Project/Area Number  09640135 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Geometry

Research Institution  Waseda University 
Principal Investigator 
UENO Kimio Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (70160190)

CoInvestigator(Kenkyūbuntansha) 
村上 斉 , 助教授 (70192771)
FUKUYAMA Masaru Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (80063741)
KOJIMA Jun Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (50063540)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥2,800,000 (Direct Cost : ¥2,800,000)
Fiscal Year 1998 : ¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 1997 : ¥1,700,000 (Direct Cost : ¥1,700,000)

Keywords  knot / link / threemanifold / quantum invariant / Vassiliev invariant / finite type invariant / CassonWalker invariant / knot cobordism / 結び目 / 絡み目 / Vassiliew不変量 / 量子不変量 / Reidemeister torsion / SeibergWitten不変量 / SeibergWitten理論 / 結び目理論 / 4次元多様体 / 3次元多様体 
Research Abstract 
I will describe my results paper by paper. In the paper (i) I gave a new elementary, combinatorial definition of the HOMFLY polynomial. In (ii) I looked at the multivariable Alexander polynomial of links from the view point of Vassiliev invariants and define a recursive definition of weight systems derived from it. In (iii) we studied a knot cobordism invariant, 4dimensional clasp number, introduced by T.Shibuya. He proved that it is greater than or equal to the 4dimensional genus and raised a problem whether there are knots which do not satisfy the equality. We gave such an example in this paper. In (iv) I studied the quantum SU(2)invariant of 3manifolds associated with the gammath root of unity. If gamma is even, it is defined for a class of the first cohomology group modulo 2. In this paper I calculated it for rational homology threespheres and for the trivial cohomology class and showed that it is a cyclotomic integer and moreover it determines the CassonWalker invariant. In (v) we introduced a filtration to the vector space spanned by all the Seifert matrices corresponding to the filtration to the vector space spanned by all the knot, which was introduced by V.Vassiliev. Moreover we clarified its relation to the Alexander polynomial. In (vi) I gave an example of hyperbolic threemanifold with trivial finite type invariants (introduced by T.Ohtsuki) up to arbitrarily given degree. In (vii) I followed (iv) and obtained a similar result in the case of nontrivial cohomology classes. Unfortunately I only showed that the invariant is a cyclotomic integer and a relation to the CassonWalker invariant is now being investigated.
