Seifert-fibered surgeries which do not arise from primitive/Seifert-fibered constructions
HTML articles powered by AMS MathViewer
- by Thomas Mattman, Katura Miyazaki and Kimihiko Motegi PDF
- Trans. Amer. Math. Soc. 358 (2006), 4045-4055 Request permission
Abstract:
We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Dean’s primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert-fibered surgeries arise from Dean’s primitive/Seifert-fibered construction. The $(-3,3,5)$-pretzel knot belongs to both of the infinite families.References
- J. Berge; Some knots with surgeries yielding lens spaces, unpublished manuscript.
- Steven A. Bleiler, Knots prime on many strings, Trans. Amer. Math. Soc. 282 (1984), no. 1, 385–401. MR 728719, DOI 10.1090/S0002-9947-1984-0728719-9
- Steven A. Bleiler, Prime tangles and composite knots, Knot theory and manifolds (Vancouver, B.C., 1983) Lecture Notes in Math., vol. 1144, Springer, Berlin, 1985, pp. 1–13. MR 823278, DOI 10.1007/BFb0075008
- Steven A. Bleiler and Craig D. Hodgson, Spherical space forms and Dehn filling, Topology 35 (1996), no. 3, 809–833. MR 1396779, DOI 10.1016/0040-9383(95)00040-2
- S. Boyer and X. Zhang, Finite Dehn surgery on knots, J. Amer. Math. Soc. 9 (1996), no. 4, 1005–1050. MR 1333293, DOI 10.1090/S0894-0347-96-00201-9
- Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR 808776
- J. Dean; Hyperbolic knots with small Seifert-fibered Dehn surgeries, Ph.D. thesis, University of Texas at Austin, 1996.
- John C. Dean, Small Seifert-fibered Dehn surgery on hyperbolic knots, Algebr. Geom. Topol. 3 (2003), 435–472. MR 1997325, DOI 10.2140/agt.2003.3.435
- Mario Eudave-Muñoz, Non-hyperbolic manifolds obtained by Dehn surgery on hyperbolic knots, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 35–61. MR 1470720, DOI 10.1090/amsip/002.1/03
- Mario Eudave-Muñoz, On hyperbolic knots with Seifert fibered Dehn surgeries, Proceedings of the First Joint Japan-Mexico Meeting in Topology (Morelia, 1999), 2002, pp. 119–141. MR 1903687, DOI 10.1016/S0166-8641(01)00114-6
- David Gabai, Foliations and the topology of $3$-manifolds. II, J. Differential Geom. 26 (1987), no. 3, 461–478. MR 910017
- Francisco González-Acuña and Hamish Short, Knot surgery and primeness, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 89–102. MR 809502, DOI 10.1017/S0305004100063969
- C. McA. Gordon, Dehn filling: a survey, Knot theory (Warsaw, 1995) Banach Center Publ., vol. 42, Polish Acad. Sci. Inst. Math., Warsaw, 1998, pp. 129–144. MR 1634453
- C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371–415. MR 965210, DOI 10.1090/S0894-0347-1989-0965210-7
- T. Mattman; The Culler-Shalen seminorms of pretzel knots, Ph.D. thesis, McGill University, Montréal, 2000.
- Katura Miyazaki and Kimihiko Motegi, Seifert fibered manifolds and Dehn surgery. II, Math. Ann. 311 (1998), no. 4, 647–664. MR 1637964, DOI 10.1007/s002080050204
- Katura Miyazaki and Kimihiko Motegi, Seifert fibered manifolds and Dehn surgery. III, Comm. Anal. Geom. 7 (1999), no. 3, 551–582. MR 1698388, DOI 10.4310/CAG.1999.v7.n3.a3
- Katura Miyazaki and Kimihiko Motegi, On primitive/Seifert-fibered constructions, Math. Proc. Cambridge Philos. Soc. 138 (2005), no. 3, 421–435. MR 2138571, DOI 10.1017/S030500410400828X
- José M. Montesinos, Surgery on links and double branched covers of $S^{3}$, Knots, groups, and $3$-manifolds (Papers dedicated to the memory of R. H. Fox), Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N.J., 1975, pp. 227–259. MR 0380802
- Kanji Morimoto, On the additivity of $h$-genus of knots, Osaka J. Math. 31 (1994), no. 1, 137–145. MR 1262793
- Kanji Morimoto, There are knots whose tunnel numbers go down under connected sum, Proc. Amer. Math. Soc. 123 (1995), no. 11, 3527–3532. MR 1317043, DOI 10.1090/S0002-9939-1995-1317043-4
- Kimihiko Motegi, Dehn surgeries, group actions and Seifert fiber spaces, Comm. Anal. Geom. 11 (2003), no. 2, 343–389. MR 2014880, DOI 10.4310/CAG.2003.v11.n2.a6
- Horst Schubert, Knoten und Vollringe, Acta Math. 90 (1953), 131–286 (German). MR 72482, DOI 10.1007/BF02392437
- J. Weeks; SnapPea: a computer program for creating and studying hyperbolic $3$-manifolds, freely available from http://thames.northnet.org/weeks/index/SnapPea.html.
- H. Zieschang; On simple systems of paths on complete pretzels, Amer. Math. Soc. Transl. 92, 127–137.
Additional Information
- Thomas Mattman
- Affiliation: Department of Mathematics and Statistics, California State University–Chico, Chico, California 95929-0525
- MR Author ID: 609682
- ORCID: 0000-0002-4900-6783
- Email: TMattman@CSUChico.edu
- Katura Miyazaki
- Affiliation: Faculty of Engineering, Tokyo Denki University, Tokyo 101-8457, Japan
- Email: miyazaki@cck.dendai.ac.jp
- Kimihiko Motegi
- Affiliation: Department of Mathematics, Nihon University, Tokyo 156-8550, Japan
- MR Author ID: 254668
- Email: motegi@math.chs.nihon-u.ac.jp
- Received by editor(s): January 20, 2003
- Received by editor(s) in revised form: June 28, 2004
- Published electronically: September 22, 2005
- Additional Notes: The first author was supported in part by grants from NSERC and FCAR
The second author was supported in part by Grant-in-Aid for Scientific Research (No. 40219978), The Ministry of Education, Culture, Sports, Science and Technology, Japan. - © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 4045-4055
- MSC (2000): Primary 57M25
- DOI: https://doi.org/10.1090/S0002-9947-05-03798-0
- MathSciNet review: 2219009
Dedicated: Dedicated to Cameron McA. Gordon on the occasion of his 60th birthday