## Algebraic & Geometric Topology

### Small Seifert fibered surgery on hyperbolic pretzel knots

Jeffrey Meier

#### Abstract

We complete the classification of hyperbolic pretzel knots admitting Seifert fibered surgeries. This is the final step in understanding all exceptional surgeries on hyperbolic pretzel knots. We also present results toward similar classifications for nonpretzel Montesinos knots of length three.

#### Article information

Source
Algebr. Geom. Topol., Volume 14, Number 1 (2014), 439-487.

Dates
Revised: 29 May 2013
Accepted: 29 June 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715808

Digital Object Identifier
doi:10.2140/agt.2014.14.439

Mathematical Reviews number (MathSciNet)
MR3158766

Zentralblatt MATH identifier
1290.57014

#### Citation

Meier, Jeffrey. Small Seifert fibered surgery on hyperbolic pretzel knots. Algebr. Geom. Topol. 14 (2014), no. 1, 439--487. doi:10.2140/agt.2014.14.439. https://projecteuclid.org/euclid.agt/1513715808

#### References

• T Abe, K Kishimoto, The dealternating number and the alternation number of a closed $3$–braid, J. Knot Theory Ramifications 19 (2010) 1157–1181
• M M Asaeda, J H Przytycki, Khovanov homology: Torsion and thickness, from: “Advances in topological quantum field theory”, NATO Sci. Ser. II Math. Phys. Chem. 179, Kluwer Acad. Publ., Dordrecht (2004) 135–166
• K Baker, C M Gordon, J Luecke, Small Seifert fiber spaces from Dehn surgery, in preparation
• S Boyer, C M Gordon, X Zhang, Dehn fillings of knot manifolds containing essential once-punctured tori
• S Boyer, C M Gordon, X Zhang, Characteristic submanifold theory and toroidal Dehn filling, Adv. Math. 230 (2012) 1673–1737
• 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
• G Burde, K Murasugi, Links and Seifert fiber spaces, Duke Math. J. 37 (1970) 89–93
• J H Conway, An enumeration of knots and links, and some of their algebraic properties, from: “Computational Problems in Abstract Algebra”, Pergamon, Oxford (1970) 329–358
• P R Cromwell, Knots and links, Cambridge Univ. Press (2004)
• M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987) 237–300
• M Dehn, Über die Topologie des dreidimensionalen Raumes, Math. Ann. 69 (1910) 137–168
• C Delman, Essential laminations and Dehn surgery on $2$–bridge knots, Topology Appl. 63 (1995) 201–221
• D Eisenbud, W Neumann, Three-dimensional link theory and invariants of plane curve singularities, Annals Math. Studies 110, Princeton Univ. Press (1985)
• M Eudave-Muñoz, Non-hyperbolic manifolds obtained by Dehn surgery on hyperbolic knots, from: “Geometric topology”, (W H Kazez, editor), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 35–61
• D Futer, M Ishikawa, Y Kabaya, T W Mattman, K Shimokawa, Finite surgeries on three-tangle pretzel knots, Algebr. Geom. Topol. 9 (2009) 743–771
• D Gabai, Surgery on knots in solid tori, Topology 28 (1989) 1–6
• C M Gordon, Small surfaces and Dehn filling, from: “Proceedings of the Kirbyfest”, (J Hass, M Scharlemann, editors), Geom. Topol. Monogr. 2 (1999) 177–199
• C M Gordon, Dehn surgery and $3$–manifolds, from: “Low dimensional topology”, (T S Mrowka, P S Ozsváth, editors), IAS/Park City Math. Ser. 15, Amer. Math. Soc. (2009) 21–71
• C M Gordon, J Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989) 371–415
• W Heil, Elementary surgery on Seifert fiber spaces, Yokohama Math. J. 22 (1974) 135–139
• K Ichihara, I D Jong, Seifert fibered surgery and Rasmussen invariant
• K Ichihara, I D Jong, Cyclic and finite surgeries on Montesinos knots, Algebr. Geom. Topol. 9 (2009) 731–742
• K Ichihara, I D Jong, Y Kabaya, Exceptional surgeries on $(-2,p,p)$–pretzel knots, Topology Appl. 159 (2012) 1064–1073
• K Ichihara, H Masai, Exceptional surgeries on alternating knots
• S Kang, Examples of reducible and finite Dehn fillings, J. Knot Theory Ramifications 19 (2010) 677–694
• M Lackenby, R Meyerhoff, The maximal number of exceptional Dehn surgeries, Invent. Math. 191 (2013) 341–382
• S Lee, Lens spaces and toroidal Dehn fillings, Math. Z. 267 (2011) 781–802
• W B R Lickorish, An introduction to knot theory, Graduate Texts in Mathematics 175, Springer, New York (1997)
• W B R Lickorish, M B Thistlethwaite, Some links with nontrivial polynomials and their crossing-numbers, Comment. Math. Helv. 63 (1988) 527–539
• T Mattman, K Miyazaki, K Motegi, Seifert-fibered surgeries which do not arise from primitive/Seifert-fibered constructions, Trans. Amer. Math. Soc. 358 (2006) 4045–4055
• K Miyazaki, K Motegi, Seifert fibering surgery on periodic knots, from: “Proceedings of the First Joint Japan–Mexico Meeting in Topology”, Topology Appl. 121 (2002) 275–285
• J M Montesinos, Surgery on links and double branched covers of $S\sp{3}$, from: “Knots, groups, and $3$–manifolds (Papers dedicated to the memory of R. H. Fox)”, (L P Neuwirth, editor), Ann. of Math. Studies 84, Princeton Univ. Press (1975) 227–259
• J W Morgan, H Bass (editors), The Smith conjecture, Pure and Applied Mathematics 112, Academic Press, Orlando, FL (1984)
• K Morimoto, M Sakuma, Y Yokota, Identifying tunnel number one knots, J. Math. Soc. Japan 48 (1996) 667–688
• K Motegi, Dehn surgeries, group actions and Seifert fiber spaces, Comm. Anal. Geom. 11 (2003) 343–389
• U Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984) 209–230
• G Perelman, The entropy formula for Ricci flow and its geometric applications
• G Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• G Perelman, Ricci flow with surgery on three-manifolds
• J Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010) 419–447
• P R Turner, Calculating Bar-Natan's characteristic two Khovanov homology, J. Knot Theory Ramifications 15 (2006) 1335–1356
• S Wolfram, The Mathematica$\sp \circledR$ book, 4th edition, Wolfram Media, Champaign, IL (1999)
• Y-Q Wu, Dehn surgery on knots of wrapping number $2$
• Y-Q Wu, Immersed surfaces and Seifert fibered surgery on Montesinos knots
• Y-Q Wu, Persistently laminar branched surfaces
• Y-Q Wu, Seifert fibered surgery on Montesinos knots
• 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, Sutured manifold hierarchies, essential laminations, and Dehn surgery, J. Differential Geom. 48 (1998) 407–437
• Y-Q Wu, The classification of toroidal Dehn surgeries on Montesinos knots, Comm. Anal. Geom. 19 (2011) 305–345
• Y-Q Wu, Exceptional Dehn surgery on large arborescent knots, Pacific J. Math. 252 (2011) 219–243