Research of topological invariants of plane closed curves
Project/Area Number  09640136 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Geometry

Research Institution  MEIJO UNIVERSITY 
Principal Investigator 
OZAWA Tetsuya Meijo University, Faculty of Sci.and Tech., Professor, 理工学部, 教授 (20169288)

CoInvestigator(Kenkyūbuntansha) 
OKAMOTO Kiyosato Meijo University, Faculty of Sci.and Tech., Professor, 理工学部, 教授 (60028115)
KATO Yoshifumi Meijo University, Faculty of Sci.and Tech., Ass.Professor, 理工学部, 助教授 (40109278)
TSUKAMOTO Michiro Meijo University, Faculty of Sci.and Tech., Assistant, 理工学部, 助手 (80076637)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1998 : ¥300,000 (Direct Cost : ¥300,000)
Fiscal Year 1997 : ¥300,000 (Direct Cost : ¥300,000)

Keywords  plane closed curve / topological invariant / Vassiliev order / Bernoulli polynomial / regular homotopy / 平面閉曲線 / 位相不変量 / Vassiliev位数 / Bernoulli多項式 / 正則変形 / Bemoulli多項式 / バシリエフ・オーダー / 正規不安定曲線 
Research Abstract 
The mail purposes of this research are, firstly to obtain new topological invariants for closed plane curves, and secondly to investigate their geometric and algebraic properties. For the first purpose, we obtained two infinite series of topological invariants which are denoted by I^<epsilon>_<habeta> and St_k, where epsilon is +, 0 or , and alpha, beta, and k vary over the set of all natural numbers. One of the important results of this research was to show the order in the sense of Vassiliev of the invariants I^<epsilon>_<habeta> to be equal to alpha + 1. Establishing the independence among those invariants, we have shown that there exist, for all finite order, infinitely many algebraically independent topological invariants. For the invariants St_k, investigating the jumps of their values at the unstable curves along regular deformations, we have verified the same geometric property as the strangeness invariants obtained by V.I.Arnold, and also algebraic independence among them as well as the additivity with respect to the connected sum operation of plane curves. We have also obtained a formula explaining the relation between I^<epsilon>_<habeta> and St_k.

Report
(4results)
Research Output
(13results)