Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Exceptional surgery on knots

Authors: S. Boyer and X. Zhang
Journal: Bull. Amer. Math. Soc. 31 (1994), 197-203
MSC: Primary 57N10; Secondary 57M25
MathSciNet review: 1260518
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

  • [BH] S. Bleiler and C. Hodgson, Spherical space forms and Dehn fillings, preprint. MR 1396779 (97f:57007)
  • [BZ1] S. Boyer and X. Zhang, Finite Dehn surgery on knots, preprint. MR 1333293 (97h:57013)
  • [BZ2] -, The semi-norm and Dehn filling, preprint.
  • [CGLS] M. Culler, C. M. Gordon, J. Luecke, and P. B. Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987), 237-300. MR 881270 (88a:57026)
  • [Ga1] D. Gagai, Surgery on knots in solid tori, Topology 28 (1989), 1-6. MR 991095 (90h:57005)
  • [Ga2] -, 1-bridge braids in solid tori, Topology Appl. 37 (1990), 221-235. MR 1082933 (92b:57011)
  • [Ga3] -, Foliations and the topology of 3-manifolds. III, J. Differential Geom. 26 (1987), 479-536. MR 910018 (89a:57014b)
  • [Go1] C. M. Gordon, Dehn surgery on knots, Proceedings of the International Congress of Mathematians, Kyoto, 1990, The Mathematical Society of Japan, Tokyo, Japan. MR 1159250 (93e:57006)
  • [Go2] -, Boundary slopes and punctured tori in 3-manifolds, preprint.
  • [GLi] C. M. Gordon and R. A. Litherland, Incompressible planar surfaces in 3-manifolds, Topology Appl. 18 (1984), 121-144. MR 769286 (86e:57013)
  • [GLu1] C. M. Gordon and J. Luecke, Reducible manifolds and Dehn surgery, preprint. MR 1380506 (97b:57013)
  • [GLu2] -, Address, John Luecke, 1993 Georgia International Topology Conference, University of Georgia at Athens, 1-13 August 1993.
  • [Li] W. B. R. Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. of Math. (2) 76 (1962), 531-540. MR 0151948 (27:1929)
  • [Mi] J. Milnor, Groups which act on $ {S^n}$ without fixed points, Amer. J. Math. 79 (1957), 623-631. MR 0090056 (19:761d)
  • [R] D. Rolfsen, Knots and links, Publish or Perish, Cambridge, MA, 1979. MR 0515288 (58:24236)
  • [Sch] M. Scharlemann, Producing reducible 3-manifolds by surgery on a knot, Topology 29 (1990), 481-500. MR 1071370 (91i:57003)
  • [Ta] D. Tanguay, Chirurgie finie et noeuds rationnels, Doctoral dissertation, Université du Québec à Montréal, 1994.
  • [Th] W. Thurston, Three-dimensional manifolds, Kleinian groups, and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357-381. MR 648524 (83h:57019)
  • [Wa] A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960), 503-528. MR 0125588 (23:A2887)
  • [We] J. Weeks, Hyperbolic structures on three-manifolds, Ph.D. thesis, Princeton University, 1985.

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC: 57N10, 57M25

Retrieve articles in all journals with MSC: 57N10, 57M25

Additional Information

Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society