10.1090/S0894-0347-1989-0965210-7
- A. Hatcher and W. Thurston, Incompressible surfaces in $2$-bridge knot complements, Invent. Math. 79 (1985), no. 2, 225–246. MR 778125, DOI 10.1007/BF01388971
- John Hempel, $3$-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
- Ulrich Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984), no. 1, 209–230. MR 732067, DOI 10.2140/pjm.1984.111.209
- Martin Scharlemann, Unlinking via simultaneous crossing changes, Trans. Amer. Math. Soc. 336 (1993), no. 2, 855–868. MR 1200011, DOI 10.1090/S0002-9947-1993-1200011-3
- E. Starr, Curves in handlebodies, Thesis, UC Berkeley (1992).
- William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0
- Ying-Qing Wu, The classification of nonsimple algebraic tangles, Math. Ann. 304 (1996), no. 3, 457–480. MR 1375620, DOI 10.1007/BF01446301
- Ying-Qing Wu, Dehn surgery on arborescent knots, J. Differential Geom. 43 (1996), no. 1, 171–197. MR 1424423
- Ying Qing Wu, Incompressibility of surfaces in surgered $3$-manifolds, Topology 31 (1992), no. 2, 271–279. MR 1167169, DOI 10.1016/0040-9383(92)90020-I
Additional Information
- Ying-Qing Wu
- Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
- Email: wu@math.uiowa.edu
- Received by editor(s): March 15, 1996
- Received by editor(s) in revised form: April 17, 1997
- Published electronically: February 5, 1999
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1458339