2012 Fiscal Year Final Research Report
The number of Cromwell moves needed for unknotting an arc-presentation of the trivial knot
| Project/Area Number |
22540101
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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 | Japan Women's University |
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Principal Investigator |
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| Project Period (FY) |
2010 – 2012
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| Keywords | 結び目理論 / 自明結び目 / アーク表示 / クロムウェル変形 |
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| Research Abstract |
A knot is a circle in the 3-dimensional space R3. Every knot can be placed in a figure in the form of an open-book (a book opened so that every adjacent pair of pages are tangent to each other only in the biding) so that it intersects every page in a single arc). We call such a placement of a knot an arc-presentation. A knot is called trivial if it lies in a plane after being moved continuously. The trivial knot has an arc-presentation with two arcs. An example of infinite sequence of arc-presentations with n arcs of the trivial knot as below is given. They need linearly many exchange moves not changing the number of arcs with respect to n until they admit a merge move decreasing the number of arcs.
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