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Developing monodromy descriptions and studying two-dimensional knots and braids by using quandles

Research Project

Project/Area Number 15540077
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionHIROSHIMA UNIVERSITY

Principal Investigator

KAMADA Seiichi  Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60254380)

Co-Investigator(Kenkyū-buntansha) MATUMOTO Takao  Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (50025467)
MATSUMOTO Makoto  Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70231602)
TERAGAITO Masakazu  Hiroshima University, Graduate School of Education, Associate Professor, 大学院・教育学研究科, 助教授 (80236984)
KAWAUCHI Akio  Osaka City University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00112524)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2004: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2003: ¥1,900,000 (Direct Cost: ¥1,900,000)
Keywordsquandle / biquandle / monodromy / 2-dimensional knot / 2-dimensional braid / chart / knot
Research Abstract

Investigation on biquandles, which are extension of quandles, is useful for study on knots, two-dimensional knots, and two-dimensional braids. Biquandles of order 4 are classified ; there are 98 types up to isomorphism. The classification can be used in order to determine whether two given biquandles are isomorphic or not. Silver-Williams' invariants of virtual knots are investigated in terms of biquandles.
It is shown that there is a one-to-one correspondence between monodromy representations of any topological objects and elements of enveloping monoidal quandles. Such monodromy representations can be described by charts, and there is a method to find basic moves connecting equivalent charts. Especially, chart description method to describe genus-1 Lefschetz fibrations is established. Using this method, we can classify genus-1 Lefschetz fibrations as fibrations, and classify their total spaces as manifolds. This result was known much earlier when it is chiral and the base space is spherical. Our result gives an elementary proof of a classification theorem of genus-1 Lefschetz fibrations due to Y.Matsumoto. Chart description method to describe genus-2 Lefschetz fibrations is also studied, and it is shown that any genus-2 Lefschetz fibrations can be stabilized, even if they are chiral or achiral. Although, we can obtain chart description method to describe Lefschetz fibrations of any higher genera, we need further study for details. Enveloping monoidal quandles determine associated groups of quandles, and elements of enveloping monoidal quandle correspond to monodromy representations, and we can obtain charts from them.

Report

(3 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • Research Products

    (14 results)

All 2005 2004 2003 Other

All Journal Article (9 results) Book (2 results) Publications (3 results)

  • [Journal Article] Word representation of cords on a punctured plane2005

    • Author(s)
      S.Kamada 他
    • Journal Title

      Topology Appl. 146

      Pages: 21-50

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Enveloping monoidal quandles2005

    • Author(s)
      S.Kamada 他
    • Journal Title

      Topology Appl. 146

      Pages: 146-147

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Word representation of cords on a punctured plane2005

    • Author(s)
      S.Kamada
    • Journal Title

      Topology Appl. 146

      Pages: 21-50

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Enveloping monoidal quandles2005

    • Author(s)
      S.Kamada
    • Journal Title

      Topology Appl. 146

      Pages: 146-147

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Word representation of cords on a punctured plane2005

    • Author(s)
      S.Kamada他
    • Journal Title

      Topology Appl. 146

      Pages: 21-50

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Enveloping monoidal quandles2005

    • Author(s)
      S.Kamada他
    • Journal Title

      Topology Appl. 146

      Pages: 146-147

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Invariants of virtual braids and a remark on left stabilizations and virtual exchange moves2004

    • Author(s)
      S.Kamada
    • Journal Title

      Kobe Journal of Mathematics 21

      Pages: 33-49

    • NAID

      110001139370

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Quandle cohomology and state-sum invariants of knotted curves and surfaces2003

    • Author(s)
      J.S.Carter 他
    • Journal Title

      Trans.Amer.Math.Soc. 355

      Pages: 3947-3989

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Quandle cohomology and state-sum invariants of knotted curves and surfaces2003

    • Author(s)
      J.S.Carter
    • Journal Title

      Trans.Amer.Math.Soc. 355

      Pages: 3947-3989

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Book] Surfaces in 4-Space2004

    • Author(s)
      J.S.Carter 他
    • Total Pages
      213
    • Publisher
      Springer Verlag
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Book] Surfaces in 4-Space2004

    • Author(s)
      J.S.Carter
    • Total Pages
      213
    • Publisher
      Springer Verlag
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] J.S.Carter 他: "Quandle cohomology and state-sum invariants of knotted curves and surfaces"Trans.Amer.Math.Soc.. 355. 3947-3989 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Kamada 他: "Enveloping monoidal quandles"Topology Appl.. 予定.

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Kamada 他: "Word representation of cords on a punctured plane"Topology Appl.. 予定.

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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