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Studies on geometric properties and realization problems of invariants for classical knots and virtual knots

Research Project

Project/Area Number 20740047
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeSingle-year Grants
Research Field Geometry
Research InstitutionOsaka Electro-Communication University

Principal Investigator

NAKAMURA Takuji  Osaka Electro-Communication University, 工学部, 准教授 (60382024)

Project Period (FY) 2008 – 2010
Project Status Completed (Fiscal Year 2010)
Budget Amount *help
¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
Fiscal Year 2010: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2009: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords結び目 / Alexander多項式 / ファイバー結び目 / C_n変形 / 仮想結び目 / Miyazawa多項式 / シャープ変形 / トポロジー / 局所変形 / シャープ型結び目解消数 / 合成結び目 / Jones多項式 / 周期的結び目
Research Abstract

In this studies, we obtain several results about realizations for geometric or algebraic properties of knots and related topics from a view of local moves for knots. In fact, we construct p-periodic knots with C_n unknotting number one for any period p. We also construct composite knots with # unknotting number one. Moreover, we obtain some partial answers for a realization problem of Alexander polynomials for # unknotting number one knots. On the other hand, we construct non-classical virtual knots with the trivial Miyazawa polynomial, which is a generalization of Jones polynomial.

Report

(4 results)
  • 2010 Annual Research Report   Final Research Report ( PDF )
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • Research Products

    (17 results)

All 2011 2010 2009 2008

All Journal Article (3 results) (of which Peer Reviewed: 3 results) Presentation (14 results)

  • [Journal Article] Braidzel surfaces for fibered knots with given Alexander polynomials2009

    • Author(s)
      中村拓司
    • Journal Title

      Kobe J.Math. 26巻

      Pages: 17-28

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Braidzel surfaces for fibered knots with given Alexander polynomials2009

    • Author(s)
      NAKAMURA Takuji
    • Journal Title

      Kobe Journal of Mathematics 26

      Pages: 17-28

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Braidzel surfaces for fibered knots with given Alexander polynomials2009

    • Author(s)
      NAKAMURA Takuj
    • Journal Title

      Kobe Joumal of Mathematics (掲載確定)

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Presentation] 結び目のシャープ変形について2011

    • Author(s)
      中村拓司
    • Organizer
      日本数学会年会(中止により予稿集刊行にて発表扱い)
    • Place of Presentation
      早稲田大学
    • Year and Date
      2011-03-20
    • Related Report
      2010 Final Research Report
  • [Presentation] 結び目のシャープ変形について2011

    • Author(s)
      中村拓司
    • Organizer
      日本数学会年会
    • Place of Presentation
      早稲田大学(中止により予稿集刊行にて発表扱い)
    • Year and Date
      2011-03-20
    • Related Report
      2010 Annual Research Report
  • [Presentation] Notes on sharp moves for knots2011

    • Author(s)
      中村拓司
    • Organizer
      国際研究集会「The 7th East Asian School of Knots and Related Topics」
    • Place of Presentation
      広島大学
    • Year and Date
      2011-01-10
    • Related Report
      2010 Final Research Report
  • [Presentation] Notes on sharp moves for knots2011

    • Author(s)
      中村拓司
    • Organizer
      国際研究集会「The 7th East Asian School of Knots and Related Topics
    • Place of Presentation
      広島大学
    • Year and Date
      2011-01-10
    • Related Report
      2010 Annual Research Report
  • [Presentation] Notes on sharp moves for knots2010

    • Author(s)
      中村拓司
    • Organizer
      研究集会「E-KOOKセミナー2010」
    • Place of Presentation
      大阪市立大学
    • Year and Date
      2010-08-27
    • Related Report
      2010 Annual Research Report 2010 Final Research Report
  • [Presentation] 自明な多項式不変量を持つ仮想結び目について2010

    • Author(s)
      中村拓司
    • Organizer
      日本数学会年会
    • Place of Presentation
      慶應義塾大学
    • Year and Date
      2010-03-26
    • Related Report
      2010 Final Research Report 2009 Annual Research Report
  • [Presentation] Notes on virtual knots and their polynomial invariants2010

    • Author(s)
      中村拓司
    • Organizer
      研究会「結び目と3次元多様体」
    • Place of Presentation
      慶應義塾大学
    • Year and Date
      2010-03-18
    • Related Report
      2010 Final Research Report 2009 Annual Research Report
  • [Presentation] Notes on virtual knots with trivial polynomial invariants2010

    • Author(s)
      中村拓司
    • Organizer
      国際研究集会「The Sixth East Asian School of Knots and Related Topics」
    • Place of Presentation
      Chern Institute of Mathematics, Nankai University
    • Year and Date
      2010-01-25
    • Related Report
      2010 Final Research Report 2009 Annual Research Report
  • [Presentation] Notes on virtual knots whose polynomial invariant is trivial2009

    • Author(s)
      中村拓司
    • Organizer
      研究集会「Intelligence of Low Dimensional Topology」
    • Place of Presentation
      大阪市立大学
    • Year and Date
      2009-11-14
    • Related Report
      2010 Final Research Report 2009 Annual Research Report
  • [Presentation] C_n-moves and periodic knots2009

    • Author(s)
      中村拓司
    • Organizer
      日本数学会年会
    • Place of Presentation
      東京大学
    • Year and Date
      2009-03-26
    • Related Report
      2010 Final Research Report 2008 Annual Research Report
  • [Presentation] C_n-moves and periodic knots2009

    • Author(s)
      中村拓司
    • Organizer
      国際研究集会「The Fifth East Asian School of Knots and Related Topics」
    • Place of Presentation
      慶州文化会館(韓国)
    • Year and Date
      2009-01-14
    • Related Report
      2010 Final Research Report 2008 Annual Research Report
  • [Presentation] C_n-moves and periodic knots2009

    • Author(s)
      中村拓司
    • Organizer
      国際研究集会「Knots in Washington XXVII」
    • Place of Presentation
      George Washington university
    • Year and Date
      2009-01-11
    • Related Report
      2010 Final Research Report 2008 Annual Research Report
  • [Presentation] C_n-moves and periodic knots2008

    • Author(s)
      中村拓司
    • Organizer
      研究集会「北陸結び目セミナー」
    • Place of Presentation
      金沢大学サテライトキャンパス
    • Year and Date
      2008-11-14
    • Related Report
      2010 Final Research Report
  • [Presentation] C_n-moves and periodic knots2008

    • Author(s)
      中村拓司
    • Organizer
      研究集会「北陸結び目セミナー」
    • Place of Presentation
      金沢大学サテライトキャンハス
    • Year and Date
      2008-11-14
    • Related Report
      2008 Annual Research Report

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Published: 2008-04-01   Modified: 2016-04-21  

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