2005 Fiscal Year Final Research Report Summary
Research on finite Dehn surgeries on knots and links
| Project/Area Number |
15540061
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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 | Saitama University |
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Principal Investigator |
SHIMOKAWA Koya Saitama University, Faculty of Science, Associate Professor, 理学部, 助教授 (60312633)
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| Co-Investigator(Kenkyū-buntansha) |
MIZUTANI Tadayoshi Saitama University, Faculty of Science, Professor, 理学部, 教授 (20080492)
SAKAMOTO Kunio Saitama University, Faculty of Science, Professor, 理学部, 教授 (70089829)
NAGASE Masayoshi Saitama University, Faculty of Science, Professor, 理学部, 教授 (30175509)
EGASHIRA Shinji Saitama University, Faculty of Science, Assistant Professor, 理学部, 助手 (00261876)
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| Project Period (FY) |
2003 – 2005
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| Keywords | Knot / Dehn surgery / 3-manifold / essential surface |
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| Research Abstract |
Dehn surgery is an operation yielding 3-manifolds using knots and links in the 3-sphere. It is known that any closed orientable 3-manifold can be obtained by a Dehn surgery on a link in the 3-sphere. Knots in the 3-sphere can be classified into three types ; torus knots, satellite knots and hyperbolic knots. Since Dehn surgeries on torus knots and satellite knots have been characterized in a sense, the most interesting case now is the hyperbolic knot case. By W.Thurston's research on Dehn surgeries on hyperbolic knots, we know that most Dehn surgeries on hyperbolic knots yield hyperbolic manifolds. For the hyperbolic knot case, the number of exceptional Dehn surgeries on a hyperbolic knot, i.e. Dehn surgeries on a hyperbolic knot which yield non-hyperbolic manifolds, is known to be finite. Hence characterizing such exceptional surgeries is a very important problem. We are studying such exceptional surgeries on Montesinos knots. Since reducible surgeries and toroidal surgeries on Montesinos knots have been characterized, we studied finite surgeries and Seifert surgeries on Montesinos knots. Especially we studied some classes of Montesinos knots for which the existence of finite surgeries is undetermined and we obtained partial results. There we used a method, developed by M.Culler and P.Shalen, which uses character varieties of fundamental groups of knot complements. Unfortunately as the study has not finished, the complete characterization of such surgeries is a next project. We also obtained a new significant relation between essential surfaces in Montesinos knot exteriors and their character varieties.
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