## Algebraic & Geometric Topology

### All exceptional surgeries on alternating knots are integral surgeries

Kazuhiro Ichihara

#### Abstract

We show that all non-trivial exceptional surgeries on hyperbolic alternating knots in the 3–sphere are integral surgeries.

#### Article information

Source
Algebr. Geom. Topol., Volume 8, Number 4 (2008), 2161-2173.

Dates
Revised: 5 October 2008
Accepted: 13 October 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796930

Digital Object Identifier
doi:10.2140/agt.2008.8.2161

Mathematical Reviews number (MathSciNet)
MR2460884

Zentralblatt MATH identifier
1158.57300

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds

#### Citation

Ichihara, Kazuhiro. All exceptional surgeries on alternating knots are integral surgeries. Algebr. Geom. Topol. 8 (2008), no. 4, 2161--2173. doi:10.2140/agt.2008.8.2161. https://projecteuclid.org/euclid.agt/1513796930

#### References

• I Agol, Bounds on exceptional Dehn filling, Geom. Topol. 4 (2000) 431–449
• I Agol, Bounds on exceptional Dehn filling II
• S Boyer, X Zhang, Cyclic surgery and boundary slopes, from: “Geometric topology (Athens, GA, 1993)”, AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc., Providence, RI (1997) 62–79
• M Brittenham, Exceptional Seifert-fibered spaces and Dehn surgery on $2$–bridge knots, Topology 37 (1998) 665–672
• M Brittenham, Y-Q Wu, The classification of exceptional Dehn surgeries on 2–bridge knots, Comm. Anal. Geom. 9 (2001) 97–113
• C Delman, Essential laminations and Dehn surgery on $2$–bridge knots, Topology Appl. 63 (1995) 201–221
• C Delman, Constructing essential laminations which survive all Dehn surgeries, preprint
• C Delman, R Roberts, Alternating knots satisfy Strong Property P, Comment. Math. Helv. 74 (1999) 376–397
• M Eudave-Muñoz, Non-hyperbolic manifolds obtained by Dehn surgery on hyperbolic knots, from: “Geometric topology (Athens, GA, 1993)”, AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc., Providence, RI (1997) 35–61
• D Gabai, Problems in foliations and laminations, from: “Geometric topology (Athens, GA, 1993)”, AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc., Providence, RI (1997) 1–33
• D Gabai, U Oertel, Essential laminations in $3$–manifolds, Ann. of Math. $(2)$ 130 (1989) 41–73
• F González-Acuña, H Short, Knot surgery and primeness, Math. Proc. Cambridge Philos. Soc. 99 (1986) 89–102
• C M Gordon, J Luecke, Non-integral toroidal Dehn surgeries, Comm. Anal. Geom. 12 (2004) 417–485
• A Hatcher, U Oertel, Boundary slopes for Montesinos knots, Topology 28 (1989) 453–480
• A Hatcher, W Thurston, Incompressible surfaces in $2$–bridge knot complements, Invent. Math. 79 (1985) 225–246
• K Ichihara, Integral non-hyperbolike surgeries, J. Knot Theory Ramifications 17 (2008) 257–261
• K Ichihara, I D Jong, S Mizushima, Seifert fibered surgeries on alternating Montesinos knots, in preparation
• R Kirby (editor), Problems in low-dimensional topology, AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc., Providence, RI (1997)
• M Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000) 243–282
• M Lackenby, R Meyerhoff, The maximal number of exceptional Dehn surgeries
• W B R Lickorish, M B Thistlethwaite, Some links with nontrivial polynomials and their crossing-numbers, Comment. Math. Helv. 63 (1988) 527–539
• W Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984) 37–44
• W Menasco, M B Thistlethwaite, Surfaces with boundary in alternating knot exteriors, J. Reine Angew. Math. 426 (1992) 47–65
• R M Patton, Incompressible punctured tori in the complements of alternating knots, Math. Ann. 301 (1995) 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, Mathematics Lecture Series 7, Publish or Perish, Berkeley, CA (1976)
• P Scott, The geometries of $3$–manifolds, Bull. London Math. Soc. 15 (1983) 401–487
• M B Thistlethwaite, On the algebraic part of an alternating link, Pacific J. Math. 151 (1991) 317–333
• W P Thurston, The geometry and topology of three-manifolds (1980) Available at \setbox0\makeatletter\@url http://msri.org/publications/books/gt3m {\unhbox0
• W P Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. $($N.S.$)$ 6 (1982) 357–381
• Y-Q Wu, Dehn surgery on arborescent knots, J. Differential Geom. 43 (1996) 171–197
• Y-Q Wu, Dehn surgery on arborescent knots and links–-a survey, Chaos Solitons Fractals 9 (1998) 671–679
• Y-Q Wu, The classification of toroidal Dehn surgeries on Montesinos knots, to appear in Topology
• Y-Q Wu, Exceptional Dehn surgery on large arborescent knots