| 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)
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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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