2017 Fiscal Year Final Research Report
Cosmetic surgery conjecture for alternating knots
| Project/Area Number |
26400100
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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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 |
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| Project Period (FY) |
2014-04-01 – 2018-03-31
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| Keywords | ３次元多様体 / デーン手術 / 交代結び目 |
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| Outline of Final Research Achievements |
In the study of knots in the 3-space, their complements have played important roles. In fact, the pair of complements of equivalent knots must be homeomorphic. In contrast, Gordon and Luecke have proved that the knots with homeomorphic complements must be equivalent. The key of their proof is an operation called Dehn surgery on knots. Actually, they showed that the trivial and any non-trivial Dehn surgeries on a non-trivial knot give non-homeomorphic pair of 3-manifolds. This theorem can be generalized to the conjecture on knots in general manifolds, which is now called the Cosmetic Surgery Conjecture. In this research project, we have focused on the Dehn surgeries on alternating knots in 3-space, and obtained some partial answer to the conjecture. Also given are some related results on 3-manifolds and knots and links.
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| Free Research Field |
数物系科学
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