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1998 Fiscal Year Final Research Report Summary

STUDY ON KNOT INVARIANTS

Research Project

Project/Area Number 09640125
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

KANENOBU Taizo  Osaka City University, Faculty of Science, Associate Professor, 理学部, 助教授 (00152819)

Co-Investigator(Kenkyū-buntansha) HASHIMOTO Yoshitake  Osaka City University, Faculty of Science, Lecturer, 理学部, 講師 (20271182)
KAMADA Seiichi  Osaka City University, Faculty of Science, Associate Professor, 理学部, 助教授 (60254380)
MASUDA Mikiya  Osaka City University, Faculty of Science, Professor, 理学部, 教授 (00143371)
KAWAUCHI Akio  Osaka City University, Faculty of Science, Professor, 理学部, 教授 (00112524)
Project Period (FY) 1997 – 1998
Keywordsknot / link / Vassiliev invariant / finite type invariant / ribbon knot / HOMFLY polynomial
Research Abstract

We studied on topological invariants of knots and links, in particular, on Vassiliev invariants or finite type invariants. In a joint work with Shima and Habiro, we generalized these invariants to a higher dimension. A 2-knot in a 4-space is a ribbon 2-knot if it bounds an immersed 3-disk with only ribbon singularities. Making use of them, we gave a notion of finite type invariants for a class of ribbon 2-knots. Then we showed that each coefficient in the Taylor expansion of the normalized Alexander polynomial of a ribbon 2-knot is a Vassiliev invariant. Furthermore, Shima and Habiro have shown that the Alexander polynomial determines all the Vassiliev invariants of a ribbon 2-knot.
Next, We gave an algorithm for calculating the second degree coefficient of the Conway polynomial of a ribbon 1-knot. This yields a recursive calculation for the Vassiliev invariant of order 2 for a ribbon 2-knot. This gives a partial answer to the question : Find a recursive formula for the Alexander polynomial for a higher dimensional knot. This problem is related to the question : Does there exist an invariant for a higher dimensional knot such as the Jones polynomial for a classical knot?
We have been studying the basis of the space of Vassiliev invariants of lowere order for classical knots and links. As an application of this, we showed that some special values of the coefficient polynomials of the HOMFLY polynomial of a link are determined by the linking numbers. This generalizes a formula of Hoste and one of Lickorish and Millett.

  • Research Products

    (15 results)

All Other

All Publications (15 results)

  • [Publications] 金信 泰造: "Recursive calculation for an in variant of a ribbon knot" J.Knot Theory Ramifications. 7・8. 1093-1105 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 金信 泰造: "Vassiliev link invariants of order three" J.Knot Theory Ramifications. 7・4. 433-462 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 金信 泰造: "HOMFLY polynomials as Vassilier link invariants" Knot Theory, Banach Center Publ. 42. 165-185 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 葉広 和夫: "Finite type invariants of ribbon 2-knots" Contemporary Math.出版予定. 出版予定.

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 鎌田 聖一: "Standard forms of 3-braid 2-knots and their Alexander polynomials" Michigan Moth. J.45. 189-205 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 河内 明夫: "Floer homology of topological imitations of homdogy 3-spheres" J.Knot Theury Ramifications. 7・1. 41-60 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Kanenobu: "Recursive Calculation for an invariant of a ribbon knot" J.Knot Theory Ramifications. 7. 1093-1105 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Kanenobu, Y.Miyazawa and A.Tani: "Vassiliev link invariants of order three" J.Knot Theory Ramification. 7. 433-462 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Kanenobu and Y.Miyazawa: "H O M F L Y polynomials as Vassiliev link invariants" in "Knot Theory, " (V.F.R.Jones, J.Kania-Bartoszynska, J.H.Przytycki, P.Traczyk and V.G.Turaev, eds.) Banach Center Publ., vol.42, Institute of Mathematics, Polish Acad.Sci., Warsaw. 165-185 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Habiro, T.Kanenobu and A.Shima: "Finite type invariants of ribbon 2-knots" Contemporary Math.to appear.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Kanenobu: "Vassiliev-type invariants of theta-curve" J.Knot Theory Ramifications. 6. 455-477 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Kanenobu and Y.Marumoto: "Unknotting and fusion numbers of ribbon 2-knots" Osaka J.Math.34. 525-540 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Kanenobu: "Kauffman polynomials as Vassiliev link invariants" in "Proceedings of Knots 96, " (S.Suzuki, ed.) World Scientific Publishing Co.411-431 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Kamada: "Standard forms of 3-braid 2-knots and their Alexander polynomials" Michigan Math.J.45. 189-205 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A.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

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Published: 1999-12-08  

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