2013 Fiscal Year Final Research Report
Research on knots using Heegaard theory
| Project/Area Number |
24740041
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Geometry
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| Research Institution | Joetsu University of Education |
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Principal Investigator |
SAITO Toshio 上越教育大学, 学校教育研究科(研究院), 准教授 (90397670)
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| Project Period (FY) |
2012-04-01 – 2014-03-31
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| Keywords | ３次元多様体 / Ｈｅｅｇａａｒｄ分解 / 結び目 / タングル |
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| Research Abstract |
We studied topological and geometric properties of knots and tangles by using Heegaard theory. Our main results are the following. (1) Studying free tangles from a viewpoint of Heegaard theory, we partially generalized Morimoto's theorem on 2-tangle decompositions. (2) We defined tunnel number and bridge number of tangles as well as those of knots, and obtained relation among them. (3) We proved that there are bridge splittings of knots in a given 3-manifold with arbitrary high distance. (4) Ozawa's result on essential free 2-tangle decompositions cannot be generalized even if a knot admits an essential 2-tangle decomposition such that one of the decomposed tangles is of tunnel number one.
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