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1999 Fiscal Year Final Research Report Summary

Knot Theory and Geometry of Manifolds

Research Project

Project/Area Number 09304011
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionOsaka City University

Principal Investigator

KAWAUCHI Akio  Osaka City Univ., Dept. of Math., Prof., 理学部, 教授 (00112524)

Co-Investigator(Kenkyū-buntansha) KANENOBU Taizo  Osaka City Univ., Dept. of Math., Associate Prof., 理学部, 助教授 (00152819)
SAKUMA Makoto  Osaka Univ., Inst. Of Math., Associate Prof., 理学研究科, 助教授 (30178602)
NAKANISHI Yasutaka  Kobe Univ., Dept. of Math., Prof., 理学部, 教授 (70183514)
MATUMOTO Takao  Hiroshima Univ., Dept. of Math., Prof., 理学部, 教授 (50025467)
MATSUMOTO Yukio  Tokyo Univ., Inst. Of Math., Prof., 数理科学研究科, 教授 (20011637)
Project Period (FY) 1997 – 1999
KeywordsKnot theory / manifold / topology / exact 4-manifold / Arf invariant / hyperbolic geometry / molecular graph / DNA knot
Research Abstract

It is well-known that in order to study a geometry of manifold, it is important to study the topological structure. The study of topology is to analyze the position and the shape of a topological object, and the study of position is to analyze a pair of manifold and submanifold, represented typically by knot theory. Knot theory and related topics are studied actively for the last two decades not only abroad but also much more in Japan. During this research program, knot theory and the related studies of low dimensional manifolds have been studied by many researchers. For example, Kawauchi obtained a new concept "exact 4-manifold" by studying a surface-knot. This concept is useful to classify 4-manifolds with infinite cyclic first homology, and as a result, we see that there exists a surface-knot invariant which is analogous to the Arf invariant of a classical knot. In other related studies, hyperbolic geometry, differential topology (including handle theory, Morse theory), gauge theory, transformation theory, foliation theory, homotopy theory, real and complex singularities, dynamical systems, general topology, surface moduli have been studied. Also, a "mew applied knot theory" was searched in relations with Yang-Baxter equation (in statistical mechanics), a molecular graph (in molecular chemistry), and DNA knot (in biochemistry).

  • Research Products

    (16 results)

All Other

All Publications (16 results)

  • [Publications] Akio KAWAUCHI: "Topological imitations"Lectures oa Lnots 86. 単項本. 19-37 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Akio KAWAUCHI: "The quadratic form of a link and a Seifert matrix"Proceedings of Applied Math. Workshop. 8. 119-129 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Akio KAWAUCHI: "Floer hmology of topological imitations of homology 3-spheres"J. Knot Therovy Ramifications. 7. 41-60 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Akio KAWAUCHI: "On the fundameutal class of ocn infinite cyclic covernings"Kobe Journal of Mathematics. 15. 103-114 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Akio KAWAUCHI: "The quadratic form of a link"Contemporary Mathematics. 233. 97-116 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Akio KAWAUCHI: "Torsion linking forms on sunface-konts and exact 4-manifolds (近刊)"Proc. Conf. Knots in Hellas 1998. 単行本.

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Akio KAWAUCHI: "A Survey of Knot Theory"Birkhauser. 420 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 河内明夫: "線形代数からホモロジーへ"培風館(近刊).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Akio Kawauchi: "Topological imitations"Lectures at Knots, World Scientific Publ.. 96. 19-37 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Akio Kawauchi: "The quadratic form of a link and a Seifert matrix"Proc. Applied Math, Workshop. 8. 119-129 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Akio Kawauchi: "Floer homology of topological imitations of homology 3-spheres"J.Knot Theory Ramifications. 7. 41-60 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Akio Kawauchi: "On the fundamental class of an infinite cyclic covering"Kobe J. Math. 15. 103-114 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Akio Kawauchi: "The quadratic form of a link"Contemp.Math. 233. 97-116 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Akio Kawauchi: "Torsion linking forms on surface-knots and exact 4-manifolds"Proc.Conf.Knots in Hellas 1998, World Sci.. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Akio Kawauchi: "A Survey of Knot Theory"Birkh\"auser. 420 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Akio Kawauchi: "From Linear Algebra to Homology"Baifukan (in Japanese). to appear

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2001-10-23  

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