| Project/Area Number |
15K04881
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | Tokyo Woman's Christian University |
|---|
Principal Investigator |
Nikkuni Ryo 東京女子大学, 現代教養学部, 教授 (00401878)
|
|---|
| Project Period (FY) |
2015-04-01 – 2019-03-31
|
| Project Status |
Completed (Fiscal Year 2018)
|
| Budget Amount *help |
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
|
|---|
| Keywords | 空間グラフ / 結び目 / 絡み目 / Jones多項式 / 正則イソトピー / 結び目内在性 / 絡み目内在性 / Conway-Gordonの定理 / ハンドル体結び目 / 結び目群 / 不変量 |
|---|
| Outline of Final Research Achievements |
(1) We gave the common factor of the difference of Jones polynomials for two oriented links which are $C_n$-equivalent, (2) For a trivalent graph with certain conditions, we showed that any regular isotopy invariant of its adjusted spatial graph diagram is an ambient isotopy invariant of the spatial graph, (3) We showed that a rectilinear spatial complete bipartite graph on 3+3 vertices is totally free if each of the constituent knots is trivial, (4) We defined a preorder in the set of genus 2 handlebody-knots and exhibited a lot of ordered pairs of irreducible genus 2 handlebody-knots in the table up to 6 crossings, each of which does not admit this order, (5) We generalized the Conway-Gordon theorems for the complete graph on 6 and 7 vertices to complete graphs with arbitrary number of vertices greater than or equal to eight. For each of the above, many applications were given.
|
| Academic Significance and Societal Importance of the Research Achievements |
(1) 結び目のJones多項式の $C_n$ 変形による最小差が特徴付けられた. (2) 結び目の同型分類が正則イソトピー分類に帰着される「基本原理」が，ある種の3価空間グラフにおいても成り立つことが明らかにされた．(3) 非自明結び目を含まない線形空間 3+3 頂点完全2部グラフは「標準的」であることが明らかとなり，ランダム線形空間グラフへの応用が見出された．(4) 種数2の既約なハンドル体結び目の階層構造の理解が進んだ．(5) 2を法とした合同式であるConway-Gordonの定理が，整数上の等式として任意の頂点数の完全グラフに一般化され，線形空間グラフの理論に多くの応用が見出された．
|
