Exceptional surgery on knots
HTML articles powered by AMS MathViewer
- by S. Boyer and X. Zhang PDF
- Bull. Amer. Math. Soc. 31 (1994), 197-203 Request permission
Abstract:
Let M be an irreducible, compact, connected, orientable 3-manifold whose boundary is a torus. We show that if M is hyperbolic, then it admits at most six finite/cyclic fillings of maximal distance 5. Further, the distance of a finite/cyclic filling to a cyclic filling is at most 2. If M has a non-boundary-parallel, incompressible torus and is not a generalized 1-iterated torus knot complement, then there are at most three finite/cyclic fillings of maximal distance 1. Further, if M has a non-boundary-parallel, incompressible torus and is not a generalized 1- or 2-iterated torus knot complement and if M admits a cyclic filling of odd order, then M does not admit any other finite/cyclic filling. Relations between finite/cyclic fillings and other exceptional fillings are also discussed.References
- Steven A. Bleiler and Craig D. Hodgson, Spherical space forms and Dehn filling, Topology 35 (1996), no. 3, 809–833. MR 1396779, DOI 10.1016/0040-9383(95)00040-2
- S. Boyer and X. Zhang, Finite Dehn surgery on knots, J. Amer. Math. Soc. 9 (1996), no. 4, 1005–1050. MR 1333293, DOI 10.1090/S0894-0347-96-00201-9 —, The semi-norm and Dehn filling, preprint.
- 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
- David Gabai, Surgery on knots in solid tori, Topology 28 (1989), no. 1, 1–6. MR 991095, DOI 10.1016/0040-9383(89)90028-1
- David Gabai, $1$-bridge braids in solid tori, Topology Appl. 37 (1990), no. 3, 221–235. MR 1082933, DOI 10.1016/0166-8641(90)90021-S
- David Gabai, Foliations and the topology of $3$-manifolds. II, J. Differential Geom. 26 (1987), no. 3, 461–478. MR 910017
- 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 —, Boundary slopes and punctured tori in 3-manifolds, preprint.
- C. McA. Gordon and R. A. Litherland, Incompressible planar surfaces in $3$-manifolds, Topology Appl. 18 (1984), no. 2-3, 121–144. MR 769286, DOI 10.1016/0166-8641(84)90005-1
- C. McA. Gordon and J. Luecke, Reducible manifolds and Dehn surgery, Topology 35 (1996), no. 2, 385–409. MR 1380506, DOI 10.1016/0040-9383(95)00016-X —, Address, John Luecke, 1993 Georgia International Topology Conference, University of Georgia at Athens, 1-13 August 1993.
- W. B. R. Lickorish, A representation of orientable combinatorial $3$-manifolds, Ann. of Math. (2) 76 (1962), 531–540. MR 151948, DOI 10.2307/1970373
- John Milnor, Groups which act on $S^n$ without fixed points, Amer. J. Math. 79 (1957), 623–630. MR 90056, DOI 10.2307/2372566
- Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
- Martin Scharlemann, Producing reducible $3$-manifolds by surgery on a knot, Topology 29 (1990), no. 4, 481–500. MR 1071370, DOI 10.1016/0040-9383(90)90017-E D. Tanguay, Chirurgie finie et noeuds rationnels, Doctoral dissertation, Université du Québec à Montréal, 1994.
- 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
- Andrew H. Wallace, Modifications and cobounding manifolds, Canadian J. Math. 12 (1960), 503–528. MR 125588, DOI 10.4153/CJM-1960-045-7 J. Weeks, Hyperbolic structures on three-manifolds, Ph.D. thesis, Princeton University, 1985.
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 31 (1994), 197-203
- MSC: Primary 57N10; Secondary 57M25
- DOI: https://doi.org/10.1090/S0273-0979-1994-00516-6
- MathSciNet review: 1260518