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Studies of local moves and finite type invariants in knot theory

Research Project

Project/Area Number 16540083
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTokyo Woman's Christian University

Principal Investigator

OHYAMA Yoshiyuki  Tokyo Woman's Christian University, College of Arts and Sciences, Professor, 文理学部, 教授 (80223981)

Co-Investigator(Kenkyū-buntansha) NAKANISHI Yasutaka  Kobe University, Faculty of Science, Professor, 理学部, 教授 (70183514)
TANIYAMA Kouki  Waseda University, Faculty of Education and Integrated Arts and Sciences, Professor, 教育・総合科学学術員, 教授 (10247207)
KOBAYASHI Kazuaki  Tokyo Woman's Christian University, College of Arts and Sciences, Professor, 文理学部, 教授 (50031323)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2004: ¥1,400,000 (Direct Cost: ¥1,400,000)
Keywordsknot / local move / C_n-move / finite type invariant / Vassiliev invariant / Cn-move
Research Abstract

In 1990, Vassiliev invariants for knots were defined. They order all knot invariants and they are also called finite type invariants. The first aim of this research is to study the finite type invariants by combinatorial methods. The start point is the following result proved by Goussarov and Habiro independently ; two knots have the same Vassiliev invariants of order less than n if and only if they can be transformed into each other by a finite sequence of C_n-moves.
As a joint work with Prof. Yasutaka Nakanishi, we have that for any given pair of a natural number n and a knot K, there exist infinitely many knots whose Vassiliev invariants of order less than or equal to n and Conway polynomials coincide with those of K. In the finite type invariants, the coefficients of the Conway polynomial are not powerful to classify the knots.
C_n-moves may change the Vassiliev invariants of order n. As a joint work with Harumi Yamada, we showed that a standard C_n-move can change the coefficient of z^n by 0 or ±2. It is possible to say that we nearly cleared the relation between C_n-moves and the coefficients of the Conway polynomial.
We can define the simplicial complex for the set of knots by using C_n-moves and it is called the C_n-Gordian complex of knots. Let K be a knot and K^<C_n> the set of knots obtained from K by a single C_n-move. We showed that there are knots K_1 and K_2 such that they have the same Conway polynomial and the sets of Conway polynomials of K_1^<C_n> and those of K_2^<C_n> do not coincide, as a joint work with Prof. Yasutaka Nakanishi. This theorem are related to the C_n-Gordian complex and the Conway polynomial and we consider an expansion of the result.

Report

(4 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (11 results)

All 2006 Other

All Journal Article (11 results)

  • [Journal Article] The C_k-Gordian Complex of knots2006

    • Author(s)
      Yoshiyuki Ohyama
    • Journal Title

      Journal of Knot Theory and Its Ramifications Vol.15,No.1

      Pages: 73-80

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Knots with given finite type invariants and Conway polynomial2006

    • Author(s)
      Yasutaka Nakanishi
    • Journal Title

      Journal of Knot Theory and Its Ramifications Vol.15,No.2

      Pages: 205-215

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2005 Annual Research Report
  • [Journal Article] Local moves and Gordian complexes2006

    • Author(s)
      Yasutaka Nakanishi
    • Journal Title

      Journal of Knot Theory and Its Ramifications Vol.15,No.9

      Pages: 1215-1224

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] The C_k-Gordian complex of knots2006

    • Author(s)
      Yoshiyuki Ohyama
    • Journal Title

      Journal of Knot Theory and Its Ramifications Vol.15,No.1

      Pages: 73-80

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary 2005 Annual Research Report
  • [Journal Article] Knots with given finite type invariants and Conway polynomial2006

    • Author(s)
      Yasutaka Nakanishi, Yoshiyuki Ohyama
    • Journal Title

      Journal of Knot Theory and Its Ramifications Vol.15,No.2

      Pages: 205-215

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Local moves and Gordian complexes2006

    • Author(s)
      Yasutaka Nakanishi, Yoshiyuki Ohyama
    • Journal Title

      Journal of Knot Theory and Its Ramifications Vol.15,No.9

      Pages: 1215-1224

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A C_n-move for a knot and the coefficients of the Conway polynomial

    • Author(s)
      Yoshiyuki Ohyama
    • Journal Title

      Journal of Knot Theory and Its Ramifications (掲載決定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A C_n-move for a knot and the coefficients of the Conway polynomial

    • Author(s)
      Yoshiyuki Ohyama, Harumi Yamada
    • Journal Title

      Journal of Knot Theory and Its Ramifications (To appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A C_n-move for a knot and the coefficients of the Conway polynomial

    • Author(s)
      Yoshiyuki Ohyama
    • Journal Title

      Journal of Knot Theory and Its Ramifications (掲載決定)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The C_k-Gordian complex of knots

    • Author(s)
      Yoshiyuki Ohyama
    • Journal Title

      Journal of Knot Theory and Its Ramifications (掲載決定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Knots with given finite type invariants and Conway polynomial

    • Author(s)
      Yasutaka Nakanishi
    • Journal Title

      Journal of Knot Theory and Its Ramifications (掲載決定)

    • Related Report
      2004 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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