An upper bound for the number of elementary moves needed for unknotting an arc-presentation of the trivial knot
| Project/Area Number |
25400100
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | Japan Women's University |
|---|
Principal Investigator |
|
|---|
| Project Period (FY) |
2013-04-01 – 2016-03-31
|
| Project Status |
Completed (Fiscal Year 2015)
|
| Budget Amount *help |
¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
Fiscal Year 2015: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2013: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
|
|---|
| 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 / 位相幾何学 / 結び目理論 / レクタンギュラー表示 / マージ / エクスチェンジ / 上界 / arc表示 / exchange変形 / merge変形 / R変形 / rectangular diagram / 交差点数 |
|---|
| 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.
|
Report
(4 results)
Research Products
(6 results)
