2015 Fiscal Year Final Research Report
An upper bound for the number of elementary moves needed for unknotting an arc-presentation of the trivial knot
| Project/Area Number |
25400100
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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 | Japan Women's University |
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Principal Investigator |
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| Project Period (FY) |
2013-04-01 – 2016-03-31
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| Keywords | 結び目 / 自明結び目 / アーク表示 / クロムウェル変形 / グリッド表示 |
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| Outline of Final Research Achievements |
I performed studies below together with T. Ando, Y. Nishikawa and M. Hayashi. We showed that any knot diagram with n crossings can be deformed by an adequate ambient isotopy of the plane into a grid diagram with 2n+2 or less vertical lines, and that the system of Seifert circles of a knot diagram can be deformed by an ambient isotopy into a disjoint union of squares composed of 2 vertical lines and 2 horizontal lines, and simultaneously, arcs substituting the crossings into vertical lines. We showed that an exchange move or merge move between the top and the bottom horizontal edges of a grid diagram with n vertical edges can be realized by a sequence of no more than 3n^2-4n-4 Reidemeister moves. We calculated by computers that how many exchange moves are sufficient for deforming an arc-presentation of the trivial knot so that it admits a merge move when the number of arcs are small.
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| Free Research Field |
低次元位相幾何学
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