Non-hyperbolic Dehn surgeries on hyperbolic knots
| Project/Area Number |
13640089
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Nihon university |
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Principal Investigator |
MOTEGI Kimihiko College of Humanities and Sciences, Professor, 文理学部, 教授 (40219978)
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| Project Period (FY) |
2001 – 2002
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| Project Status |
Completed (Fiscal Year 2002)
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| 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)
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| Keywords | Dehn surgery / hyperbolic knot / Seifert fiber space / primitive / Seifert construction / symmetry of knots / tunnel number / primitive / toroidal manifold / seifert-fibered construction |
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| 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
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