Relations about geometric characteristics of surface knots and invariants
| Project/Area Number |
16540082
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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 | Tokai University |
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Principal Investigator |
SHIMA Akiko Tokai University, School of Science, Associated Professor (50317765)
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| Project Period (FY) |
2004 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥3,110,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥210,000)
Fiscal Year 2007: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2006: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
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| Keywords | topology / surface link / chart / surface braid / surface knot / crossing / minimal / bigon / triple point |
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| Research Abstract |
Kamada define charts to investigate embedded surfaces in 4-space. A chart is a oriented graph in the plane such that each edge are labeled by one of integers from 1 to n-1, and each vertex is degree 1, 4 or 6 satisfying some conditions. A vertex of degree 1 is called a black vertex, a vertex of degree 4 is a crossing, and a vertex of degree 6 is a white vertex. There are C-moves between charts such that C-moves do not change the ambient isotopy classes of the associated surfaces in 4-space. A chart is a ribbon if it can be moved a chart without white vertices by C-moves.
Kamada showed that any 3-chart is a ribbon chart. We show that if any n-chart contains at most two crossings, and if its associated surface in 4-space is a disjoint union of spheres, then the chart is a ribbon chart. To show this theorem, we show that any minimal generalized n-chart contains exactly two crossings, then the chart contains at least 4n-10 black vertices. It is well known that the associated surface of any chart in 4-space is one sphere, then the chart contains exactly 2n-2 black vertices. We have the desired result. To investigate the number of black vertices, we define tangles for charts. We found some condition reducing white vertices for tangles.
We investigate charts containing exactly 4, 5 or 7 white vertices. If a chart contains exactly 5 or 7 white vertices, then we can cancel some white vertex by C-moves.
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Report
(5 results)
Research Products
(42 results)
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[Presentation] On charts with two crossings2008
Author(s)
Teruo, Nagase, Akiko, Shima
Organizer
The 4th East Asian School of Knots, Links and Related Topics
Place of Presentation
The University of Tokyo
Year and Date
2008-06-22
Description
「研究成果報告書概要(欧文)」より
Related Report
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[Presentation] チャートの分解2007
Author(s)
志摩 亜希子, 永瀬 輝男
Organizer
4次元トポロジー
Place of Presentation
広島大学
Year and Date
2007-01-30
Description
「研究成果報告書概要(和文)」より
Related Report
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[Presentation] チャートを用いた結び目曲面の考察2006
Author(s)
志摩 亜希子, 永瀬 輝男
Organizer
第3回「トポロジー, 代数幾何蔵王セミナー」
Place of Presentation
蔵王ハイツ
Year and Date
2006-07-28
Description
「研究成果報告書概要(和文)」より
Related Report
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[Presentation] Charts with at most one crossing2006
Author(s)
Teruo, Nagase, Akiko, Shima
Organizer
Seminar of Divisions -Monodromy-chart in Hakone 2006
Place of Presentation
Hakone Lake Hotel
Year and Date
2006-06-29
Description
「研究成果報告書概要(欧文)」より
Related Report
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