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Dehn surgery on knots in solid tori
creating essential annuli

Authors: Chuichiro Hayashi and Kimihiko Motegi
Journal: Trans. Amer. Math. Soc. 349 (1997), 4897-4930
MSC (1991): Primary 57M25
MathSciNet review: 1373637
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Abstract: Let $M$ be a $3$-manifold obtained by performing a Dehn surgery on a knot in a solid torus. In the present paper we study when $M$ contains a separating essential annulus. It is shown that $M$ does not contain such an annulus in the majority of cases. As a corollary, we prove that symmetric knots in the $3$-sphere which are not periodic knots of period $2$ satisfy the cabling conjecture. This is an improvement of a result of Luft and Zhang. We have one more application to a problem on Dehn surgeries on knots producing a Seifert fibred manifold over the $2$-sphere with exactly three exceptional fibres.

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

  • 1. Boyer, S. and Zhang, X., The semi-norm and Dehn filling, (preprint).
  • 2. Culler, M., Gordon, C.McA., Luecke, J. and Shalen, P., Dehn surgery on knots, Ann. of Math. 125 (1987), 237-300; 127 (1988), 663. MR 88a:57026; MR 89c:57015
  • 3. Gabai, D., Foliations and the topology of 3-manifolds. II, J. Diff. Geom. 26 (1987), 461-478. MR 89a:57014a
  • 4. -, Foliations and the topology of 3-manifolds. III, J. Diff. Geom. 26 (1987), 479-536. MR 89a:57014b
  • 5. -, Surgery on knots in solid tori, Topology 28 (1989), 1-6. MR 90h:57005
  • 6. González-Acuña, F. and Short, H., Knot surgery and primeness, Math. Proc. Camb. Phil. Soc. 99 (1986), 89-102. MR 87c:57003
  • 7. Gordon, C.McA., Dehn surgery and satellite knots, Trans. Amer. Math. Soc. 275 (1983), 687-708. MR 84d:57003
  • 8. -, Dehn surgery on knots, Proceedings ICM Kyoto 1990 (1991), 555-590. MR 93e:57006
  • 9. -, Boundary slopes of punctured tori in 3-manifolds, Trans. Amer. Math. Soc. (to appear).
  • 10. Gordon, C.McA. and Litherland, R., Incompressible planar surfaces in 3-manifolds, Topology Appl. 18 (1984), 121-144. MR 86e:57013
  • 11. Gordon, C.McA. and Luecke, J., Only integral surgeries can yield reducible manifolds, Math. Proc. Camb. Phil. Soc. 102 (1987), 97-101. MR 89a:57003
  • 12. -, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), 371-415. MR 90a:5700ba
  • 13. -, Reducible manifolds and Dehn surgery, Topology 35 (1996), 385-409. CMP 96:10
  • 14. -, Dehn surgery on knots, Conference in Montreal (1995).
  • 15. Hayashi, C. and Motegi, K., Only single twist on unknots can produce composite knots, Trans. Amer. Math. Soc. 349 (1997), 4465-4479.
  • 16. Hayashi, C. and Shimokawa, K., Symmetric knots satisfy the cabling conjecture, Math. Proc. Camb. Phil. Soc. (to appear).
  • 17. Jaco, W, Lectures on three-manifold topology, CBMS Regional Conf. Ser. Math., no. 43. Amer. Math. Soc., 1980. MR 81k:57009
  • 18. -, Adding a $2$-handle to a $3$-manifold: an application to property R, Proc. Amer. Math. Soc. 92 (1984), 288-292. MR 86b:57006
  • 19. Kouno, M., Motegi, K. and Shibuya, T., Twisting and knot types, J. Math. Soc. Japan 44 (1992), 199-216. MR 93e:57011
  • 20. Luft, E. and Zhang, X., Symmetric knots and the cabling conjecture, Math. Ann. 298 (1994), 489-496. MR 95f:57014
  • 21. Mathieu, Y., Sur des noeuds qui ne sont pas déterminés par leur complément et problémes de cirurgie dans les variétés de dimension 3, Thèse, L'Université de Provence (1990).
  • 22. Miyazaki, K. and Motegi, K., Seifert fibred manifolds and Dehn surgery, Topology 36 (1997), 579-603. CMP 97:03
  • 23. -, Seifert fibred manifolds and Dehn surgery II, (preprint).
  • 24. Morgan, J. and Bass, H. (editors), The Smith conjecture, Pure and Applied Math., vol. 112, Academic Press, 1984. MR 86i:57002
  • 25. Scharlemann, M., Sutured manifolds and generalized Thurston norms, J. Diff. Geom. 29 (1987), 557-614. MR 90c:57021
  • 26. Wu, Y.-Q., Incompressibility of surfaces in surgered $3$-manifolds, Topology 31 (1992), 271-279. MR 94e:57027

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Additional Information

Chuichiro Hayashi
Affiliation: Department of Mathematics, Faculty of Science, Gakushuin University, Mejiro 1-5-1, Toshima-ku, Tokyo 171, Japan

Kimihiko Motegi
Affiliation: Department of Mathematics, College of Humanities & Sciences, Nihon University Sakurajosui 3-25-40, Setagaya-ku, Tokyo 156, Japan

Keywords: Dehn surgery, essential annulus, cabling conjecture, Seifert fibred manifold, Scharlemann cycle
Received by editor(s): June 28, 1995
Received by editor(s) in revised form: January 30, 1996
Additional Notes: The first author was supported in part by Fellowships of the Japan Society for the Promotion of Science for Japanese Junior Scientists, and the second author was supported in part by Grant-in-Aid for Encouragement of Young Scientists 07740077, The Ministry of Education, Science and Culture.
Article copyright: © Copyright 1997 American Mathematical Society

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