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Study of Knot theory from Geometric Structure

Research Project

Project/Area Number 17540059
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionYamagata University

Principal Investigator

UCHIDA Yoshiaki  Yamagata University, Faculty of Science, Departrnent of Mathematical Sciences, Associate Professor (80280890)

Co-Investigator(Kenkyū-buntansha) ASHIKAGA Tadashi  Tohoku Gakuin University, Faculty of Engineerurg, Professor (90125203)
TORISU Ichiro  Naruto University of Education, 学校教育学部, Associate Professor (50323134)
Project Period (FY) 2005 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥2,940,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥240,000)
Fiscal Year 2007: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2005: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordsknot / covering space / Seifert fiberd space / link / Riemann surface / Dedekind sum / Legendrian knot / strong trivial / マルコフ同値 / 不変量 / 結び目理論 / 多様体 / 被膜空間
Research Abstract

1. Uchida studies (1) the three-fold irregular branched covering space of three-dimensional sphere branched over three-bridge knot and three-braid knot And he showed that the covering space is a lens space of type L (n, 1) and L (n, m), respectively. (2) lf a torus knot has a three-fold irregular branched covering space, then its type is T (2x, 3y), where x and y are co-prime integers. Its covering space is a Seifert fibered space of type M (β_1/2x, β_1/x, β_2/Y), where β_1 and β_2 are integers with 2xβ_2+3yβ_1=±1. Moreover for T (2, x) and T (3, x), he proved this theorem by using a knot diagram.
2. Ashikaga studies topology and algebraic geometry. (1) He describes the recent development of study of degenerate families of Riemann surfaces. This field is located at the boundary area where topology, algebraic geometry complex analysis and others get complicated each other. (2) He propose a certain formula of Dedekind sum by using two mutually distinct methods. The one is an elementary number theoretic method and the other is a geometric method. Moreover he re-prove the reciprocity law by the formula.
3. Torisu gives (1) some examples of links having 2-adjacency relation and he also reports a recent study of 2-bridge links with 2-adjacency relation. (2) He gives a necessary condition for a two-bridge knot or link S (p, q) to be 2-adjacent to another two-bridge knot or link S (r, s). In particular he shows that if the trivial knot or link is 2-adjacent to S (p, q), then S (p, q) is trivial, that if S (p, q) is 2-adjacent to its mirror image, then S (p, q) is amphicheiral, and that for a prime integer p, if S (p, q) is 2-adjacent to S (r, s), then S (p, q)=S (r, s) or S (r, s)=S (1, 0). (3) He present a off Lain family of strongly 1-trivial Montesinos knots, and show that if a well known conjecture on Seifert surgery is valid, then the family contains all strongly 1-trivial Montesinos knots.

Report

(4 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • Research Products

    (68 results)

All 2008 2007 2006 2005 Other

All Journal Article (17 results) (of which Peer Reviewed: 9 results) Presentation (49 results) Remarks (2 results)

  • [Journal Article] On 2-adjacency relation of links2007

    • Author(s)
      Ichiro, Torisu
    • Journal Title

      the Proceedings of the International Workshop on Knot Theory for Scientific Objects

      Pages: 277-284

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On 2-adjacency relation of links2007

    • Author(s)
      Ichiro Torisu
    • Journal Title

      the Proceedings of the International Workshop on Knot Theory for Scientific Objects

      Pages: 277-284

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Journal Article] Vatrious aspects of degenerate of Riemann surfaces2006

    • Author(s)
      Tadashi, Ashikaga・Hisaaki, Endo
    • Journal Title

      Sugaku Expositions, Amer. Math. Soc. 19

      Pages: 171-196

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Two-bridge links with strong triviality2006

    • Author(s)
      Ichiro, Torisu
    • Journal Title

      Tokyo Journal of Mathematics 29

      Pages: 233-237

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Various aspects of degenerate families of Riemann surfaces2006

    • Author(s)
      Tadashi Ashikaga and Hisaaki Endo
    • Journal Title

      Sugaku Expositions Amer. Math. Soc. 19

      Pages: 171-196

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Two-bridge links with strong triviality2006

    • Author(s)
      Ichiro Torisu
    • Journal Title

      Tokyo Journal of Mathematics 29

      Pages: 233-237

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On three-fold irregular branched covering over closed three-braid and three-bridge knots2005

    • Author(s)
      Yoshiaki, Uchida
    • Journal Title

      Proceedings of International Conference on Topology in Matsue, Topology and its Applications 146-147

      Pages: 189-194

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] A note on strongly n-trivial links2005

    • Author(s)
      Ichiro, Torisu
    • Journal Title

      Journal of Knot Theory and Its Ramifications 14

      Pages: 565-569

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] A note on strongly n-trivial links2005

    • Author(s)
      Ichiro Torisu
    • Journal Title

      Journal of Knot Theory and Its Ramifications 14

      Pages: 565-569

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On three-fold irregular branched covering over closed three-braid and three-bridge knots2005

    • Author(s)
      Yoshiaki Uchida
    • Journal Title

      Proceedings of international Conference on Topology in Matsue, Topology and its Applications 146-147

      Pages: 189-194

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On 2-adjacency relation of two-bridge knots and links

    • Author(s)
      Ichiro, Torisu
    • Journal Title

      Journal of the Australian Mathematical Society (To appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On strongly l-trivial Montesinos knot

    • Author(s)
      Ichiro., Torisu
    • Journal Title

      Hiroshima Mathematical Journal (To appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On three-fold irregular branched covering over closed three-braid and three-bridge knots

    • Author(s)
      Yoshiaki Uchida
    • Journal Title

      Proceedings of International Conference on Topology in Matsue, Topology and its Applications vol.146-147

      Pages: 189-194

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On 2-adjacency relation of two-bridge knots and links

    • Author(s)
      Ichiro Torisu
    • Journal Title

      to appear in Journal of the Australian Mathematical Society

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On strongly 1-trivial Montesinos knot

    • Author(s)
      Ichiro Torisu
    • Journal Title

      to appear in Hiroshima Mathematical Journal

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On 2-adjacency relation of two-bridge knots and links

    • Author(s)
      Ichiro Torisu
    • Journal Title

      Journal of the Australian Mathematical Society (To appear)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On strongly 1-trivial Montesinos knot

    • Author(s)
      Ichiro Torisu
    • Journal Title

      Hiroshima Mathematical Journal (To appear)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Presentation] Remarks on Horikawa index2008

    • Author(s)
      足利 正
    • Organizer
      代数幾何ミニワークショップ
    • Place of Presentation
      兵庫県八千代町
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Invariants of fiber germs and surface singularities2008

    • Author(s)
      足利 正
    • Organizer
      分岐被覆に関連する代数幾何とトポロジー(第2回)
    • Place of Presentation
      首都大学東京
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Remarks on Horikawa index2008

    • Author(s)
      Tadashi Ashikaga
    • Organizer
      Mini workshop "Algebraic Geometry"
    • Place of Presentation
      Yachiyo town Hyogo
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Invariants of fiber germs and surface singularities2008

    • Author(s)
      Tadashi Ashikaga
    • Organizer
      Workshop "Branched covering with Algebraic Geometry and Topology II"
    • Place of Presentation
      Tokyo Metropolitan University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Many non-conjugate braids whose closures represent same knot via covering space2007

    • Author(s)
      内田吉昭
    • Organizer
      Knotting Mathematics and Art
    • Place of Presentation
      University of South Florida
    • Year and Date
      2007-11-03
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] 組み紐上分岐する三次元球面の3重分岐被覆空間2007

    • Author(s)
      内田吉昭
    • Organizer
      第一回「分岐被覆に関連する代数幾何とトポロジー」ワークショップ(東北結び目セミナーと共催)
    • Place of Presentation
      東北学院大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 被覆空間から見た組み紐表示2007

    • Author(s)
      内田吉昭
    • Organizer
      東北結び目セミナー
    • Place of Presentation
      国際蔵王高原ホテル
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Many non-conjugate braids whose closures represent same knot via covering space2007

    • Author(s)
      内田吉昭
    • Organizer
      International Conference on Topology and its Applications 2007
    • Place of Presentation
      京都大学数理解析研究所
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] 射影直線束内のNon-Galois被覆曲面の標準解消2007

    • Author(s)
      足利 正
    • Organizer
      分岐被覆に関連する代数幾何とトポロジー(第1回)
    • Place of Presentation
      東北学院大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Dedekind和の相互律とモノドロミー2007

    • Author(s)
      足利 正
    • Organizer
      代数幾何ミニワークショップ
    • Place of Presentation
      兵庫県八千代町
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Strongly n-trivial Montesinos knots2007

    • Author(s)
      鳥巣伊知郎
    • Organizer
      東北結び目セミナー
    • Place of Presentation
      国際蔵王高原ホテル
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Legendrian twist knot の分類について2007

    • Author(s)
      鳥巣伊知郎
    • Organizer
      研究集会「接触構造、特異点、微分方程式及びその周辺」
    • Place of Presentation
      旭川ときわ市民ホール
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Branched covering space branched along braid2007

    • Author(s)
      Yoshiaki Uchida
    • Organizer
      Workshop "Branched covering with Algebraic Geometry and Topology I"
    • Place of Presentation
      Tohoku Gakuin University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Many non-conjugate braids whose closures represent same knot via covering space2007

    • Author(s)
      Yoshiaki Uchida
    • Organizer
      Tohoku Knot Seminar
    • Place of Presentation
      Kokusai Zao Kogen Hotel
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Many non-conjugate braids whose closures represent same knot via covering space2007

    • Author(s)
      Yoshiaki Uchida
    • Organizer
      Knotting Mathematics and Art
    • Place of Presentation
      University of South Florid
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Many non-conjugate braids whose closures represent same knot via covering space2007

    • Author(s)
      Yoshiaki Uchida
    • Organizer
      International Conference on Topology and its applications 2007
    • Place of Presentation
      RIMS
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Deneration of non-Galois covering in projective space bundle2007

    • Author(s)
      Tadashi Ashikaga
    • Organizer
      Workshop "Branched covering with Algebraic Geometry and Topology I"
    • Place of Presentation
      Tohoku Gakuin University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] The reciprocity law of the Dedekind sum and monodromy2007

    • Author(s)
      Tadashi Ashikaga
    • Organizer
      Mini Workshop "Algebraic Geometry"
    • Place of Presentation
      Yachiyo town Hyogo
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Strongly n-trivial Montesinos knots2007

    • Author(s)
      Ichiro Torisu
    • Organizer
      Tohoku Knot Seminar
    • Place of Presentation
      Kokusai Zao Kogen Hotel
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Classification of Legendrian twist knot2007

    • Author(s)
      Ichiro Torisu
    • Organizer
      Workshop "Contact structure, singular points, differential equations"
    • Place of Presentation
      Asahikawa Tokiwa Hall
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 射影直線東内のNon・Galois被覆曲面の標準解消2007

    • Author(s)
      足利 正
    • Organizer
      会岐被覆に関連する代数幾何とトポロジー(第1回)
    • Place of Presentation
      東北学院大学
    • Related Report
      2007 Annual Research Report
  • [Presentation] Branched covering branched along tricoloring thorus knots2006

    • Author(s)
      内田吉昭
    • Organizer
      東北結び目セミナー
    • Place of Presentation
      国際蔵王高原ホテル
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Local signatute of fibered complex surfaces via monodromy and stable reduction2006

    • Author(s)
      足利 正
    • Organizer
      特異点と代数幾何ワークショップ
    • Place of Presentation
      東京理科大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Local invariants of pencils of curves2006

    • Author(s)
      足利 正
    • Organizer
      代数幾何学と複素力学系研究会
    • Place of Presentation
      京阪奈プラザホテル
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 符号数・モノドロミー・Dedekind2006

    • Author(s)
      足利 正
    • Organizer
      分裂族・チャート ・モノドロミー箱根セミナー
    • Place of Presentation
      小田急箱根レイクホテル
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Dedeind和とモノドロミー2006

    • Author(s)
      足利 正
    • Organizer
      代数幾何ミニワークショップ
    • Place of Presentation
      兵庫県八千代町
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Local sigitature and Horikawa index of pencils of curves2006

    • Author(s)
      足利 正
    • Organizer
      代数幾何と位相幾何の周辺研究集会
    • Place of Presentation
      京都大学数理解析研究所
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On strongly n-trivial knots2006

    • Author(s)
      鳥巣伊知郎
    • Organizer
      上智トポロジー研究集会プレセミナー
    • Place of Presentation
      上智大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 写像類群とcontact open book2006

    • Author(s)
      鳥巣伊知郎
    • Organizer
      伊豆トポロジーセミナー
    • Place of Presentation
      伊豆瑞光園
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On 2-adjacency relation of two-bridge knots and links2006

    • Author(s)
      鳥巣伊知郎
    • Organizer
      International Workshop on Knot Theory for Scientific Objects
    • Place of Presentation
      大阪市立大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Branched covering space branched along tricoloring torus knots2006

    • Author(s)
      Yoshiaki Uchida
    • Organizer
      Thhoku Knot Seminar
    • Place of Presentation
      Kokusai Zao Kogen Hotel
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Local signature of fiberd complex surfaces via monodromy and stable reduction2006

    • Author(s)
      Tadashi Ashikaga
    • Organizer
      Workshop "Singular point and Algebraic Geometry"
    • Place of Presentation
      Tokyo University of Science
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Local invariants of pencils of curves2006

    • Author(s)
      Tadashi Ashikaga
    • Organizer
      Workshop "Algebraic Geometry and Complex Dynamical Systems"
    • Place of Presentation
      Keihanna-plaza Hotel
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Signature・Monodromy・Dedekind2006

    • Author(s)
      Tadashi Ashikaga
    • Organizer
      Hakone Seminar "Splitting family・Chart・"Monodromy"
    • Place of Presentation
      Hakone Lake Hotel
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Dedekind sum and monodromy2006

    • Author(s)
      Tadashi Ashikaga
    • Organizer
      Mini Workshop "Algebraic Geometry"
    • Place of Presentation
      Yachiyo town Hyogo
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Local signature and Horikawa index of pencils of2006

    • Author(s)
      Tadashi Ashikaga
    • Organizer
      Workshop "Algebraic Geometry and Topology"
    • Place of Presentation
      RIMS
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On strongly n-trivial knots2006

    • Author(s)
      Ichiro Torisu
    • Organizer
      Jochi Topology preseminar
    • Place of Presentation
      Sophia University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Mapping class group and contact open book2006

    • Author(s)
      Ichiro Torisu
    • Organizer
      Izu Topology Seminar
    • Place of Presentation
      Izu Zuikoen
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On 2-adjacency relation of two-bridge knots and links2006

    • Author(s)
      Ichiro Torisu
    • Organizer
      International workshop on Knot Theory for Scientific Object
    • Place of Presentation
      Osaka City University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Localization of signature2005

    • Author(s)
      足利 正
    • Organizer
      Workshop on Singularity theory
    • Place of Presentation
      東京理科大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Horika.wa index of maximal-gonal fibrations of odd genus2005

    • Author(s)
      足利 正
    • Organizer
      Algebraic Geomerty in East Asia II
    • Place of Presentation
      Hanoi,ベトナム
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 2-component transverse links2005

    • Author(s)
      鳥巣伊知郎
    • Organizer
      東北結び目セミナー
    • Place of Presentation
      国際蔵王高原ホテル
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] transverse Hopf link の分類2005

    • Author(s)
      鳥巣伊知郎
    • Organizer
      日本数学会
    • Place of Presentation
      岡山大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Transverse Hopf link の分類2005

    • Author(s)
      鳥巣伊知郎
    • Organizer
      低次元トポロジーセミナー
    • Place of Presentation
      大阪大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Localization of signature2005

    • Author(s)
      Tadashi Ashikaga
    • Organizer
      Workshop on Singularity
    • Place of Presentation
      Tokyo University of Science
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Horikawa index of maximal-gonal fibrations of odd genus2005

    • Author(s)
      Tadashi Ashikaga
    • Organizer
      Algebraic Geometry in East Asia II
    • Place of Presentation
      Hanoi
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 2-component transverse links2005

    • Author(s)
      Ichiro Torisu
    • Organizer
      Tohoku Knot Seminar
    • Place of Presentation
      Kokusai Zao Kogen Hotel
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Classification of taranvers Hopf link2005

    • Author(s)
      Ichiro Torisu
    • Organizer
      MSJ meeting
    • Place of Presentation
      Okayama University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Classification of taranvers Hopf link2005

    • Author(s)
      Ichiro Torisu
    • Organizer
      Low Dimensional Topology Seminar
    • Place of Presentation
      Osaka University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Remarks] 「研究成果報告書概要(和文)」より

    • URL

      http://sci.kj.yamagata-u.ac.jp/~yuuchida/index.html

    • Related Report
      2007 Final Research Report Summary
  • [Remarks]

    • URL

      http://sci.kj.yamagata-u.ac.jp/~yuchida/index.html

    • Related Report
      2007 Annual Research Report

URL: 

Published: 2005-04-01   Modified: 2016-04-21  

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