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Seifert-fibered surgeries which do not arise from primitive/Seifert-fibered constructions

Authors: Thomas Mattman, Katura Miyazaki and Kimihiko Motegi
Journal: Trans. Amer. Math. Soc. 358 (2006), 4045-4055
MSC (2000): Primary 57M25
Published electronically: September 22, 2005
MathSciNet review: 2219009
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Abstract: We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Dean's primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert-fibered surgeries arise from Dean's primitive/Seifert-fibered construction. The $(-3,3,5)$-pretzel knot belongs to both of the infinite families.

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  • 1. J. Berge; Some knots with surgeries yielding lens spaces, unpublished manuscript.
  • 2. S. A. Bleiler; Knots prime on many strings, Trans. Amer. Math. Soc. 282 (1984), 385-401. MR 0728719 (85h:57003)
  • 3. S. A. Bleiler; Prime tangles and composite knots, Lect. Notes in Math., vol. 1144, Springer-Verlag, 1985, pp. 1-13. MR 0823278 (87e:57006)
  • 4. S. Bleiler and C. Hodgson; Spherical space forms and Dehn filling, Topology 35 (1996), 809-833. MR 1396779 (97f:57007)
  • 5. S. Boyer and X. Zhang; Finite surgery on knots, J. Amer. Math. Soc. 9 (1996), 1005-1050. MR 1333293 (97h:57013)
  • 6. G. Burde and H. Zieschang; Knots, de Gruyter Studies in Mathematics 5, 1985. MR 0808776 (87b:57004)
  • 7. J. Dean; Hyperbolic knots with small Seifert-fibered Dehn surgeries, Ph.D. thesis, University of Texas at Austin, 1996.
  • 8. J. Dean; Small Seifert-fibered Dehn surgery on hyperbolic knots, Algebraic and Geometric Topology 3 (2003), 435-472. MR 1997325 (2004m:57009)
  • 9. M. Eudave-Muñoz; Non-hyperbolic manifolds obtained by Dehn surgery on a hyperbolic knot, In: Studies in Advanced Mathematics vol. 2, part 1, (ed. W. Kazez), 1997, Amer. Math. Soc. and International Press, pp. 35-61. MR 1470720 (98i:57007)
  • 10. M. Eudave-Muñoz; On hyperbolic knots with Seifert fibered Dehn surgeries, Topology Appl. 121 (2002), 119-141. MR 1903687 (2003c:57005)
  • 11. D. Gabai; Foliations and the topology of 3-manifolds III, J. Diff. Geom. 26 (1987), 479-536. MR 0910018 (89a:57014b)
  • 12. F. González-Acuña and H. Short; Knot surgery and primeness, Math. Proc. Camb. Phil. Soc. 99 (1986), 89-102. MR 0809502 (87c:57003)
  • 13. C. McA. Gordon; Dehn Filling; a survey, Knot theory (Warsaw, 1995), 129-144, Banach Center, Publ. 42, Polish Acad. Sci., Warsaw, 1998. MR 1634453 (99e:57028)
  • 14. C. McA. Gordon and J. Luecke; Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), 371-415. MR 0965210 (90a:57006a)
  • 15. T. Mattman; The Culler-Shalen seminorms of pretzel knots, Ph.D. thesis, McGill University, Montréal, 2000.
  • 16. K. Miyazaki and K. Motegi; Seifert fibered manifolds and Dehn surgery II, Math. Ann. 311 (1998), 647-664. MR 1637964 (99g:57010)
  • 17. K. Miyazaki and K. Motegi; Seifert fibered manifolds and Dehn surgery III, Comm. Anal. Geom. 7 (1999), 551-582. MR 1698388 (2000d:57030)
  • 18. K. Miyazaki and K. Motegi; On primitive/Seifert-fibered constructions, Math. Proc. Camb. Phil. Soc. 138 (2005), 421-435. MR 2138571
  • 19. J. M. Montesinos; Surgery on links and double branched coverings of $S^3$, Ann. Math. Studies 84 (1975), 227-260. MR 0380802 (52:1699)
  • 20. K. Morimoto; On the additivity of h-genus of knots, Osaka J. Math. 31 (1994), 137-145. MR 1262793 (95d:57006)
  • 21. K. Morimoto; There are knots whose tunnel numbers go down under connected sum, Proc. Amer. Math. Soc. 123 (1995), 3527-3532. MR 1317043 (96a:57022)
  • 22. K. Motegi; Dehn surgeries, group actions and Seifert fiber spaces, Comm. Anal. Geom. 11 (2003), 343-389. MR 2014880 (2004m:57015)
  • 23. H. Schubert; Knoten und Vollringe, Acta Math. 90 (1953), 131-286.MR 0072482 (17:291d)
  • 24. J. Weeks; SnapPea: a computer program for creating and studying hyperbolic $3$-manifolds, freely available from
  • 25. H. Zieschang; On simple systems of paths on complete pretzels, Amer. Math. Soc. Transl. 92, 127-137.

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Additional Information

Thomas Mattman
Affiliation: Department of Mathematics and Statistics, California State University–Chico, Chico, California 95929-0525

Katura Miyazaki
Affiliation: Faculty of Engineering, Tokyo Denki University, Tokyo 101-8457, Japan

Kimihiko Motegi
Affiliation: Department of Mathematics, Nihon University, Tokyo 156-8550, Japan

Keywords: Dehn surgery, hyperbolic knot, Seifert fiber space, primitive/Seifert-fibered construction
Received by editor(s): January 20, 2003
Received by editor(s) in revised form: June 28, 2004
Published electronically: September 22, 2005
Additional Notes: The first author was supported in part by grants from NSERC and FCAR
The second author was supported in part by Grant-in-Aid for Scientific Research (No. 40219978), The Ministry of Education, Culture, Sports, Science and Technology, Japan.
Dedicated: Dedicated to Cameron McA. Gordon on the occasion of his 60th birthday
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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