Algebraic & Geometric Topology

Small Seifert-fibered Dehn surgery on hyperbolic knots

John C Dean

Full-text: Open access

Abstract

In this paper, we define the primitive/Seifert-fibered property for a knot in S3. If satisfied, the property ensures that the knot has a Dehn surgery that yields a small Seifert-fibered space (i.e. base S2 and three or fewer critical fibers). Next we describe the twisted torus knots, which provide an abundance of examples of primitive/Seifert-fibered knots. By analyzing the twisted torus knots, we prove that nearly all possible triples of multiplicities of the critical fibers arise via Dehn surgery on primitive/Seifert-fibered knots.

Article information

Source
Algebr. Geom. Topol., Volume 3, Number 1 (2003), 435-472.

Dates
Received: 9 August 2002
Revised: 7 February 2003
Accepted: 4 April 2003
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882379

Digital Object Identifier
doi:10.2140/agt.2003.3.435

Mathematical Reviews number (MathSciNet)
MR1997325

Zentralblatt MATH identifier
1021.57002

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M27: Invariants of knots and 3-manifolds

Keywords
Dehn surgery hyperbolic knot Seifert-fibered space exceptional surgery

Citation

Dean, John C. Small Seifert-fibered Dehn surgery on hyperbolic knots. Algebr. Geom. Topol. 3 (2003), no. 1, 435--472. doi:10.2140/agt.2003.3.435. https://projecteuclid.org/euclid.agt/1513882379


Export citation

References

  • I Agol, Topology of hyperbolic 3-manifolds, Ph.D. thesis, University of California, San Diego (1998)
  • J Berge, Some knots with surgeries yielding lens spaces, unpublished manuscript
  • S Bleiler, C Hodgson, Spherical space forms and Dehn filling, Topology 35 (1996) 809–833
  • M Boileau, M Rost, H Zieschang, On Heegaard Decompositions of Torus Knot Exteriors and Related Seifert Fibre Spaces, Math. Ann. 279 (1988) 553–581
  • S Boyer, X Zhang, Finite Dehn surgery on knots, J. Amer. Math. Soc. 9 (1996) 1005–1050
  • Mark Brittenham, Ying-Qing Wu, The classification of exceptional Dehn surgeries on 2-bridge knots, Comm. Anal. Geom. 9 (2001) 97–113
  • M Cohen, W Metzler, A Zimmermann, What Does a Basis of ${F}(a,b)$ Look Like?, Math. Ann. 257 (1981) 435–445
  • DJ Collins, Presentations of the Amalgamated Free Product of two Infinite Cycles, Math. Ann. 237 (1978) 233–241
  • M Culler, C McA Gordon, J Luecke, PB Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987) 237–300
  • J Dean, Hyperbolic knots with small Seifert-fibered Dehn surgeries, Ph.D. thesis, University of Texas at Austin (1996)
  • M Eudave-Muñoz, On hyperbolic knots with Seifert fibered Dehn surgeries, preprint
  • M Eudave-Muñoz, Non-hyperbolic manifolds obtained by Dehn surgery on hyperbolic knots, from: “Proceedings of the Georgia International Topology Conference” (1993) To appear
  • R Fintushel, R Stern, Constructing lens spaces by surgery on knots, Math. Z. 175 (1980) 33–51
  • F González-Acuña, H Short, Knot surgery and primeness, Math. Proc. Cambridge Philos. Soc. 99 (1986) 89–102
  • C McA Gordon, J Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989) 371–415
  • Marc Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000) 243–282
  • WBR Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. of Math. 76 (1962) 531–540
  • T Mattman, K Miyazaki, K Motegi, Seifert fibered surgeries which do not arise from primitive Seifert-fibered constructions, preprint
  • K Miyazaki, K Motegi, Seifert Fibred Manifolds and Dehn surgery, Topology 36 (1997) 579–603
  • Katura Miyazaki, Kimihiko Motegi, Seifert fibered manifolds and Dehn surgery. III, Comm. Anal. Geom. 7 (1999) 551–582
  • L Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971) 734–745
  • RP Osborne, H Zieschang, Primitives in the Free Group on Two Generators, Invent. Math. 63 (1981) 17–24
  • J Stallings, Constructions of fibred knots and links, from: “Algebraic and geometric topology”, Proc. Sympos. Pure Math. 32, Amer. Math. Soc., Providence, R.I. (1978) 55–60
  • F Waldhausen, Gruppen mit Zentrum und 3-dimensionale Mannigfaltigkeiten, Topology 6 (1967) 505–517
  • AH Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960) 503–528
  • H Zieschang, On simple systems of paths on complete pretzels, Amer. Math. Soc. Transl. 92 (1970) 127–137
  • H Zieschang, Generators of the Free Product with Amalgamation of two Infinite Cyclic Groups, Math. Ann. 227 (1977) 195–221