Transactions of the American Mathematical Society

Seifert-fibered surgeries which do not arise from primitive/Seifert-fibered constructions

Author(s): Thomas Mattman; Katura Miyazaki; Kimihiko Motegi
Journal: Trans. Amer. Math. Soc. 358 (2006), 4045-4055.
MSC (2000): Primary 57M25
Posted: September 22, 2005
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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

-pretzel knot belongs to both of the infinite families.

1.: J. Berge; Some knots with surgeries yielding lens spaces, unpublished manuscript.
2.: S. A. Bleiler; Knots prime on many strings, Trans. Amer. Math. Soc. 282 (1984), 385-401. MR 0728719 (85h:57003)
3.: S. A. Bleiler; Prime tangles and composite knots, Lect. Notes in Math., vol. 1144, Springer-Verlag, 1985, pp. 1-13. MR 0823278 (87e:57006)
4.: S. Bleiler and C. Hodgson; Spherical space forms and Dehn filling, Topology 35 (1996), 809-833. MR 1396779 (97f:57007)
5.: S. Boyer and X. Zhang; Finite surgery on knots, J. Amer. Math. Soc. 9 (1996), 1005-1050. MR 1333293 (97h:57013)
6.: G. Burde and H. Zieschang; Knots, de Gruyter Studies in Mathematics 5, 1985. MR 0808776 (87b:57004)
7.: J. Dean; Hyperbolic knots with small Seifert-fibered Dehn surgeries, Ph.D. thesis, University of Texas at Austin, 1996.
8.: J. Dean; Small Seifert-fibered Dehn surgery on hyperbolic knots, Algebraic and Geometric Topology 3 (2003), 435-472. MR 1997325 (2004m:57009)
9.: M. Eudave-Muñoz; Non-hyperbolic manifolds obtained by Dehn surgery on a hyperbolic knot, In: Studies in Advanced Mathematics vol. 2, part 1, (ed. W. Kazez), 1997, Amer. Math. Soc. and International Press, pp. 35-61. MR 1470720 (98i:57007)
10.: M. Eudave-Muñoz; On hyperbolic knots with Seifert fibered Dehn surgeries, Topology Appl. 121 (2002), 119-141. MR 1903687 (2003c:57005)
11.: D. Gabai; Foliations and the topology of 3-manifolds III, J. Diff. Geom. 26 (1987), 479-536. MR 0910018 (89a:57014b)
12.: F. González-Acuña and H. Short; Knot surgery and primeness, Math. Proc. Camb. Phil. Soc. 99 (1986), 89-102. MR 0809502 (87c:57003)
13.: C. McA. Gordon; Dehn Filling; a survey, Knot theory (Warsaw, 1995), 129-144, Banach Center, Publ. 42, Polish Acad. Sci., Warsaw, 1998. MR 1634453 (99e:57028)
14.: C. McA. Gordon and J. Luecke; Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), 371-415. MR 0965210 (90a:57006a)
15.: T. Mattman; The Culler-Shalen seminorms of pretzel knots, Ph.D. thesis, McGill University, Montréal, 2000.
16.: K. Miyazaki and K. Motegi; Seifert fibered manifolds and Dehn surgery II, Math. Ann. 311 (1998), 647-664. MR 1637964 (99g:57010)
17.: K. Miyazaki and K. Motegi; Seifert fibered manifolds and Dehn surgery III, Comm. Anal. Geom. 7 (1999), 551-582. MR 1698388 (2000d:57030)
18.: K. Miyazaki and K. Motegi; On primitive/Seifert-fibered constructions, Math. Proc. Camb. Phil. Soc. 138 (2005), 421-435. MR 2138571
19.: J. M. Montesinos; Surgery on links and double branched coverings of , Ann. Math. Studies 84 (1975), 227-260. MR 0380802 (52:1699)
20.: K. Morimoto; On the additivity of h-genus of knots, Osaka J. Math. 31 (1994), 137-145. MR 1262793 (95d:57006)
21.: K. Morimoto; There are knots whose tunnel numbers go down under connected sum, Proc. Amer. Math. Soc. 123 (1995), 3527-3532. MR 1317043 (96a:57022)
22.: K. Motegi; Dehn surgeries, group actions and Seifert fiber spaces, Comm. Anal. Geom. 11 (2003), 343-389. MR 2014880 (2004m:57015)
23.: H. Schubert; Knoten und Vollringe, Acta Math. 90 (1953), 131-286.MR 0072482 (17:291d)
24.: J. Weeks; SnapPea: a computer program for creating and studying hyperbolic -manifolds, freely available from http://thames.northnet.org/weeks/index/SnapPea.html.
25.: H. Zieschang; On simple systems of paths on complete pretzels, Amer. Math. Soc. Transl. 92, 127-137.

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Thomas Mattman
Affiliation: Department of Mathematics and Statistics, California State University--Chico, Chico, California 95929-0525
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
Email: motegi@math.chs.nihon-u.ac.jp

DOI: 10.1090/S0002-9947-05-03798-0
PII: S 0002-9947(05)03798-0
Keywords: Dehn surgery, hyperbolic knot, Seifert fiber space, primitive/Seifert-fibered construction
Received by editor(s): January 20, 2003
Received by editor(s) in revised form: June 28, 2004
Posted: 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.
Dedicated: Dedicated to Cameron McA. Gordon on the occasion of his 60th birthday
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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