Algebraic & Geometric Topology

On hyperbolic 3–manifolds realizing the maximal distance between toroidal Dehn fillings

Hiroshi Goda and Masakazu Teragaito

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Abstract

For a hyperbolic 3–manifold M with a torus boundary component, all but finitely many Dehn fillings on the torus component yield hyperbolic 3–manifolds. In this paper, we will focus on the situation where M has two exceptional Dehn fillings, both of which yield toroidal manifolds. For such situation, Gordon gave an upper bound for the distance between two slopes of Dehn fillings. In particular, if M is large, then the distance is at most 5. We show that this upper bound can be improved by 1 for a broad class of large manifolds.

Article information

Source
Algebr. Geom. Topol., Volume 5, Number 2 (2005), 463-507.

Dates
Received: 11 January 2005
Revised: 13 April 2005
Accepted: 29 April 2005
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796419

Digital Object Identifier
doi:10.2140/agt.2005.5.463

Mathematical Reviews number (MathSciNet)
MR2153119

Zentralblatt MATH identifier
1082.57011

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M50: Geometric structures on low-dimensional manifolds

Keywords
Dehn filling toroidal filling knot

Citation

Goda, Hiroshi; Teragaito, Masakazu. On hyperbolic 3–manifolds realizing the maximal distance between toroidal Dehn fillings. Algebr. Geom. Topol. 5 (2005), no. 2, 463--507. doi:10.2140/agt.2005.5.463. https://projecteuclid.org/euclid.agt/1513796419


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References

  • S. Boyer, C. McA. Gordon and X. Zhang, Dehn fillings of large hyperbolic $3$-manifolds, J. Differential Geom. 58 (2001), 263–308.
  • S. Boyer and X. Zhang, Reducing Dehn filling and toroidal Dehn filling, Topology Appl. 68 (1996), 285–303.
  • M. Culler, C. McA. Gordon, J. Luecke and P. Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987), 237–300.
  • C. McA. Gordon, Dehn filling: a survey, In Knot theory (Banach Center Publ., 1998), 129–144.
  • C. McA. Gordon, Boundary slopes on punctured tori in $3$-manifolds, Trans. Amer. Math. Soc. 350 (1998), 1713–1790.
  • C. McA. Gordon, Small surfaces and Dehn fillings, In Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr. 2, (1999), 177–199.
  • C. McA. Gordon and J. Luecke, Dehn surgeries on knots creating essential tori, I, Comm. Anal. Geom. 3 (1995), 597–644.
  • C. McA. Gordon and J. Luecke, Toroidal and boundary-reducing Dehn fillings, Topology Appl. 93 (1999), 77–90.
  • C. McA. Gordon and J. Luecke, Dehn surgeries on knots creating essential tori, II, Comm. Anal. Geom. 8 (2000), 671–725.
  • C. McA. Gordon and Y. Q. Wu, Toroidal and annular Dehn fillings, Proc. London Math. Soc. 78 (1999), 662–700.
  • C. Hayashi and K. Motegi, Only single twists on unknots can produce composite knots, Trans. Amer. Math. Soc. 349 (1997), 4465–4479.
  • S. Lee, S. Oh and M. Teragaito, Reducing Dehn fillings and small surfaces, preprint.
  • S. Oh, Reducible and toroidal $3$-manifolds obtained by Dehn fillings, Topology Appl. 75 (1997), 93–104.
  • S. Oh, Reducing spheres and Klein bottles after Dehn fillings, Canad. Math. Bull. 46 (2003), 265–267.
  • T. M. Price, Homeomorphisms of quaternion space and projective planes in four space, J. Austral. Math. Soc. 23 (1977), 112–128.
  • M. Teragaito, Creating Klein bottles by surgery on knots, J. Knot Theory Ramifications 10 (2001), 781–794.
  • M. Teragaito, Distance between toroidal surgeries on hyperbolic knots in the $3$-sphere, to appear in Trans. Amer. Math. Soc.
  • Y. Q. Wu, Dehn fillings producing reducible manifolds and toroidal manifolds, Topology 37 (1998), 95–108.
  • Y. Q. Wu, Sutured manifold hierarchies, essential laminations, and Dehn surgery, J. Differential Geom. 48 (1998), 407–437.