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Non-hyperbolic Dehn surgeries on hyperbolic knots

Research Project

Project/Area Number 13640089
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionNihon university

Principal Investigator

MOTEGI Kimihiko  College of Humanities and Sciences, Professor, 文理学部, 教授 (40219978)

Project Period (FY) 2001 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 2002: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2001: ¥1,000,000 (Direct Cost: ¥1,000,000)
KeywordsDehn surgery / hyperbolic knot / Seifert fiber space / primitive / Seifert construction / symmetry of knots / tunnel number / primitive / toroidal manifold / seifert-fibered construction
Research Abstract

Thurston's hyperbolic Dehn surgery theorem asserts that if a knot K in the 3-sphere is hyperbolic (i.e., the complement of K admits a complete hyperbolic structure of finite volume), then all but finitely many Dehn surgeries on K yield hyperbolic 3-manifolds. Then it is important to describe non-hyperbolic surgeries on hyperbolic knots. It is known that any non-hyperbolic surgery is a reducing surgery, a toroidal surgery, a Seifert fibered surgery, or a surgery producing a counter example to Geometrization conjecture.
In this research we focus on Seifert fibered surgeries. Seifert fibered surgeries on torus knots can be naturally explained by considering how Seifert fibrations of the exterior extends over the surgered manifold. Are there any natural explanation for Seifert fibered surgeries on hyperbolic knots? Berge gave an explicit construction which yields several infinite families of knots each admitting a lens space Dehn surgery. It is conjectured that any lens space surgery can be … More explained by Berge's primitive/primitive construction. The natural generalization of primitive /primitive construction was done by Dean, which explains many Seifert fibered surgeries on hyperbolic knots. In 1996, Gordon conjectured that any Seifert fibered surgery can be explained by Dean's primitive/Seifert-fibered construction. Recently Eudave-Munoz has shown that all known examples of Seifert fibered surgeries constructed by the Montesinos trick can be explained by primitive/Seifert-fibered construction. In this research, as a joint work with Thomas Mattman and Katura Miyazaki, 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. Very recently we are interested in the questions : If K has a Seifert fibered surgery, then is K embedded in a genus 2 Heegaard surface of the 3-sphere? Less

Report

(3 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • Research Products

    (20 results)

All Other

All Publications (20 results)

  • [Publications] Katura Miyazaki: "Seifert fibering surgery on periodic knots"Topology Appl.. 121. 275-285 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Katura Miyazaki: "Crossing change and exceptional Dehn surgery"Osaka J.Math.. 40. 1-5 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kazuhiro Ichihara: "Stably filling curves on a surface"Kobe J.Math.. 19. 61-66 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kimihiko Motegi: "Dehn surgeries, group actions and Seifert fiber spaces"Comm.Anal.Geom.. (出版受理).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kimihiko Motegi: "An experimental study of Seifert fibered Dehn surgery via SnapPea"Interdisciplinary Information Sciences. (出版受理).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Mohamed AitNouh: "Obtaining graph knots by twisting unknots"Topology Appl.. (出版受理).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Miyazaki and K. Motegi: "Seifert fibering surgery on periodic knots"Topology Appl.. 121. 275-285 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Miyazaki and K. Motegi: "Crossing change and exceptional Dehn surgery"Osaka J. Math.. 40. 1-5 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Ichihara and K. Motegi: "Stably filling curves on a surface"Kobe J. Math.. 19. 61-66 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Motegi: "Dehn surgeries, group actions and Seifert fiber spaces"Comm. Anal. Geom.. To appear in.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Motegi: "An experimental study of Seifert fibered Dehn surgery via SnapPea"Interdisciplinary Information Sciences. To appear in.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Aitnouh, D. Matignon and K. Motegi: "Obtaining graph knots by twisting unknots"Topology Appl.. To appear in.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Katura Miyazaki: "Seifert fibering surgery on periodic knots"Topology Appl.. 121. 275-285 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Katura Miyazaki: "Crossing change and exceptional Dehn surgery"Osaka J. Math.. 40. 1-5 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Kazuhiro Ichihara: "Stably filling curves on a surface"Kobe J. Math.. 19. 61-66 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Kimihiko Motegei: "Dehn surgeries, group actions and Seifert fiber spaces"Comm. Anal. Geom.. 11. 343-389 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Kimihiko Motegei: "An experimental study of Seifert fibered Dehn surgery via SnapPea"Interdisciplinary Information Sciences. 9. 95-125 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Mohamed AitNouh: "Obtaining graph knots by twisting unknots"Topology Appl.. (出版受理).

    • Related Report
      2002 Annual Research Report
  • [Publications] Katura Miyazaki: "Seifert fibering surgery on periodic knots"Topology Appl.. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] Kimihiko Motegi: "Dehn surgeries, group actions and Seifert fiber spaces"Comm. Anal. Geom.. (to appear).

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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