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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)
大嶋 秀明  大阪市立大学, 理学部, 助教授 (70047372)
Project Period (FY) 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)
Keywordsknot / link / Vassiliev invariant / finite type invariant / ribbon knot / HOMFLY polynomial / Vassilier invariant / ribbon knot / Kauffman polynomial / 係数多項式 / 結び目 / 2本橋結び目 / 2次元リボン結び目 / C多項式 / アレキサンダー多項式 / Vassiliev不変量
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.

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (28 results)

All Other

All Publications (28 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
    • Related Report
      1998 Final Research Report Summary
  • [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
    • Related Report
      1998 Final Research Report Summary
  • [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
    • Related Report
      1998 Final Research Report Summary
  • [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
    • Related Report
      1998 Final Research Report Summary
  • [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
    • Related Report
      1998 Final Research Report Summary
  • [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
    • Related Report
      1998 Final Research Report Summary
  • [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
    • Related Report
      1998 Final Research Report Summary
  • [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
    • Related Report
      1998 Final Research Report Summary
  • [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
    • Related Report
      1998 Final Research Report Summary
  • [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
    • Related Report
      1998 Final Research Report Summary
  • [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
    • Related Report
      1998 Final Research Report Summary
  • [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
    • Related Report
      1998 Final Research Report Summary
  • [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
    • Related Report
      1998 Final Research Report Summary
  • [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
    • Related Report
      1998 Final Research Report Summary
  • [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
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 金信 泰造: "Recursive calculation for an invariant of a ribbon knot" J.Knot Theory Ramifications. 7・8. 1093-1105 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 金信 泰造: "Vassilier link invariants of order three" J.Knot Theory Ramifications. 7・4. 433-462 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 金信 泰造: "HOMFLY polynomials as Vassilier link invariants" Knot Theory,Banach Center Publ.Warsawa. 42. 165-185 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 葉広 和夫: "Finite type invariants of ribbon 2-knots" Contemporary Math.出版予定.

    • Related Report
      1998 Annual Research Report
  • [Publications] 鎌田 聖一: "Standard forms of 3-braid 2-knots and their Alexarder polynomials" Michigan Math J.45. 189-205 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 河内 明夫: "Floer homology of topological imitations of homology 3-spheres" J.Knot Theory Ramifications. 7・1. 41-60 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Kanenobu,T.: "Kauffmon polynomial as Vassiliev link invariants" Proc.of Knots96,World Sci.Pub.Co.411-431 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Kanenobu,T.: "Vassiliev-type invariants of a theta-curve" J.Knot.Theory Ramifications. 6・4. 455-477 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Kanenobu,T.-Marumoto,Y.: "Unknotting and fusion numbers of ribbon 2-knots" Osaka J.Math.34・3. 525-540 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Kanenobu,T.-Miyazawa,Y、Tani,A: "Vassiliev link invariants of order three" J.Knot Theory Ramifications. (発表予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] Kanenobu,T.-Miyazawa,Y.: "HOMFLY polynomials as Vassiliev link invariants" Banach Center Publications. 42(発表予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] Kamada,S.: "Surfaces in 4-space : a view of normal forms and braidings" Lectures at Knots'96,World Sci.Co.39-71 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] C.C.アダムス,金信泰造(訳): "結び目の数学" 培風館, 309 (1998)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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