Non-hyperbolic Dehn surgeries on hyperbolic knots
| Project/Area Number |
15540095
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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 Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (40219978)
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| Project Period (FY) |
2003 – 2004
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| Project Status |
Completed (Fiscal Year 2004)
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| Budget Amount *help |
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2003: ¥1,000,000 (Direct Cost: ¥1,000,000)
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| Keywords | hyperbolic knot / Dehn surgery / exceptional surgery / longitudinal surgery / fibered knot / surface bundle over the circle / Seifert fiber space / Nielsen-Thurston types / primitive / Seifert構成 / 結ぴ目の対称性 |
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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 counterexample 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 manifolds. 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.
Dean introduced a primitive/Seifert-fibered construction, which is a natural modification of Berge's construction and provides infinite families of knots each of which admits Seifert fibered surgery. We determine non-hyperbolic, primitive/Seifert-fibered knots and show that for each such knots any integral, small Seifert surgery arises from a primitive/Seifert-fibered construction. We also study longitudinal exceptional surgeries on hyperbolic knots.
Gabai found a hyperbolic, fibered knot in the 3-sphere on which a longitudinal surgery produces a toroidal manifold, and now it is known that there are infinitely many such hyperbolic, fibered knots. On the other hand, there have been no known examples of hyperbolic, fibered knots with longitudinal, Seifert fibered surgeries, and Teragaito asks if there are no such examples. We give an answer this question by constructing an infinite family of hyperbolic, fibered knots each of which admits a longitudinal, Seifert fibered surgery.
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