Proceedings of the Japan Academy, Series A, Mathematical Sciences

Toroidal Seifert fibered surgeries on alternating knots

Kazuhiro Ichihara and In Dae Jong

Full-text: Open access

Abstract

We give a complete classification of toroidal Seifert fibered surgeries on alternating knots. Precisely, we show that if an alternating knot $K$ admits a toroidal Seifert fibered surgery, then $K$ is either the trefoil knot and the surgery slope is zero, or the connected sum of a $(2,p)$-torus knot and a $(2,q)$-torus knot and the surgery slope is $2(p+q)$ with $|p|, |q| \ge 3$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci. Volume 90, Number 3 (2014), 54-56.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
http://projecteuclid.org/euclid.pja/1393510215

Digital Object Identifier
doi:10.3792/pjaa.90.54

Mathematical Reviews number (MathSciNet)
MR3178485

Zentralblatt MATH identifier
1295.57019

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

Keywords
Seifert fibered surgery toroidal surgery alternating knot

Citation

Ichihara, Kazuhiro; Jong, In Dae. Toroidal Seifert fibered surgeries on alternating knots. Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 3, 54--56. doi:10.3792/pjaa.90.54. http://projecteuclid.org/euclid.pja/1393510215.


Export citation

References

  • S. Boyer, Dehn surgery on knots, in Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 165–218.
  • S. Boyer and X. Zhang, Cyclic surgery and boundary slopes, in Geometric topology (Athens, GA, 1993), 62–79, AMS/IP Stud. Adv. Math., 2.1, Amer. Math. Soc., Providence, RI, 1997.
  • M. Eudave-Muñoz, On hyperbolic knots with Seifert fibered Dehn surgeries, Topology Appl. 121 (2002), no. 1–2, 119–141.
  • F. González-Acuña and H. Short, Knot surgery and primeness, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 89–102.
  • C. McA. Gordon and J. Luecke, Seifert fibered surgeries on hyperbolic knots, Abstracts Amer. Math. Soc. 20 (1999), 405.
  • K. Ichihara and I. D. Jong, Toroidal Seifert fibered surgeries on Montesinos knots, Comm. Anal. Geom. 18 (2010), no. 3, 579–600.
  • K. Ichihara and H. Masai, Exceptional surgeries on alternating knots, arXiv:1310.3472.
  • K. Ichihara, K. Motegi and H.-J. Song, Seifert fibered slopes and boundary slopes on small hyperbolic knots, Bull. Nara Univ. Ed. Natur. Sci. 57 (2008), no. 2, 21–25.
  • W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), no. 1, 37–44.
  • K. Miyazaki and K. Motegi, Seifert fibred manifolds and Dehn surgery, Topology 36 (1997), no. 2, 579–603.
  • K. Motegi, Dehn surgeries, group actions and Seifert fiber spaces, Comm. Anal. Geom. 11 (2003), no. 2, 343–389.
  • U. Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984), no. 1, 209–230.
  • R. M. Patton, Incompressible punctured tori in the complements of alternating knots, Math. Ann. 301 (1995), no. 1, 1–22.
  • G. Perelman, The entropy formula for the Ricci flow and its geometric applications..
  • G. Perelman, Ricci flow with surgery on three-manifolds..
  • G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds..
  • D. Rolfsen, Knots and links, Publish or Perish, Berkeley, CA, 1976.
  • W. P. Thurston, The geometry and topology of 3-manifolds, Lecture notes, Princeton University, 1978, electronic version available at http://www.msri.org/publications/books/gt3m.
  • W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381.